Handbook of
Exact
Solutions for
Ordinary
Differential
Equations
Andrei D. Polyanin
Valentin F. Zaitsev
CRC Press
Boca Raton New York London Tokyo

Library of Congress Cataloging-in-Pnblication Data
Polyanin, A. D. (Andrei Dmitrievich)
Handbook of exact solutions for ordinary differential equations /
Andrei D. Polyanin, Valentin F. Zaitsev.
p. cm.
Includes bibliographical references and index.
ISBN 0-8493-9438-4 (alk. paper)
1. Differential equations — Numerical solutions. I. Zaitsev,
V.R (Valentin R) П. Title.
QA372.P725 1995
515'.352—dc20 95-10217
CIP
This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with
permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish
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Direct all inquiries to CRC Press, Inc., 2000 Corporate Blvd.,N.W, Boca Raton, Florida 33431.
© 1995 by CRC Press, Inc.
No claim to original U.S. Government works
International Standard Book Number 0-8493-9438-4
Library of Congress Card Number 95-10217
Printed in the United States of America 234567890
Printed on acid-free paper

FOREWORD
Exact solutions have always played and still play an important role in properly un-
understanding the qualitative features of many phenomena and processes in various fields of
natural science.
Equations of applied and theoretical physics often contain parameters or functions which
are found experimentally and therefore are not stringently fixed. At the same time, equa-
equations that model real phenomena and processes must be sufficiently simple to make possible
their analysis and solution. It is natural to adopt, as one of feasible criterions of simplicity,
the requirement that the model equation admits a solution in a closed form.
It should be noted that even exact solutions of nonlinear equations (including those with-
without a clear physical sense and which do not correspond to real phenomena and processes)
play an important role of "test" problems for verifying the correctness and assessment of
accuracy of various numerical, asymptotic, and approximate methods. Moreover, the model
equations and problems admitting exact solutions serve as the basis for the development of
new numerical, asymptotic, and approximate methods, which, in turn, enable us to study
more complicated problems having no analytical solution.
This book contains nearly 5000 ordinary differential equations and their solutions. The
total number of linear and nonlinear equations is several times greater than those found in
any other text. The table below compares data presented in this book with those of currently
available handbooks concerning the general number of concrete second- and higher-order
nonlinear ordinary differential equations analyzed.
The order of equations E. Kamke A976) M. Murphy A960) This book
Second order
Third order
Fourth order
Higher order
Total number of equations
249
13
3
3
268
315
22
3
9
349
1228
587
75
160
2045
When selecting the material, the authors gave preference to the following two types of
equations:
1. Equations that traditionally attracted the attention of many researchers: those of the
simplest appearance but involving the most difficulties for integration (Abel equations,
Emden—Fowler equations, Painleve equations, etc.).
2. Equations that encountered in various applications (in the theory of heat and mass
transfer, nonlinear mechanics, hydrodynamics, the theory of nonlinear oscillations, the
theory of combustion, chemical engineering science, etc.).
Special attention is paid to equations containing arbitrary functions. The other equa-
equations contain one or more arbitrary parameters (i.e., actually, this book deals with whole
families of ordinary differential equations) which can be fixed by a reader at will. Many
solutions have been obtained just recently with the aid of new (discrete group) methods
described in other books by the authors A993, 1994).
When compiling this book, the handbooks by E. Kamke A976), M. Murphy A960), and
D. Zwillinger A989) were partly used in which one can find basic notions and definitions
of the theory of ordinary differential equations, apart from concrete equations. In these
handbooks, classical and some new methods of solving differential equations are described
as well—see also the books by E.L. Ince A964), P.J. Olver A986), and N.H. Ibragimov
© 1995 by CRC Press, Inc.

A993). The latter books give a great number of references to the original papers and
books by other authors, which are devoted to exact solutions and methods of the theory of
ordinary differential equations.
In addition, when describing solutions of linear ordinary differential equations, which
are connected to higher transcendental functions (Bessel, Legendre, Mathieu, hypergeomet-
ric, etc.), the handbooks by G. Beitmen and A. Erdeii A953-1955), M. Abramowitz and
LA. Stegun A964) were used.
In some sections of this book, asymptotic solutions of some classical equations of non-
nonlinear mechanics and theoretical physics are also given, which are discussed in the books by
J.D. Cole A968), M.V. Fedoryuk A983), and A.H. Nayfeh A973, 1971) in detail.
The detailed table of contents enables a reader to quickly navigate through this book
in searching for desired equations.
The authors hope that this book will be helpful for a wide range of scientists, lec-
lectures, engineers, and students engaged in the fields of mathematics, physics, mechanics,
and chemical engineering science.
Andrei D. Polyanin
Valentin F. Zaitsev
© 1995 by CRC Press, Inc.

Some Remarks and Notation
1. In this book, in the original equations the independent variable is denoted by x, and
the dependent one is denoted by y. In the given solutions, the symbols C, Co, Ci, C2,
stand for arbitrary integration constants.
dxi d xi d xi
2. The following notation is used for derivatives: y'x = ——, yxx = —r^, y'xxx = -f!f>
(XX (XX (XX
Ух'ххх = Vt, and Ухп) = V1- with n > 5.
Ухххх dx4 Ух dxn
/ d \n
3. In some cases, we use the operator notation (j—r- ) g which is defined by the
V ax J
recurrence relation
4. In some sections of the book (see, for example, 1.3, 2.3-2.6, 3.2-3.4), for the sake of
brevity, solutions are represented as several formulae containing the terms with the signs
"±" and "=F." By this is meant two formulae—one correspond to the upper signs, and
another corresponds to the lower signs. For example, the solution of equation 1.3.1.6 can
be written in the parametric form
ж = а/-1ехр(тт2), у = а/-1[ехр(тт2)±2т/], where /= /*ехр(тт2) dr-C, А = т2а2.
This is equivalent to that the solutions of equation 1.3.1.6 are given by the formulae
x = af~x exp(—t2), y = af~x [exp(—т2) + 2т/], where /= / exp(—т2) dr — C, A = —2a2
and
ж = а/-1ехр(т2), у = а/-1[ехр(т2)-2т/], where /= /*ехр(т2) dr - С, А = 2a2.
5. When referencing to a particular equation, the notation like .1.2.5" stands for
"equation 5 in Subsection 4.1.2."
6. The book includes two supplements that provide a reader with useful information on
some elementary and special functions which appear in solutions of the differential equation
outlined.
7. References that may be helpful for a reader are given at the end of the book.
© 1995 by CRC Press, Inc.

THE AUTHORS
Andrei D. Polyanin, Ph.D., D.Sc, is a noted scientist in the fields of ordinary dif-
differential equations, engineering and applied mathematics, heat and mass transfer, nonlinear
mechanics, and chemical engineering science.
Professor Polyanin graduated from the Faculty of Mechanics and Mathematics of the
Moscow State University in 1974 and received his Candidate of Sciences (Ph.D.) degree in
1981 (at the Institute for Problems in Mechanics of the U.S.S.R. Academy of Sciences).
His Ph.D. thesis was devoted to the asymptotic analysis of the problems of heat and mass
transfer. In 1986, Professor Polyanin received his Doctor of Sciences degree; his D.Sc. thesis
was dedicated to the mass and heat exchange between reacting particles and fiow.
Since 1975, Professor Polyanin is a member of staff of the Institute for Problems in
Mechanics of the Russian Academy of Sciences.
Professor Polyanin has made important contributions to new approximate analitical
methods in the theory of heat and mass transfer, hydrodynamics, and chemical engineering
science, as well as to new methods of the theory of ordinary differential equations. In 1991
he was awarded a Chaplygin Prize of the U.S.S.R. Academy of Sciences for his research in
mechanics.
Professor Polyanin has published more than 100 research papers and seven books. He
is also an author of three patents.
Valentin F. Zaitsev, Ph.D., D.Sc, is a noted scientist in the fields of ordinary
differential equations, mathematical physics, and nonlinear mechanics.
Professor Zaitsev graduated from the Radio Electronics Faculty of the Leningrad Po-
litechnical Institute (now Saint-Petersburg Technical University) in 1969 and received his
Candidate of Sciences (Ph.D.) degree in 1983 (at the Leningrad State University). His
Ph.D. thesis was devoted to the group approach to the study of some classes of ordinary
differential equations. In 1992, Professor Zaitsev received his Doctor of Sciences degree;
his D.Sc. thesis was dedicated to the discrete-group analysis of the ordinary differential
equations.
Since 1971, Professor Zaitsev is in the Research Institute for Computational Mathe-
Mathematics and Control Processes of the St .-Petersburg State University. He is also a Professor
at the Russian State Pedagogical University (St.-Petersburg), the Orel State Pedagogical
Institute, and Orel State Politechnical Institute.
Professor Zaitsev has made important contributions to new methods in the theory of
ordinary differential equations. He is an author of more than 70 scientific publications,
including five monographs and the patent.
Zaitsev V.F. also read the theoretical course at the Leningrad Conservatory, later par-
participating in development of mathematical methods in musical sciences.
© 1995 by CRC Press, Inc.

CONTENTS
1. First Order Differential Equations
1.1. Simplest Equations with Arbitrary Functions Integrable in a Closed Form
1.1.1. Equations of the Form y'x = f(x)
1.1.2. Equations of the Form y'x = f(y)
1.1.3. Separable Equations y'x = f(x)g(y)
1.1.4. Linear Equation g{x)y'x = fi(x)y + fo(x)
1.1.5. Bernoulli Equation g(x)y'x = fi(x)y + fn(x)yn
1.1.6. Homogeneous Equation y'x = f(y/x)
1.2. Riccati Equations: g(y)yx = h{x)y2 + fi(x)y + fo(x)
1.2.1. Preliminary Comments
1.2.2. Equations Containing Power Functions
1.2.3. Equations Containing Exponential Functions
1.2.4. Equations Containing Hyperbolic Functions
1.2.5. Equations Containing Logarithmic Functions
1.2.6. Equations Containing Trigonometric Functions
1.2.7. Equations Containing Inverse Trigonometric Functions
1.2.8. Equations Containing Arbitrary Functions
1.2.9. Some transformations
1.3. Abel Equations of the Second Kind
1.3.1. Equations of the Form yy'x — у = f(x)
1.3.2. Equations of the Form yy'x = f(x)y + 1
1.3.3. Equations of the Form yy'x = fi(x)y + fo(x)
1.3.4. Equations of the Form [gi(x)y + go(x)]y'x = /г(ж)у2 + fi(x)y + fo(x)
1.3.5. Some Types of First and Second Order Equations Reducible to Abel
Equations of the Second Kind
1.4. Equations Containing Polinomial Functions of у
1.4.1. Abel Equations of the First Kind y'x = /з(х)у3+ f2(x)y2+ fi(x)y+fo(x)
1.4.2. Equations of the Form (А22У2 + A12xy + Ацх2 + A0)y'x =
В22У2 + B12xy + Вцх2 + Bo
1.4.3. Equations of the Form (А22У2 + A12xy + Ацх2 + A2y + Aix)y'x =
В22У2 + B12xy + Вцх2 + B2y + Bix
1.4.4. Equations of the Form (А22У2 + Ai2xy + Ацх2 + A2y + Axx + A0)y'x =
В22У2 + B12xy + Вцх2 + B2y + Bix + Bo
1.5. Nonlinear Equations of the Form f(x,y)y'x = g(x,y) Containing Arbitrary
Parameters
1.5.1. Equations Containing Power Functions
1.5.2. Equations Containing Exponential Functions
1.5.3. Equations Containing Hyperbolic Functions
1.5.4. Equations Containing Logarithmic Functions
1.5.5. Equations Containing Trigonometric Functions
1.5.6. Equations Containing Combinations of Exponential, Hyperbolic,
Logarithmic, and Trigonometric Functions
© 1995 by CRC Press, Inc.

1.6. Equations Not Solved for Derivative
1.6.1. Equations of the Second Degree in y'x
1.6.2. Equations of the Third Degree in y'x
1.6.3. Equations of the Form (y'x)k = f(y) + g(x)
1.6.4. Other equations
1.7. Equations of the Form F(x,y)y'x = G(x,y) Containing Arbitrary Functions
1.7.1. Equations Containing Power Functions
1.7.2. Equations Containing Exponential and Hyperbolic Functions
1.7.3. Equations Containing Logarithmic Functions
1.7.4. Equations Containing Trigonometric Functions
1.7.5. Equations Containing Combinations of Exponential, Logarithmic, and
Trigonometric Functions
1.8. Equations of the Form F(x,y,y'x) = 0 Not Solved for the Derivative and
Containing Arbitrary Functions
1.8.1. Some Equations
1.8.2. Some Transformations
2. Second Order Differential Equations
2.1. Linear Equations
2.1.1. Preliminary Comments
2.1.2. Equations Containing Power Functions
2.1.3. Equations Containing Exponential Functions
2.1.4. Equations Containing Hyperbolic Functions
2.1.5. Equations Containing Logarithmic Functions
2.1.6. Equations Containing Trigonometric Functions
2.1.7. Equations Containing Inverse Trigonometric Functions
2.1.8. Equations Containing Combinations of Exponential, Logarithmic,
Trigonometric, and Other Functions
2.1.9. Equations Containing Arbitrary Functions
2.1.10. Some Transformations
2.1.11. Asymptotic Solutions
2.1.12. Series Solutions
2.2. Autonomous Equations yxx = F(y,y'x)
2.2.1. Equations of the Form y?x - y'x = f(y)
2.2.2. Equations of the Form y'^x + f(y)y'x +y = 0
2.2.3. Lienard Equations y'^ + f(y)y'x + g(y) = 0
2.2.4. Rayleigh Equations у%х + f(y'x) + g(y) = 0
2.3. Emden—Fowler Equation y'^ = Axnym
2.3.1. Exact Solutions
2.3.2. First Integrals (Conservations Laws)
2.3.3. Some Formulas and Transformations
2.4. Equations of the Form y'^ = A^y™1 + A2xn2ym2
2.4.1. Classification Table
2.4.2. Exact Solutions
2.5. Generalized Emden—Fowler Equation y'^ = Axnym(y'xI
2.5.1. The Classification Table
2.5.2. Exact Solutions
2.5.3. Some Formulae and Transformations
© 1995 by CRC Press, Inc.

2.6. Equations of the Form y%x = AlXniymi(y'x)h + A2xn2ym2(y'x)h
2.6.1. Modified Emden—Fowler Equation ylx = A^x'^y^ + A2xnym .
2.6.2. Equations of the Form ylx = (Aixniymi + А2хП2ут2)(у'хI
2.6.3. Equations of the Form ylx = <jAxnym(y'xI + Axn-Iym+I{y'xf~1
2.6.4. Other Equations (l\ ф l2)
2.7. Equations of the Form yxx = f(x)g(y)h(y'x)
2.7.1. Equations of the Form yxx = f(x)g(y)
2.7.2. Equations Containing Power Functions (h ф const)
2.7.3. Equations Containing Exponential Functions (h ф const)
2.7.4. Equations Containing Hyperbolic Functions (h ф const)
2.7.5. Equations Containing Trigonometric Functions (h ф const)
2.7.6. Some Transformations
2.8. Some Nonlinear Equations with Arbitrary Parameters
2.8.1. Equations Containing Power Functions
2.8.2. Painleve Equations
2.8.3. Equation Containing Exponential Functions
2.8.4. Equations Containing Hyperbolic Functions
2.8.5. Equations Containing Logarithmic Functions
2.8.6. Equations Containing Trigonometric Functions
2.8.7. Equations Containing the Combinations of Exponential, Hyperbolic,
Logarithmic, and Trigonometric Functions
2.9. Equations Containing Arbitrary Functions
2.9.1. Equations of the Form F(x, y)^ + G(x, у) = О
2.9.2. Equations of the Form F(x, y)^ + G(x, y)y'x + H(x, у) = О
2.9.3. Equations of the Form F(x,y)y'x'x + Еш=оСт(х,у)(у'хГ = О
(M = 2, 3, 4)
2.9.4. Equations of the Form F(x, y, y'x)yxx + G(x, y, y'x) = 0
2.9.5. Equations of the Form F(x, y, yx, y'xx) = 0
2.9.6. General Equations Admitting the Order Reduction
2.9.7. Some Transformations
3. Third Order Differential Equations
3.1. Linear Equations
3.1.1. Preliminary Comments
3.1.2. Equations Containing Power Functions
3.1.3. Equations Containing Exponential Functions
3.1.4. Equations Containing Hyperbolic Functions
3.1.5. Equations Containing Logarithmic Functions
3.1.6. Equations Containing Trigonometric Functions
3.1.7. Equations Containing Inverse Trigonometric Functions
3.1.8. Equations Containing Combinations of Exponential, Logarithmic,
Trigonometric, and Other Functions
3.1.9. Equations Containing Arbitrary Functions
3.2. Equations of the Form Ух"хх = Axayl3{y'xI {y'^f
3.2.1. Preliminary Comments. Classification Table
3.2.2. Equations of the Form y'^ = Ay?
3.2.3. Equations of the Form y'^ = AxayP
3.2.4. Equations with \j\ + \6\ ф О
3.2.5. Some Transformations
© 1995 by CRC Press, Inc.

3.3. Equations of the Form y^'xx = f(y)g(y'x)h(y'x'x)
3.3.1. Equations Containing Power Functions
3.3.2. Equations Containing Exponential Functions
3.3.3. Other Equations
3.4. Some Nonlinear Equations with Arbitrary Parameters
3.4.1. Equations Containing Power Functions
3.4.2. Equations Containing Exponential Functions
3.4.3. Equations Containing Hyperbolic Functions
3.4.4. Equations Containing Logarithmic Functions
3.4.5. Equations Containing Trigonometric Functions
3.5. Nonlinear Equations Containing Arbitrary Functions
3.5.1. Equations of the Form F(x, y)yxxx + G(x, y)=0
3.5.2. Equations of the Form F(x, y, yx)yxxx + G(x, y, y'x) = 0
3.5.3. Equations of the Form F(x, y, y'x)y^xx + G(x, y, y'x)y'^ + H(x, y,y'x)=0
3.5.4. Equations of the Form F(x,y,yx)y^xx + ЕаС«(х,У,У'х)(УхХ = 0
3.5.5. Equations of the Form F(x, y, y'x, yxx)yxxx + G(x, y, y'x,yxx) = 0
4. Fourth Order Differential Equations
4.1. Linear Equations
4.1.1. Preliminary Comments
4.1.2. Equations Containing Power Functions
4.1.3. Equations Containing Exponential, Hyperbolic, and Logarithmic
Functions
4.1.4. Equation Containing Trigonometric Functions
4.1.5. Equations containing arbitrary functions
4.1.6. Asymptotic Solutions
4.2. Nonlinear Equations
4.2.1. Equation Containing Power Functions
4.2.2. Equations Containing Exponential, Hyperbolic, Logarithmic, and
Trigonometric Functions
4.2.3. Equations Containing Arbitrary Functions
5. Higher Orders Differential Equations
5.1. Linear Equations
5.1.1. Preliminary Comments
5.1.2. Equations Containing Power Functions
5.1.3. Equations Containing Exponential Functions
5.1.4. Equations Containing Trigonometric Functions
5.1.5. Equations Containing Arbitrary Functions
5.1.6. Asymptotic Solutions
5.2. Nonlinear Equations
5.2.1. Equations Containing Power Functions
5.2.2. Equations Containing Exponential Functions
5.2.3. Equations Containing Hyperbolic Functions
5.2.4. Equations Containing Logarithmic Functions
5.2.5. Equations Containing Trigonometric Functions
5.2.6. Equations Containing Arbitrary Functions
© 1995 by CRC Press, Inc.

Supplement 1. Some Elementary Functions and Their Properties
1.1. Trigonometric Functions
1.1.1. Simplest Relations
1.1.2. Relations Between Trigonometric Functions of Identical Argument
1.1.3. Reduction Formulae
1.1.4. Addition Formulae
1.1.5. Addition and Subtraction of Trigonometric Functions
1.1.6. Product of Trigonometric Functions
1.1.7. Powers of Trigonometric Functions
1.1.8. Trigonometric Functions of Multiple Arguments
1.1.9. Euler and de Moivre Formulae, Relation to Hyperbolic Functions
1.1.10. Differentiation and Integration Formulae
1.2. Hyperbolic Functions
1.2.1. Simplest Relations
1.1.2. Relations Between Hyperbolic Functions of Identical Argument
1.2.3. Addition Formulae
1.2.4. Addition and Subtraction of Hyperbolic Functions
1.2.5. Product of Hyperbolic Functions
1.2.6. Powers of Hyperbolic Functions
1.2.7. Hyperbolic Functions of Multiple Argument
1.2.8. Relation to Trigonometric Functoins
1.2.9. Differentiation and Integration Formulae
1.3. Inverse Trigonometric Functions
1.3.1. Simplest Relations
1.3.2. Relation Between Inverse Trigonometric Functions
1.3.3. Addition and Subtraction of Inverse Trigonometric Functions
1.3.4. Differentiation and Integration Formulae
1.4. Inverse Hyperbolic Functions
1.4.1. Relation to Logarithmic Functions and Simplest Relations
1.4.2. Relations Between Inverse Hyperbolic Functions
1.4.3. Addition and Subtraction of Inverse Hyperbolic Functions
1.4.4. Differentiation and Integration Formulae
1.5. Some Conventional Symbols
1.5.1. Factorial
1.5.2. Binomial Coefficients
1.5.3. Pochhammer Symbol
Supplement 2. Some Special Functions
2.1. Gamma-function
2.2. Bessel functions Jv and Yv
2.2.1. Basic Formulae
2.2.2. Bessel functions for v = ±n ± \; n = 0, 1, 2,
2.2.3. Wronskians an Similar Formulae
2.2.4. Integral Representation
2.2.5. Integrals with Bessel Functions on Closed Intervals
2.2.6. Asymptotic Expansion, as |ж| —> oo
© 1995 by CRC Press, Inc.

2.3. Modified Bessel Functions Iv and Kv
2.3.1. Basic Formulae
2.3.2. Modified Bessel Functions for v = ±n ± \\ n = 0, 1, 2,
2.3.3. Wronskians and Similar Formulae
2.3.4. Integral Representation
2.3.5. Integrals with Modified Bessel Functions on Closed Intervals
2.3.6. Asymptotic Expansion, as |ж| —>• oo
2.4. Degenerate Hypergeometric Functions
2.4.1. Definitions
2.4.2. Basic Properties
2.4.3. Integral Representation
2.4.4. Integrals with Degenerate Hypergeometric Functions
2.4.5. Asymptotic Expansion, as |ж| —>• oo
2.5. Legendre Functions
2.5.1. Definitions
2.5.2. Trigonometric Expansions
2.5.3. Some Relations
2.5.4. Integral Representation
2.6. The Weierstrass function p
2.6.1. Definitions
2.6.2. Some Properties
References
© 1995 by CRC Press, Inc.

Chapter 1
First Order
Differential Equations
1.1. Simplest Equations with Arbitrary Functions Integrable
in a Closed Form*
1.1.1. Equations of the Form y'x = f{x)
Solution:** y = ff(x)dx + C.
1.1.2. Equations of the Form yx = f(y)
Solution: x = / —V + С.
J f(y)
Particular solutions: у = А^, where A^ are roots of the algebraic equation /(Л^,) = О.
1.1.3. Separable Equations y'x = f(x)g(y)
Solution: / -^- = f f(x)dx + C.
J g(y) J
Particular solutions: у = A^, where A^ are roots of the algebraic equation g{A}c) = 0.
Remark. The equation of the form fi(x)gi(y)y'x = /2(^M2B/) is reduced to the form
1.1.3 by dividing both sides by
1.1.4. Linear Equation g(x)yx = fi(x)y + /0(cc)
Solution:
„ p , F I —F JOyE) i i r/ •. / Jl\%)
у = Се + e e ———- ax, where b (x) = / ——-
J 9\x) J g(x)
dx.
* Special cases of equations 1.1.1—1.1.5 for concrete functions /, /0, /1, fa, and g are
not discussed in this book; such cases can be readily recognized by the appearance of
equations investigated, and the solution can be obtained using the general formulae given
in Section 1.1.
** Hereinafter we shall often use the term "solution" to mean "general solution."
© 1995 by CRC Press, Inc.

1.1.5. Bernoulli Equation g{x)y'x = h{x)y + fn{x)yn
Here, n is an arbitrary number.
The substitution w(x) = y1~n leads to a linear equation:
Solution:
F F [ рЩ4 / Ц^ dx.
= CeF + A - n)eF [ е~рЩ4- dx, where F(x) = A - n) / Ц
J g(x) J g
g(x)
1.1.6. Homogeneous Equation y'x = f(y/x)
The substitution u(x) = y/x leads to an equation with separation of variables: xu'x =
f(u) - u.
—
f(u) - и
Particular solutions: у = А^х, where A^ are roots of the algebraic equation
Ak-f(Ak) = 0.
Solution: / —— = In I ж I + C.
J f()
1.2. Riccati Equations: g{y)y'x = hi^y1 + fi(x)y + fo(x)
1.2.1. Preliminary Comments
For /2 = 0, we obtain a linear equation (see 1.1.3), and for /0 = 0, we have the Bernoulli
equation (see 1.1.4 with n = 2), whose solutions were given previously. Below we discuss
equations with /0/2 / &
1. Given a particular solution yo = yo{x) of the Riccati equation, the general solution
can be written as
J
g(x)
where
Ф(ж) =
-l
To the particular solution yo(x) corresponds С = oo.
2. The substitution
u(x) = expl — / —у dx
reduces the general Riccati equation to a second order linear equation:
g2f2<x + g[f2g'x - дШ'х ~ Л/2К + Mh = 0,
which often may be easier to solve than the original Riccati equation.
3. Let <7 = /2 = 1) fi(x) and fo(x) be polynomials. If the degree of the polynomial
A = /j2 — 2(fi)'x — 4/o is odd, the Riccati equation can not possess a polynomial solution.
© 1995 by CRC Press, Inc.

If the degree of A is even, the equation involved may possess only the following polynomial
solutions:
where [л/А ] denotes an integer rational part of the expansion of л/А in decreasing powers
of x (for example, [s/x2 — 2x + 3 ] = x — 1).
4. The general Riccati equation, with the aid of the substitution
Р® 4 A)
is reduced to the canonical form
w's=w2 + Ф(?), B)
where function Ф is defined by the formula
u /I 1 9 1 OP Л \ T? I OF
4 Z Z i*2 4 \ Г2 I ? Г'.
(prime denotes differentiation with respect to ?).
Substitution A) is determined by function ip = ip(^) which may be arbitray. For a specific
original Riccati equation, different functions ip in A) will generate different functions Ф in
equation B).
In the special case where the original equation has the canonical form
Ух = У2 +
transformation A) is written as
and the transformed equation B) is determined by function Ф:
If the original Riccati equation is integrable by quadrature, we may obtain, specifying
different functions ip, a variety of different integrable equations of the form B). In Subsection
1.2.8, some useful transformations are given for specific functions ip.
5. The transformation (ip, ipi, тр2, фз, and ф^ are arbitrary functions)
У =
reduces the general Riccati equation to the Riccati equation.
© 1995 by CRC Press, Inc.

1.2.2. Equations Containing Power Functions
1. y'x = ay2 + bx + с
For b = 0, we have an equation of the form 1.1.2. For ЬфО, the substitution bt = bx + c
leads to an equation of the form 1.2.2.4: y't = ay2 + bt.
2- У'Х=У2 ~ a2x2 + So-
Particular solution: yo = ax — х~г.
3- V'x = V2 + ax2 + bx + c.
This is a special case of equation 1.2.2.9 with a = 0, /3 = 0.
4- y'x = ay2 + bxn.
Special Riccati equation, n is an arbitrary number.
1 u'
Solution: у = —, where
а и
q =
2
Jm and Ym are Bessel functions, n ф —2. With n = —2, see equation 1.2.2.36.
5. y'x=y2 + anxn~x - a2x2n.
Particular solution: yo = axn.
6- y'x = ay2 + bx2n + cxn-1.
For n = —1, we have 1.2.2.36. For n ф —1, the substitution
?x
n+l
leads to an equation of the form 1.2.2.25:
^+^2 +
7. y'x=y2 + axny + ах™'1.
Particular solution: yo = —1/x.
8. y'x=y2 + axny + bxn-1.
The substitution у = —и'х/и leads to a second order linear equation of the form
2.1.2.42:
ulx - axnu'x + bxn~1u = 0.
9- V'x = У2 + (otx + C)y + ax2 + bx + с
The substitution у = —и'х/и leads to a second order linear equation of the form
2.1.2.28:
u" - (ax + /3)u' + (ax2 + bx + c)u = 0.
© 1995 by CRC Press, Inc.

10. y'x = у2 + axny - abxn - b2.
Particular solution: yo = b.
11. y'x = axny2 + bx-n~2.
Solution:
du J~a „,-, n + 1
+C, where и = ^xn+1y, Ц =
12. y'x = axny2 + bxm.
1°. For n ф —1, the substitution ? = rrn+1 leads to an equation of the form 1.2.2.4:
2°. For n = —1 and m ф —1, the transformation ? = rrm+1, w = —1/y yields an
equation of the form 1.2.2.4:
/ fr 2 a
c m+1 ra+1
3°. For n = m = —1, the original equation is an equation with separation of variables.
In this case we have
ay2 + b
The transformation
?= —, и = — [(ex + dJy + c(cx + d)], where A = ad — be,
CX ~~\~ (t I-\
leads to an equation of the form 1.2.2.4: u'F = u2 + kA~2?n.
14. y'x = axny2 + bmx™-1 - ab2xn+2m.
Particular solution: yo = bxm.
15. y'x = -(n + l)xny2 + axn+m+1y-axm.
Particular solution: yo = x~n~ .
16. y'x = axny2 + bxmy + bcxm - ac2xn.
Particular solution: yo = —с-
17. y'x = axny2 -axn(bxm+ c)y + bmxm-x
Particular solution: y0 = bxm + с
18. y'x = -anxn-1y2 + cxm(axn + b)y - cx
Particular solution: yo = (axn + b)~x.
© 1995 by CRC Press, Inc.

19. y'x = axny2 + bxmy + ckx1*-1 - bcxm+k - ac2xn+2k.
Particular solution: yo = ex .
20. y'x = (ax2n + bxn~^)y2 + с
The substitution у = —1/w leads to an equation of the form 1.2.2.8:
21. xy'x = ay2 +by + cx2b.
The transformation t = xb, w = x~by leads to an equation with separation of variables:
bw'f = aw2 + с
22. xy'x = ay2 +by + cxn.
The transformation ? = xb, r/ = yx~b reduces this equation to the special Riccati
equation 1.2.2.4:
г,'=±тJ + ^Г, where m=-?-2.
23. xy'x = ay2 + (n + bxn)y + cx2n.
The substitution у = wxn leads to an equation with separation of variables:
w'x = xn~1(aw2 + bw + c).
24. xy'x = xy2 + ay + bxn.
The substitution у = —и'х/и leads to a second order linear equation of the form
2.1.2.62: xu"xx - au'x + bxnu = 0.
25. xy'x + a3xy2 + a2y + сцх + o0 = 0.
The substitution a^y = u'x/u leads to a second order linear equation of the form
2.1.2.59: xuxx + a<i%)!x + аз(а\х + ao)u = 0.
xx
26. xy'x = axny2 + by + cx~n.
The substitution w = yxn leads to an equation with separation of variables: xw'x
aw2 + (b + n)w + с
27. xy'x = axny2 +my- ab2xn+2m.
Particular solution: yo = bxm.
28. xy'x = x2ny2 + (m - n)y + x2m.
Solution: у = xm~n tan I + C).
© 1995 by CRC Press, Inc.

29. xy'x = axny2 + by + cxm.
The transformation ? = xn~b, r\ = yxb leads to the special Riccati equationl.2.2.4:
2 k , , m-n-2b
(n + b)rj? = arj2 + c?k, where к =
n + b
30. xy'x = ax2ny2 + (bxn - n)y + с
For n = 0, this is an equation with separation of variables. For n^O, the solution is
/dw „ „ „
r—; = x + C, where w = yx .
awz + bw + с
31. xy'x = ax2n+my2 + (bxn+m - n)y + cxm.
The substitution w = yxn leads to an equation with separation of variables:
w'x=xn+m-1{aw2 + bw + c).
32. (о2ж + Ь2)(у'х + Xy2) + @1Ж + bi)y + aox + b0 = 0.
The substitution Ay = u'x/u leads to a second order linear equationof the form
2.1.2.103:
(a2x + b<2)v!'xx + (aix + h)u'x + X(aox + bo)u = 0.
33. (ax + c)y'x = a(ay + bxJ + /3(ay + bx) - bx + 7.
The substitution t = ay + bx leads to a linear equation with respect to x = x(t):
(aat2 + /3at + ja + bc)x't = ax + с
34. 2x2y'x = 1y2 +xy- 2a2x.
Particular solution: yo = a\fx.
35. 2x2y'x = 2y2 + 3xy — 2a2x.
IT
Particular solution: yo = a\fx——.
36. x2y'x = ax2y2 + b.
Solution:
1
where A is a root of the quadratic equation aX + A + b = 0.
37. x2y'x = ax2y2 + bxy + с
The substitution w = xy leads to an equation with separation of variables:
xw'x = aw2 + (b + l)w + с
38. x2y'x = x2y2 - a2x4 + o(l - 2b)x2 - b(b + 1).
Particular solution: yo = ax + Ьх~г.
© 1995 by CRC Press, Inc.

39. x2y'x = cx2y2 + (ax2 + bx)y + ax2 + f3x + 7.
The substitution cy = —u'x/u leads to a second order linear equation of the form
2.1.2.134:
x2uxx — x(ax + b)u'x + c(ax2 + fix + j)u = 0.
40. x2y'x = ax2y2 + bxn + с
Having set w = xy + A, where Л is a root of the quadratic equation a A2 — A + с = 0,
we arrive at an equation of the form 1.2.2.22: xw'x = aw2 + A — 2aA)w + bxn.
1 —n2
41. x2y'x = x2y2-\ \-ax2m(bxm + c)n.
The transformation
w =
^ху + ^rx
bm 2bm
leads to an equation of the form 1.2.2.4: w'? = w2 + a(bm)~2^n.
42. x2y'x = ax2y2 + bxy + cxm + s.
The substitution ay = —u'x/u leads to a second order linear equation of the form
2.1.2.127: x2u%x - bxu'x + a(cxm + s)u = 0.
43. x2y'x = ax2y2 + bxy + cx2m + sxm.
The substitution ay = —u'x/u leads to a second order linear equation of the form
2.1.2.128:
x2uxx - bxu'x + axm(cxm + s)u = 0.
44. x2y'x = cx2y2 + (axn + b)xy + ax2n + /3xn + 7.
The substitution cy = —u'x/u leads to a second order linear equation of the form
2.1.2.141:
x2uxx - (axn + b)xu'x + c(ax2n + fix11 + i)u = 0.
45. x2y'x = (ax2n + /3xn + i)y2 + (axn + b)xy + ex2.
The substitution у = —1/w leads to an equation of the form 1.2.2.44:
x2w'x = cx2w2 - (axn + b)xw + ax2n + fixn + 7.
46. (ж2 - l)y'x + X(y2 - Ixy + 1) = 0.
The substitution у = x -\ т—г- leads to an equation of the same form:
A A u(x)
(x2 - l)u'x + (A - l)(u2 - 2xu + 1) = 0.
If A = n is a positive integer, then by using the above substitution, the original
equation can be reduced to an equation of the same form, wherein A = 1, i.e., to an
equation of the form 1.2.2.49 with a = 1, b = —1.
© 1995 by CRC Press, Inc.

47. (ax2 + Ь)у'х + ay2 + (Зху -\ (о + /3) = 0.
a
Particular solution: yo = x-
a
48. (ax2 + b)y'x + ay2 + /Зху + 7 = 0.
The substitution у = x -—i- leads to an equation of the same form:
a u(x)
a + /3
(ax2 + b)u'x + Cy - ——— b\u2 + Ba + /3)xu + a = 0.
a
49. (ax2 + b)y'x + y2- Ixy + A - a)x2 - b = 0.
adx X
—о—r
ax2 + b
50. (ax2 + bx + c)y'x = y2 + BXx + b)y + Л(Л - a)x2 + \x.
Particular solutions:
yo = -\x + A, where A = \(-Ъ± \/Ъ2 -4/л- 4Ас).
51. (ax2 + bx + c)y'x = y2 + (ax + ц)у — X2x2 + X(b — ц)х + Ac.
Particular solution: yo = Аж.
52.
Particular solution: y$ = Xx.
53. (a2x2 + b2x + c2)y'x = y2 + (a-yx + b\)y + a0x2 + box + c0.
Let A and C be roots of the system of the quadratic equations
A2 + A(ai - a2) + a0 = 0, C2 + (ЗЪг + c0 - Xc2 = 0,
that are solved consecutively (in the general case there are four roots). If some of
roots satisfy the condition 2A/3 + A&i + Ca\ + bo — Xb2 = 0, the original equation
posesses a paticular solution: yo = Xx + C.
54. (x -a)(x- b)y'x + y2 + k(y + x - a)(y + x - b) = 0.
To the case of к = 0 corresponds an equation with separation of variables. To the
case of к = — 1 corresponds a linear equation. For к ф —1 and 0, with the aid of the
substitution ku(x) = у + k(y + x), we obtain the general solution:
у + k(y + x - a) ( x — a \k _
у + k(y + x — b)
r—^ r + —^— = C if a = b.
у + k(y + x - a) x -a
© 1995 by CRC Press, Inc.

55. (с2ж2 + b2x + а2)(у'х + Ау2)
oi)y + о0 = 0.
The substitution Ay = и'х/и leads to a second order linear equation of the form
2.1.2.166:
(c2x2 + b2x + a2)uxx + (hx + ai)u'x + Xaou = 0.
56. x3y'x = ax3y2 + (bx2 + c)y + sx.
The substitution ay = —u'x/u leads to a second order linear equation of the form
2.1.2.170:
x3uxx - (bx2 + c)u'x + asxu = 0.
57. x3y'x = ax3y2 +x(bx + c)y + ax +/3.
The substitution ay = —ux/u leads to a second order linear equation of the form
2.1.2.173:
x3u" - x(bx + c)u' + a(ax + /3)u = 0.
58. x(x2 + a){y'x + Ay2) + (bx2 + c)y + sx = 0.
The substitution Ay = ux/u leads to a second order linear equation of the form
2.1.2.177:
x(x2 + a)uxx + (bx2 + c)u'x + Xsxu = 0.
59. x2(x + a)(y'x + Ay2) + x(bx + c)y + ax + C = 0.
The substitution Ay = u'x/u leads to a second order linear equation of the form
2.1.2.181:
x2(x + a)uxx + x(bx + c)u'x + X(ax + C)u = 0.
60. (ax2 + bx + c)(xy'x - y) - y2 + x2 = 0.
Solution: In
y-x
y + x
dx
ax2 + bx + с
61. ж4у' = —ж4у2 — о2.
1 а . а .
Solution: у = 1 — tan( \- С¦).
хх2 х
62. ж2 (ж2 + а)(у'х + Ау2) + ж(Ьж2 + с)у + s = 0.
The substitution Ay = u'x/u leads to a second order linear equation of the form
2.1.2.206:
x2(x2 + a)uxx + x(bx2 + c)u'x + Xsu = 0.
63. ож2(ж - lJ(y^, + Ay2) + Ьж2 + ex + s = 0.
The substitution Ay = ux/u leads to a second order linear equation of the form
2.1.2.205:
ax2(x — lJuxx + X(bx2 + ex + s)u = 0.
© 1995 by CRC Press, Inc.

64. a(x2 - 1J(у'х + Лу2) + Ьх(х2 - 1)у + сх2 + dx + s = 0.
The substitution Ay = u'x/u leads to a second order linear equation of the form
2.1.2.213:
a(x2 - lJuxx + bx(x2 - l)u'x + X(cx2 +dx + s)u = 0.
65. (ax2 + bx + cJ(yfx + y2) + A = 0.
The substitution y = u'x/u leads to a second order linear equation of the form 2.1.2.220:
(ax2 + bx + cJu"xx + Аи = О.
66. xn+1y'x = ax2ny2 + bxny + cxm + d.
Having set w = xny + A, where A is a root of the quadratic equation a A2 — (b + n)A +
d = 0, we arrive at an equation of the form 1.2.2.22:
xw'x = aw2 + (n + b- 2aA)w + cxm.
67. x(axk + b)y'x = axny2 + (/3 - anxk)y + ^x~n.
The transformation t = xny, z = x~k leads to an equation with separation of variables:
[at2 + (p + bn)t + j}z't = -k(bz + a).
68. x2(axb - l){y'x + Ay2) + (pxb + q)xy + rxb + s = 0.
The substitution Ay = ux/u leads to a second order linear equation of the form
2.1.2.238:
x2(axb - l)uxx + (pxb + q)xu'x + X(rxb + s)u = 0.
69. (axn + bxm + c)y'x = cy2 - bxm~1y + axn~2.
Particular solution: yo = —1/x.
70. (axn + bxm + c)y'x = axn~2y2 + bxm~1y + с
Particular solution: yo = x.
71. (axn + bxm + c)y'x = axky2 + /3xsy - a\2xk + /3Xxs.
Particular solution: yo = —A.
72. (axn + bxm + c)(xy'x - y) + sxk(y2 - Xx2) = 0.
Particular solutions: yo = ±жл/Х
73. (axn + bxm + c)(y'x - y2) + an(n - l)xn~2 + bm(m - l)xm~2 = 0.
_, . , , . апхп~г + Ьтхт~г
Particular solution: yo = .
y axn + bxm + с
© 1995 by CRC Press, Inc.

1.2.3. Equations Containing Exponential Functions
1. y'x = ay2 + beXx.
The substitution t = eXx leads to an equation of the form 1.2.2.22: Xty't = ay2 + bt.
2. y'x=y2 + a\eXx - a2e2Xx.
Particular solution: yo = aeXx ¦
3. y'x = cry2 + a + beXx + ce2Xx.
The substitution ay = —ux/u leads to a second order linear equation of the form
2.1.3.5: <ж + a(a + beXx + ce2Xx)u = 0.
4. y' = cry2 -\- ay -\- bex -\- с
The substitution ay = —u'x/u leads to a second order linear equation of the form
2.1.3.10: v!'xx - au'x + a(bex + c)u = 0.
5. y'x = y2 + by + o(A - b)eXx - a2e2Xx.
Particular solution: yo = ae .
6. y'x=y2 + aeXxy - abeXx - b2.
Particular solution: yo = b.
7. y'x=y2 + ae2Xx(eXx + b)n - -^A2.
The transformation
leads to an equation of the form 1.2.2.4: w'? = w2 + a\~2^n.
8. y'x=y2 + ae8Xx + beeXx + ce4Xx - A2.
The transformation
i
leads to an equation of the form 1.2.2.3: w'L = w2 + BA) 2(a?2 + b?, + c).
9. y'x = y2 + axeXxy + aeXx.
Particular solution: yo = — 1/x.
10. y'x = aeXxy2 + be~Xx.
/dz
—т: 7 — = x + C, where z = eXxy.
az2 + Xz + b y
11. y'x = aekxy2 + besx, к ф 0.
The substitution t = ekx leads to an equation of the form 1.2.2.4: ky't = ay2 + bts~k.
© 1995 by CRC Press, Inc.

12. y'x = Ъе»ху2 + aXeXx — a
Particular solution: yo = aeXx.
13. y'x = aeXxy2 + by + ce~Xx.
The substitution z = eXxy leads to an equation with separation of variables: z'x =
az2 + (b + \)z + с
14. y'x = ae^y2 + Xy - ab2e^+2X">x.
Particular solution: yo = beXx.
15. y'x = eXxy2
Particular solution: yo = ~Xe~Xx.
16. y'x = -XeXxy2 +
Particular solution: yo = e~Xx¦
17. y'x = ae^y2 + abe^x+^xy - bXeXx.
Particular solution: yo = — beXx.
18. y'x = aekxy2 + by + cesx + de~kx.
The substitution t = ekx leads to an equation of the form 1.2.2.42:
kt2y't = at2y2 + bty + Sk+sVk + d.
19. y'x = aeBX+^xy2 + [be^x+^x - X]y
The substitution w = eXxy leads to an equation with separation of variables:
20. y'x = aekxy2 + by + ceknx
The substitution t = ekx leads to an equation of the form 1.2.2.43:
kt2y't = at2y2 + bty + ctn+1 + dt2(~n+1\
21. y'x = e^iy - beXxJ + ЬХеХх.
Particular solution: yo = beXx.
22. y'x = aeXxy2 + Ьпхп~г — ab2eXxx2n.
Particular solution: yo = bxn.
23. y'x = eXxy2 + axny + aXxne~Xx.
Particular solution: yo = — Xe~Xx.
© 1995 by CRC Press, Inc.

24. y'x = -XeXxy2 + axneXx - axn.
= e~Xx.
Particular solution: yo
25. y'x = aeXxy2 - abxneXxy
Particular solution: yo = bxn.
26. y'x = axny2 + bXeXx - ab2xne2Xx.
Particular solution: yo = beXx.
27. y'x = axny2 +Xy- ab2xne2Xx.
Particular solution: yo = beXx.
28. y'x = axny2 - abxneXxy + ЬХеХх.
Particular solution: yo = beXx.
29. y'x = -(k + l)xky2 + axk+1eXxy - aeXx.
Particular solution: yo = x~ .
30. y'x = axny2 - axn(beXx + c)y + ЬХеХх.
Particular solution: y0 = beXx + с
31. y'x = axne2Xxy2 + (bxneXx - X)y + cxn.
The substitution w = eXxy leads to an equation with separation of variables:
wx=xnex(aw +bw + c).
32. y'x = aeXx(y-Ьхп - сJ+ Ьпхп-г.
Particular solution: yo = bxn + с
33. xy'x = aeXxy2 + ky + ab2x2keXx.
Solution: у = bxk tan(a6 / xk~1eXx dx + C).
J
34. xy'x = ax2neXxy2 + (bxneXx - n)y + ceXx.
Solution: / r— = xn~1eXx dx + C, where w = xny.
J awz + bw + с J
35. (aeXx + be^x + c)y'x = y2 + kevxy — m2 + kmevx.
Particular solution: yo = —то.
36. (aeXx + Ъе»х + с)(у'х - у2) + аХ2еХх + b^e^ = 0.
a\eXx
Particular solution: yo
aeXx
© 1995 by CRC Press, Inc.

37. y'x = у2 + 2a\xeXx2 - a2e2Xx\
Particular solution: yo = o,eXx .
38. y'x = ae-Xx2y2 + Xxy + ab2.
Solution: у = beXx2/2 tan(ab f e~Xx2/2 dx + C
39. y'x = axny2 + Xxy + ab2xneXx2.
Solution: у = beXx2/2 tan(a6 f xneXx2/2 dx + C}.
40. x (y — у ) = a + b exp — ) + с exp .
\X' V X >
The transformation ? = 1/x, w = —x2y — x leads to an equation of the form 1.2.3.3:
го' =U;2+a + 6efe«+ce2fe«.
1.2.4. Equations Containing Hyperbolic Functions
1. y'x = ay2 + /3 + 7 cosh x.
The transformation x = 2t, ay = —u'x/u leads to the modified Mathieu equation
2.1.4.1:
u"t — (a — 2qcosh2t)u = 0, where a = — 4a/?, q = 2aj.
2. y'x = y2 — a2 + oAsinh(Aa;) — a2 sinh2(Aa;).
Particular solution: yo = acosh(Xx).
3. y'x=y2 - X2 + acoshn(Xx) sinh-"-4(Aa;).
The transformation
? = coth(Arr), w = —— sinh (Xx)y — sinh(Aa;) cosh(Aa;)
A
leads to an equation of the form 1.2.2.4: w'? = w2 + A~2?n.
4. y'x=y2+ aX-a(a + X)tanh2(Xx).
Particular solution: yo = atanh(Aa;).
5. y'x =y2 + 3oA- A2 - o(o + A)tanh2(Aa;).
Particular solution: yo = atanh(Aa;) — Acoth(Aa;).
6. y'x =y2+ аХ —а(а +X)coth2(Xx).
Particular solution: yo = йсоШ(Аж).
© 1995 by CRC Press, Inc.

7. y'x = у2 - X2+ 3aX-a(a +X)coth2(Xx).
Particular solution: yo = йсоШ(Аж) — Atanh(Arr).
8. y'x=y2 - 2Л2 tanh2(A«) - 2A2 соШ2(Аж).
Particular solution: yo = Atanh(Arr) + Acoth(Arr).
9. y'x = y2 + aX + bX — lab — a(a + X) tanh2(A#) - b(b + X) coth2(Aa;).
Particular solution: yo = atanh(Aa;) + 6coth(Arr).
10. y'x = Xsinh(Xx)y2 — Xsinh3(Xx).
Particular solution: yo = cosh(Aa;).
11. y'x = a sinh(Aa;)y2 + b sinh(Aa;) cosh"(Aa;).
The transformation ^ = cosh(Arr), w = — у leads to an equation of the form 1.2.2.4:
w? = w2 + ab\-2in.
12. y'x = ocosh(Aa;)y2 + bcosh(Aa;) sinh"(Aa;).
The transformation ^ = sinh(Aa;), w = —у leads to an equation of the form 1.2.2.4:
w? = w2 + abX~2in.
13. y'x = [osinh2(Aa;) — X]y2 — osinh2(Aa;) + X - a.
Particular solution: yo = coth(Aa;).
14. y'x = [осоэЬ2(Аж) - X]y2 + a + X - осоэЬ2(Аж).
Particular solution: yo = tanh(Aa;).
15. 1y'x = [a — X + ocosh(A«)]y2 + a + A — осоэЬ(Аж).
Particular solution: yo = tanh I I.
1.2.5. Equations Containing Logarithmic Functions
1. y'x = у2 + о ln(/3x)y - ab ln(/3x) - b2.
Particular solution: yo = b.
2. y'x=y2 + ax lnm (bx)y + a lnm(bx).
Particular solution: yo = —1/x.
3. y'x = axny2-abxn+1lnxy +
Particular solution: yo = bxlnx.
© 1995 by CRC Press, Inc.

4. у'х = -(га + 1)ж"у2 + ахп+1(\пх)ту - аAпж)"\
Particular solution: yo = х~п~ .
5. у'х = аAпх)пу2 + Ътхт-Х - аЬ2х2т(\пх)п.
Particular solution: yo = Ьхт.
6. у'х = оAп х)пу2 - аЬх(\п ж)™+1у + Ып ж + Ь.
Particular solution: yo = bxlnx.
7. у^ = оAпж)'г(у-Ьжгг -сJ + Ьгажгг-1.
Particular solution: yo = &жп + с.
8. у^ = аAпх)пу2 + ЬAпх)ту + Ьс(\пх)т - ас2(\пх)п.
Particular solution: yo = ~с-
9. жу^, = ау2 + Ып ж + с.
The substitution ж = е* leads to an equation of the form 1.2.2.1: y? = ay2 + bt + с
10. xy'x = ay2 + Ыпк x + cln2k+2 x.
The substitution ? = In ж leads to an equation of the form 1.2.2.6 with к = n — 1:
y't = ay2 + btk+ct2k+2.
11. жу^, = (oy + bin жJ.
/dz
—^—r + С) where z = ay + b In x.
az2 + b
12. xy'x = xy2 - A2 \n2 (fix) + A.
Particular solution: yo = Aln(ftx).
13. жу^ = жу2 - A2x In2fc(^) + Ak \пк-г(Cх).
Particular solution: yo = Aln (Cx).
14. жу^, = ож"у2 + Ь + ab2xn In2 ж.
Particular solution: yo = b In x.
15. жу^ = а 1п"г(Лж)у2 + fey + о62ж2'г 1п"г(Лж).
Solution: у = 6rrfe tan [a& / ж'0 lnm(Arr) dx + c\.
16. xy'x = ахп(у + ЫпхJ-b.
Solution: 3_ + ±xn = C
у + Ыпх п
© 1995 by CRC Press, Inc.

17. xy'x = ax2n(\nx)y2 + (bxn In x — n)y + с In ж.
Solution: / t ; = xn~1lnx dx + C, where w = xny.
J awz + bw + с J
18. x2y'x = x2y2 + о In2 ж + bin ж + с
The transformation ? = In ж, w = жу + -j leads to an equation of the form 1.2.2.3:
w^ = w2 + a?2 + &? + с - -L.
19. ж2у4 = ж2у2 + о(Ыпж + с)"+^.
The transformation
1
leads to an equation of the form 1.2.2.4: wi = w2 + ab~2?n.
20. x2y'x = a2x2y2 -xy + b2 In™ ж.
The substitution а2у = —и'х/и leads to an equation of the form a second order linear
equation of the form 2.1.5.24: x2v!'xx + xu'x + (abJ lnm xu = 0.
21. ж21п(ож)(у^-у2) = 1.
Particular solution: уд = [жк^аж)].
22. (о In ж + b)y'x = у2 + c(ln x)ny - Л2 + ЛсAп ж)".
Particular solution: yo = —A.
23. (о1пж + Ь)у^ = (\nx)ny2 + су -Л2 (In ж)" + сЛ.
Particular solution: yo = ~^-
1.2.6. Equations Containing Trigonometric Functions
1. y'x = ay2+C+fsin(\x).
The substitution 2i = 2Аж + тг leads to an equation of the form 1.2.6.2: Xy'x = ay2 +
/3 + 7 cos t.
2. y'x = ay2 + /3 + 7 cos ж.
The transformation ж = 2i, ay = —u'x/u leads to the Mathieu equation 2.1.6.4:
uxx + (a — 2q cos 2t)u = 0, where a = Aaj3, q = —
3. y'x = y2 — a2 + аЛэт(Лж) + а2 эт2(Лж).
Particular solution: yo = —асов(Аж).
4. y'x = у2 — a2 + оЛсоз(Лж) + о2 соэ2(Лж).
Particular solution: уо = авш(Аж).
© 1995 by CRC Press, Inc.

5. y'x = у2 + X2 + csinn(Xx) cos-"-4(Aa;).
This is a special case of equation 1.2.6.6 with a = 0, b = тг/2.
6. y'x = y2 + A2 + csinn(Xx + a) sin~™-4(Aa; + b).
The transformation
вш(Аж + a) вш2(Аж + &
ч~8ш(Аж + &)' sinF-a) LA ' v ' ;J
leads to an equation of the form 1.2.2.4:
w'f = w2 + A?n, where A = c[AsinF — a)]~2.
7. y'x = y2 + as\n(Cx)y + absin(Cx) — b2.
Particular solution: yo = —b.
8. y'x = y2 + ax sin71 (bx)y + a sin71 (bx).
Particular solution: yo = —1/x.
9. y'x = y2 -\- aX + o(A — o) tan2(Aa;).
Particular solution: yo = atan(Arr).
10. y'x=y2 + X2+ 3aX + a(X-a) tan2(Xx).
Particular solution: yo = atan(Arr) — Acot(Aa;).
11. y'x = y2 + oA + o(A — o) cot2(Aa;).
Particular solution: yo = — acot(Arr).
12. y'x=y2 + X2+ ЗаХ + а(Х-а) cot2(Xx).
Particular solution: yo = Atan(Arr) — acot(Arr).
13- y'x = ay2+ btanxy+ c.
Having set ay = —u'x/u, we obtain a second order linear equation of the form 2.1.6.29:
u'xx ~ Ь tan x u'x + acu = 0-
14. y'x = ay2 + lab tan x у + b(ab — 1) tan2 x.
The substitution и = у + b tan x leads to an equation of the form 1.1.2: u'x = av? + b.
15. y'x = y2 — у tan x + o(l — o) cot2 x.
Particular solution: yo = —a cot ж.
16. y'x = y2 — my tan x + b2 cos2r™ x.
Solution: у = -b cosm ж cot (b / cosm xdx + Cj.
© 1995 by CRC Press, Inc.

17. y'x = у2 - 2acot(ax)y + b2 - a2.
Particular solution: yo = acot(ax) — bcot(bx).
18. y'x =y2 + my cot x + b2(sin xJm.
Solution: у = -b sinm x cot (b / sinm xdx + C
19. y'x=y2 - 2A2 tan2(Aa;) - 2A2 cot2(Aa;).
Particular solution: yo = Acot(Arr) — Atan(Arr).
20. y'x=y2 + Xa + Xb + lab + o(A - o) tan2(Aa;) + b(A - b) cot2(Aa;).
Particular solution: yo = atan(Aa;) — 6cot(Aa;).
21. y'x=y2 + ax tanm(bx)y + a tanm(bx).
Particular solution: yo = —1/ж.
22. y'x=y2 — -i-A2 - -|А^ап2(Аж) + асоэ2(Аж) эт^Аж).
у sin(Arr)
2 4
The transformation
= sin(Arr), w=
Асов(Аж) 2сов2(Аж)
leads to an equation of the form 1.2.2.4: w'e = w2 + a\~2(n.
23. y'x = A sin(Аж)у2 + A sin3 (Аж).
Particular solution: yo = — cos(Arr).
24. y^ = Асоэ(Аж)у2 + Асоэ3(Аж).
Particular solution: yo = sin(Aa;).
25. 2y^, = [A + о — оэ1п(Аж)]у2 + A — о — оэт(Аж).
/ Аж 7Г \
Particular solution: yo = tan I 1 1.
26. 2у^, = [А + о + осоэ(Аж)]у2 + А — о + осоэ(Аж).
Particular solution: yo = tan I J.
27. y'x = [А + оэт2(Аж)]у2 + A - а + аэт2(Аж).
Particular solution: yo = — cot(Aa;).
28. y'x = [А + осоэ2(Аж)]у2 + А- о + осоэ2(Аж).
Particular solution: yo = tan(Arr).
© 1995 by CRC Press, Inc.

29. y'x = asin(\x)y2 + bsin(Xx) cosn(Xx).
The transformation ? = cos(Arr), w = —— у leads to an equation of the form 1.2.2.4:
2 2
30. y'x = Asin(Aa;)y2 + acosn(Xx)y - ocos"~1(Aa;).
Particular solution: yo = l/cos(Aa?).
31. y'x = a cos(\x)y2 + bcos(Xx) sin"(Aa;).
The transformation ? = sin(Aa;), w = —у leads to an equation of the form 1.2.2.4:
w? = w2 + abX~2in.
32. y'x = Asin(Aa;)y2 + osin(Aa;)y — otan(Aa;).
Particular solution: yo = 1/сов(Аж).
33. y'x = \sin(\x)y2 + axncos(\x)y - axn.
Particular solution: yo = 1/сов(Аж).
34. y'x = -(к + l)xky2 + oa;fc+1(sina;)r"y - a(sinx)m.
Particular solution: yo = x~k~x.
35. y'x = -(k + l)xky2 + axk+1(tanx)my - a(tana;)m.
Particular solution: yo = x~k~x.
36. y'x = osinfc(Aa; + ц)(У ~ bxn - cJ
Particular solution: yo = bxn + с
37. y'x = atann(Xx)y2 - ab2 tann+2(\x) + b\tan2(\x) + bX.
Particular solution: yo = 6tan(Arr).
38. y'x = atank(Xx + ц)(у - bxn - сJ + Ъпхп~г.
Particular solution: yo = bxn + с
39. xy'x = asinm(Xx)y2 + ky + ab2x2k sinm(Xx).
Solution: у = bxk tan lab / хк~г sinm(Arr)drr + d\.
40. xy'x = atanm(Xx)y2 + ky + ab2x2k tan™(Aa;).
Solution: у = bxk tan lab / xk~x tanm(Arr)drr + d\.
© 1995 by CRC Press, Inc.

41. sin"+1 Bx)y'x = ay2 sin2" x + b cos2" ж.
The substitution z = у tan™ ж leads to an equation of the form an equation with
separation of variables: 2™ smBx)z'x = az2 + n2n+1z + b.
42. [аэт(Аж) + b]y'x = y2 + сэт(/хж)у — d2 + cd sin(^x).
Particular solution: yo = —d.
43. [otan(Aa;) + b]y'x = у2 + кЬап{цх)у — d2 + kdtan(nx).
Particular solution: yo = —d.
1.2.7. Equations Containing Inverse Trigonometric Functions
> In the equations 1-9, function arccosrr may be substituted for arcsin ж.
1. y'x = y2 + A(arcsina;)Tly — a2 + oA(arcsina;)Tl.
Particular solution: уд = —а.
2. y^ = у2 -|- Аж(arcsin ж)"у + А(агсэ1пж)гг.
Particular solution: yo = —1/ж.
3- У^ = -(fe + 1)sbV + А(агс5тж)гг(ж'г+1у - 1).
Particular solution: yo = x~ ~ .
4. y^, = А(агсэтж)ггу2 + ay + ab — Ь2А(агсэ1пж)гг.
Particular solution: yo = —b.
5. y^, = А(агсэтж)ггу2 — ЬАж"г(агсэтж)ггу + bmxm~1.
Particular solution: yo = bxm.
6. y'x = А(агс5тж)ггу2 + /Зтж™-1 - A/32s2m(arcsina;)n.
Particular solution: yo = /Зжт.
7. y'x = А(агс5тж)гг(у - ожт - bJ + атх.
Particular solution: yo = axm + b.
8. жу^, = А(агсэтж)ггу2 + fey + АЬ2ж2'г(агсз1пж)гг.
Solution: у = bxk tan\\b / ж'0 (arcsinж)" da; + С\.
9. жу^, = (ax2ny2 + bxny + c) (arcsin x)m — ny.
The substitution z = xny leads to an equation with separation of variables: z'x
ж™ (arcsin x)m(az2 + bz + c).
© 1995 by CRC Press, Inc.

> In the equations 10-18, function arccot ж may be substituted for arctan ж.
10. y'x = y2 + A(arctana;)"y - o2 + oA(arctana;)".
Particular solution: yo = —a.
11. y'x = у2 + Лж (arctan ж) "у + А^г^апж)™.
Particular solution: yo = —1/x.
12. y'x = -(fc + l)xky2 + Л(аг^апж)гг(ж'г+1у - 1).
Particular solution: yo = x~k~x.
13. y'x = A(arctanx)V + ay + ab — Ь2Л(аг^апж)".
Particular solution: yo = —b.
14. y'x = A(arctanx)V — b\xm (arctan ж)"у + bmx
Particular solution: yo = bxm.
.m — l
15. y'x = Л(arctanж)rгy2 + bmxm~x - Xb2x2m(arctanx)n.
Particular solution: yo = bxm.
16. y'x = Л(arctaпж)rг(y - axm - bJ + атхт~1.
Particular solution: yo = axm + b.
17. жу^ = Л(arctaпж)rгy2 + fey + Xb2x2k (arctan x)n.
Solution: y = bxktan\\b I xk~1(&rct&nx)ndx + c\.
18. жу^, = (ax2ny2 + bxny + c)(arctan ж) - ny.
The substitution z = xny leads to an equation with separation of variables: z'x
xn~x (arctan x)m(az2 + bz + c).
1.2.8. Equations Containing Arbitrary Functions
Notation: f = f(x) and g = g(x) are arbitrary functions; a, b, n, and A are arbitrary
parameters.
i- vL = v2 + fv - o2 - o/.
Particular solution: yo = a.
2- y'x = fy2 + ay-ab- b2f.
Particular solution: yo = b.
© 1995 by CRC Press, Inc.

3. Ух = У2 + xfy + f.
Particular solution: yo = —1/x.
4- y'x = fy2 - axnfy + anx™-1.
Particular solution: yo = axn.
5. y'x = fy2 + anx™-1 - a2x2nf.
Particular solution: yo = axn.
Particular solution: yo = х~п~г.
7 1 о *> *>эт о
1"?| Т11 —1— Т7?| —1— fill* I
Solution with a > 0: у = y/axn ta,n(y/a / x71'1 f dx + Cj.
Solution with a < 0: у = л/|а| ж™ tanhf-W|a| I xn~1fdx + C\.
8. жу^ = ж2"/у2 + (axnf - n)y + bf.
The substitution z = xny leads to an equation with separation of variables: z'x
9- y'x = fy2 +gy- a2f - ag.
Particular solution: yo = a.
10. y'x = fy2 +gy + anx™-1 - axng - a2fx2n.
Particular solution: yo = axn.
11. y'x = fy2 - axngy + anx^1 + a2x2n(g - /).
Particular solution: yo = axn.
12. y'x = -fxy2 + fgy - g.
Particular solution: yo = 1//.
13. y'x = fy2 - fgy + g'x-
Particular solution: yo = g.
14. yx=g(y-fJ+fx.
Particular solution: yo = /.
Particular solution: yo = —g/f-
© 1995 by CRC Press, Inc.

16.
f2v'x - f'xV2 + 9(У - f) = 0.
Particular solution: yo = /•
18.
19.
Particular solution: yo = —f'x/f-
y' = aeXxy2 + aeXxfy + Xf.
Particular solution: yo = e
a
~Xx
y'x = fy2 - aeXxfy + aXeXx.
Particular solution: yo = ae .
20. y'x = fy2 + aXeXx - a2e2Xxf.
„\x
21.
22.
y'x = fy2 + Xy + ae2Xxf.
Solution with a > 0: y=^/aeXx tan Ufa \ eXx f dx + c).
Solution with a < 0: у = J\a\ eXx tanh(-J\a\ / eXxfdx + i
y'x = fy2 - f(aeXx + b)y + aXeXx.
Particular solution: yo = aeXx + b.
23. y'x = eXxfy2 + (o/ - X)y + be~Xxf.
The substitution z = eXxy leads to an equation with separation of variables: z'x =
f{x){z2+az + b).
y'x = fy2 +gy + aXeXx - aeXxg - a2e2Xxf.
Particular solution: yo = aeXx.
24.
25.
26.
27.
y'x = fy2 - aeXxgy + aXeXx + a2e2Xx (g - /).
Particular solution: yo = aeXx.
y'x = fy2 + 2aXxeXx2 - a2fe2Xx\
Particular solution: yo = ae .
y'x = fy2 + Xxy + afeXx2.
Solution with a > 0: у = у/аеХх2/2 tan(Va / eXx2/2f dx +
Solution with a < 0: у = ^J\a\eXx2/2 tanh(-^J\a\ f eXx2/2fdx + С1
© 1995 by CRC Press, Inc.

28- У'х = fLv2 + aeXxfy + aeXx.
Particular solution: yo = — 1//.
29. vL = fv2 + aLv -
Solution with a > 0: у = у/а е9 Ьа,п(у/а / fe9dx + C
J
Solution with a < 0: у = J\a\ e9 tanh(- J\a\ / fe9 dx
30- VL = fV2 ~ a tanh2 (Лж) (о/ + A) + oA.
Particular solution: yo = atanh(Aa;).
31. y'x = fy2-acoth2(Xx)(af + X) + a\.
Particular solution: yo = acoth(Aa;).
32. y'x = fy2 - a2f + aAsinh^) - о2/51пЬ2(Аж).
Particular solution: yo = acosh(Xx).
33. xy'x = fy2 + a + a2f(\nxJ.
Particular solution: yo = alnx.
34. xy'x = f(y + a In жJ - а.
Solution:
у + a In x
f
J
35. y^, = fy2 — ax In xfy + о In ж + а.
Particular solution: yo = axlnx.
36- v'x = —о In ж у2 + о/(ж1пж — х)у — f.
Particular solution: yo = —;—: —.
а(жшж — х)
37. у'х = А 5т(Аж)у2 + / со3(Аж)у - /.
Particular solution: yo = ——-.
сов(Аж)
38. у'х = fy2 - a2f + aXsin(Xx) + a2f sin2(Аж).
Particular solution: yo = —acos(Xx).
39. y'x = fy2 — a2f + aXcos(Xx) + a2fcos2(Xx).
Particular solution: yo = asin(Xx).
40. y'x = fy2 - с^ап2(Аж)(а/ - A) + oA.
Particular solution: yo = atan(Arr).
41. y'x = /y2-ocot2(Aж)(o/- A) + oA.
Particular solution: yo = — acot(Arr).
© 1995 by CRC Press, Inc.

1.2.9. Some transformations
Notation: f, g, and h are arbitrary functions of a complex argument which is written
in parentheses following the function name (the argument is a function of x).
The transformation ? = ax + b, и = у/a leads to the equation u'^ = и +
The transformation ? = 1/x, w = —x2y — x leads to the equation wi = w2 +
, „ 1 / ax -\- b
3- y'x = v + f(
(ex + aL V ex + a
The transformation
I h 1
, w = — [(ex + dJy + c(cx + d)], where A = ad — be,
leads to a simpler equation: w'? = w2
4. x2y'x = x4f(x)y2 + 1.
The substitution и = = leads to the equation u' = u2 + fix).
x2y x x y '
5. x2y'x = x2y2 + -Ц^- + x2nf(axn + b).
X1~n 1 — П
The transformation ? = axn-\-b, w = у-\ x~n leads to a simpler equation:
an Ian
«;? = w2 + (an)-2f(i).
6- y'x = f(x)y2+g(x)y + h(x).
The substitution у = —1/w leads to an equation of the same form: w'x = h(x)w2 —
g(x)w + f(x).
The transformation ? = eXx, w = \c~Xxy — \e~Xx leads to a simpler equation:
, _ 2 A2 e2Xx / aeXx + b
8- Ух-У ~ — + (ceAa; + dy f
The transformation
аеЛа; + 6 (ceAa; + dJ <?e2Xx - d? . , ,
t = —: -, w = -—A , . у H —г , where A = ad — be,
ceXx+d AXeXx 2&eXx
leads to a simpler equation: w'e = w2
© 1995 by CRC Press, Inc.

9. y'x = у2 - A2 + 3тЬ-4(АЖ)/(соШ(Аж)).
The transformation
= соШ(Аж), ад = —- sinh2(Aa;)y — — sinhBAa;)
A 2
leads to a simpler equation: ад! = ад2 + A~2/(?).
10. y'x = y2 - A2 + cosh~4(Aa;)/(tanh(Aa;)).
The transformation
= tanh(Arr), w = — cosh2(Xx)y + — sinhBAa;)
A 2
+ A
leads to a simpler equation: w'e = w + A
11. xyx = xy + f(alnx + b) + .
x 1
The transformation ? = a\nx + Ъ, w = —у -\ leads to a simpler equation:
a 2a
2 2
12. y'x = y2 + A2 + sin-4(Aaj)/(cot(Aaj)).
The transformation
С = cot(Arr), w = - sin2(Arr) \^- + cot(Arr)
LA
leads to a simpler equation: w'c = w + A~ /(^).
13. y^ = у2 + А2 + со3-4(Аж)/^ап(Аж)).
The transformation
? = tan(Arr), ад = cos2(Aa?) —— tan(Arr)
leads to a simpler equation: ад! = ад + А
„ „ . / sin(Aa; + o)
= y2 + A2 + 3т-4(Аж + b)f(y '
„ „ . / si
14. y^ = y2 + A2 + 3т-4(Аж + b)f(-
sin(Aa; + b)
The transformation
sin(Aa; + a) sin2(Aa: + 6) \y
sin (Arr + b) г у
=—^1 г— —+cot(Aa
sinF - a) L А ч
sin(Aa: + b) ' sinF — a)
leads to a simpler equation: ад! = ад2 + [AsinF — a)]~2/(?).
© 1995 by CRC Press, Inc.

TABLE 1.1
Solvable Abel equations of the form yy'x — у = sx + Ax"
A is an arbitrary parameter
m
arbitrary
—7
-4
-5/2
-2
-2
-5/3
-5/3
-5/3
-7/5
s
2(m + l)
(m + 3J
15/4
6
12
0
2
-3/16
-9/100
63/4
-5/36
Equation
1.3.1.10
1.3.1.56
1.3.1.54
1.3.1.47
1.3.1.33
1.3.1.19
1.3.1.30
1.3.1.23
1.3.1.48
1.3.1.27
m
-1
-1/2
-1/2
-1/2
-1/2
0
0
1/2
2
2
s
0
-2/9
-4/25
0
20
arbitrary
0
-12/49
-6/25
6/25
Equation
1.3.1.16
1.3.1.26
1.3.1.22
1.3.1.32
1.3.1.55
1.3.1.2
1.3.1.1
1.3.1.53
1.3.1.45
1.3.1.46
1.3. Abel Equations of the Second Kind
1.3.1. Equations of the Form yy'x — у = f{x)
Preliminary comments. For the sake of convenience, in Tables 1.1—1.4 are listed
all the Abel equations discussed in Section 1.3. Tables 1.1—1.3 classify the Abel equations
wherein functions / are of the same form; Table 1.4 gives the other Abel equations. In
Table 1.1, equations are arranged in accordance with the growth of parameter m. In Table
1.2, equations are arranged in accordance with the growth of parameter s. In Table 1.3,
equations are arranged in accordance with the growth of parameter p. In the rightmost
columns of the tables are indicated equation numbers where the corresponding solutions
are written out.
Below in this section are given all the Abel equations united into groups wherein all
the solutions are expressed in terms of the same functions. A notation is given before each
group.
In most cases the solutions are presented in the parametric form
x = F1(t,C), y = F2(T,C),
where т is a parameter, С is an arbitrary constant.
1- УУХ -У = A.
Solution: x = у - A In \y + A\ + C.
2- УУ'Х — у = Ах + В, АфО.
Solution in the parametric form:
т dr
x = С exp I —
т2-т-А
Г> У = CVexpl —
т dr
г2-т-А
© 1995 by CRC Press, Inc.

TABLE 1.2
Solvable Abel equations of the form yy'x — у = sx-\-GA(ax1^2 -\- CA-\-fA2x~1^2),
A is an arbitrary parameter
s
arbitrary ф 0
2(m-l)
(m-3J
-1/4
-30/121
-12/49
-12/49
-12/49
-12/49
-12/49
-12/49
-12/49
-12/49
-12/49
-6/25
-6/25
-28/121
-2/9
-2/9
-2/9
-10/49
-4/25
-4/25
0
0
0
0
0
0
0
2
2
20
a
arbitrary
2
(m-3J
1/4
3/242
arbitrary
1/98
6/49
2/49
4/49
1/49
6/49
2/49
1
2/25
6/25
2/121
arbitrary
arbitrary
6
2/49
arbitrary
1/50
arbitrary
arbitrary
n + 2
n + 2
1
2
arbitrary
2
2
arbitrary
a
0
m(m + 3)
1
21
arbitrary
25
1
5
-10
5
-3
1
3/49 + 3B
2
2
5
0
0
0
4
0
7
0
1
1
1
-1
1
0
-10
10
0
p
arbitrary
4m2 + 3m + 9
5
35
0
41
8
34
27
262
23
166
12/49-15B/2
19
7
106
arbitrary
0
1
61
0
49
0
2
2(n + 2)
2(n + 2)
2
4
arbitrary
19
31
0
7
0
3m(m + 3)
3
6
0
10
5
15
10
65
12
55
15/196+ 75B/16
6
4
15
arbitrary
arbitrary
2
12
arbitrary
6
arbitrary
arbitrary
(n + l)(n + 3)
2n + 3
0
3
0
30
30
arbitrary
Equation
1.3.1.2
1.3.1.12
1.3.1.17
1.3.1.29
1.3.1.53
1.3.1.25
1.3.1.38
1.3.1.24
1.3.1.31
1.3.1.52
1.3.1.28
1.3.1.58
1.3.1.64
1.3.1.20
1.3.1.39
1.3.1.21
1.3.1.3
1.3.1.26
1.3.1.11
1.3.1.57
1.3.1.22
1.3.1.59
1.3.1.32
1.3.1.36
1.3.1.34
1.3.1.35
1.3.1.37
1.3.1.4
1.3.1.1
1.3.1.50
1.3.1.49
1.3.1.55
3. УУ'Х-У= —l
1°. Solution in the parametric form with A > 0:
Bk - 1)Стк -(к-2)т-к-1]2 (к- 1JСтк+1
х = а\± ; „ ,. v—-4 I , у=-Ъа^- '
Стк +т
Стк + т + 1
where А = -|а(/с2 - к + 1), В = J-a3/2B/c - 1)(/с - 2)(к + 1).
© 1995 by CRC Press, Inc.

TABLE 1.3
Solvable Abel equations of the form yy'x — у = sx + otAxp + CA2xq,
A is an arbitrary parameter
p
-1
1
L
-1
-3/5
-5/11
-1/3
-1/3
-1/3
-1/3
-1/5
0
2
q
-3
9
О
-3
-7/5
-13/11
-5/3
-5/3
-5/3
-5/3
-4/5
-1/2
3
s
arbitrary
2m + 1
4m2
0
-5/36
-33/196
-3/16
-3/16
-3/16
15/4
-10/49
-2/9
4/9
a
1
i
L
1
arbitrary
286,4/3
arbitrary
3
5
6
13,4/5
arbitrary
2
/?
-1
i
-1
arbitrary
-770 A/9
arbitrary
-12
-12
-3
-7,4/20
arbitrary
2
Equation
1.3.1.5
1 9 1 19
.L.G. L. LO
1.3.1.7
1.3.1.63
1.3.1.69
1.3.1.61
1.3.1.40
1.3.1.15
1.3.1.60
1.3.1.68
1.3.1.3
1.3.1.14
4.
2°. Solution in the parametric form with A < 0:
x = С[2Ае"Лт - (CiA - 3C2w) sinwr - CO + C2A) coswt]2,
у = 6?{(C2 + Cf)w2 - [Ci(A2 - w2) - 2C2wA]e-AT sinwr
A + C2(A2 - ш2)]е"Л
where A=±aCu>2-\2), B=^ra\(9u>2+5\2), ^ = a(e~XT+C1smu>T+C2cosu>T)~2.
3°. See 1.3.1.26 for the solution with A = 0.
titi* лi — О \ (т* -'¦/ ¦" I Л A I Q Л д rp
Solution in the parametric form:
y = ±aL±(R2±L±+T),
= arctan т — С, Д_|
T-l
where
dr
f dr _ 1
1-T2
5- yy^ "" У = ^ж "I" Вх~г —
Solution in the parametric form:
= — In
T + l
1+T
-a
l-T
-C, R- =
X =
(V_yW
W J
,-V2 _/ V \l/2
'f — )
© 1995 by CRC Press, Inc.

TABLE 1.4
Other solvable Abel equations of the form yy'x — у = /(ж)
Function f(x)
Ax"-1 - kBxk + kB2x2k~1
lr2 9
X 625 A
±x-^3 + \a2x^3-^-A^3
6 x 1 7 1ж1/3 1 31 12ж/3 10° А^х~5/3
25 ' 5 ' 3 3
-—x + ax1'3 + b + ex-1'3 + dx-2'3
(coefficients a, 6, c, and <i are related by an equality)
21 x + 7 A2(l23x-^7 + 280Ax-^7 400 A2x^'7)
к
VAx2 + Bx + C
A
Vx2 + 4A
3 9a2 - 6rr2
32 X 6AVx2 + a2
3 6rr2 + 5a2
8 16-\/ж2 + а2
3 6rr2 + 9A
—x -\ ,
8 16Va;2 + A
9 ЗОж2 + 33,4
—x -\
32 64 y/x2 + A
^+B4-f)
^Ht)-']
a2Xe2Xx - a(b\ + l)eXx + b
a2Xe2Xx + a\xeXx + beXx
2a AsinBAa:) + 2asin(Aa:)
Equation
1.3.1.6
(particular solution)
1.3.1.44
1.3.1.66
1.3.1.67
1.3.1.65
1.3.1.70
1.3.1.63
1.3.1.18
1.3.1.43
1.3.1.21
1.3.1.41
1.3.1.42
1.3.1.8
1.3.1.9
1.3.1.73
(particular solution)
1.3.1.74
(particular solution)
1.3.1.75
(particular solution)
© 1995 by CRC Press, Inc.

where
V =
(т2 + т — А) ехр
(т2+т-А)
/=A
¦ arctan ¦
2т+ 1
/=A
2т + 1 - л/Д 1
2т + 1 + v^ J
W =
C + 2B
/ exp -
•J L
2 . 2r+lu
, arctan —, ат
/-A
C + 2B exp -
2t
dT
2т + 1
if A<0,
if A = 0,
if A>0,
if A<0,
if A = 0,
if A>0,
where A = 4A + 1.
Particular solution: y0 = x — Bxk .
kB
yy'x - У = Ax-1 - A2x~3.
Solution in the parametric form:
x =
у =
- In
- In
- CI'2 - \
- In
where A = a2/2.
Solution in the parametric form:
x = In
AB
A\n т + Vt2 + AB\ + С
УУ'Х-У = A(e2xlA - 1).
Solution in the parametric form:
Л i i
A In t + Vt2+AB +C
У = т , . A.
Vt2 + AB
x = Aln
- (arctan т — C)
y= — [т + (т2 - 1)(arctanт - С)].
> In the solutions of equations 10-15, the following notation is used:
E
ml
= J
^ dr -C, Em = Em0 =
±
- C,
Rm = \/1±тт+1, Fm = RmEm - т.
© 1995 by CRC Press, Inc.

Solution in the parametric form:
ra + 3
m + 6 1 ~, / i \
x = т-аЕт т, у = aEm [ RmEm -\ -r I,
m — 1 V m — 1 /
, . m+l/m-
where A = ±
11- УУ'Х-У = -fx + 6A2(l + 2Ax-1/2), A>0.
Solution in the parametric form:
x = А2В.-^Е-2{Н2Е ± 6т1/2J, у = -12A2R-4E-2(R2E - 2т),
where E = ?-1/2,3/2, R = R-i/2-
2(ra - 1)
12- yy*-y=j^wx
1A
[m(m + 3)ж1/2 + Dm2 + 3m + 9) A + 3m(m + 3)A2x~1/2].
+
уТГЬ — О )
Solution in the parametric form:
( ТТЬ —
- 3)RmEm + Зт]2,
У = -^rT-2Em[±(m - 1)тт+1Еп - 2Em + 2rRm],
ТТЬ — о
where A = — -
m-3
, 2m +1 1 ¦> ч
13. yy'x-y= ———x + Ax - A2x~3
Solution in the parametric form:
X — —T It П, , у — — T Л ?/ |T — 1 ^F \^П + L)T JrLm-C/j-,
a 2ma
where a2 = —2mA, E = Ет<3р.
14. yy'x - у = ±x + 2Ax2 + 2A2x3.
Solution in the parametric form:
1 _,„ 1 _,,
15. уу'х -у= --^х + БАх-1/3 -12А2х~5/3.
Solution in the parametric form:
x = ar^E-^F3'2, у = ^T^E-^F-^iF2 - 2tF -
where A = a4/3/24, E = E_b/3, F = F_5/3.
© 1995 by CRC Press, Inc.

> In the solutions of equations 16-18, the following notation is used:
f=f ехр(тт2) dr-C, g = 2т \ f ехр(тт2) dr - c\ ± ехр(тт2).
УУХ — У = Ax-1.
Solution in the parametric form:
x = аГ1 ехр(тт2), у = af~x [ехр(тт2) ± 2т/],
where A = =F2a2.
17- УУх — У=—\х + \А(х1/2 + ЬА-
Solution in the parametric form:
x = -j-r [3 ± 8т/ехр(±т2)]2, у = af ехр(±т2) [Bт2 ±1)/ ехр(±т2) ± т]
where A = \\fa.
18- УУ'Х-У = ±"
2o2
л/ж2 ± 8o2
Solution in the parametric form:
x = ±a(fg)-1(g2 т 2/2), у = a(/ff)-1[exp(Tr2)<? - 2/2].
> In the solutions of equations 19-21, the following notation is used:
E= x/t{t + 1) - ln\C(Vr + V7TT
'r + 1 „ л /т + 1
R =
F = 1-
Solution in the parametric form:
X = —1
243
-^-т-RE), where A = —a3
6 1A
20. yy'x-y= x H Bж1/2 + 19А + 6A2x~1/2).
2o 2o
Solution in the parametric form:
ж = атEД?;-ЗтJ, у = 5ат~3Е[Bт + 3)E - 2t2R], where A =
a
21. yy' - у = — x-\ \Jx2 + O2
8 8 16л/ж2 + о2
Solution in the parametric form:
a E2 - 2t2F a
x = —— 1=—, y =
2 - E2
2л/2
4V2
© 1995 by CRC Press, Inc.

> In the solutions of equations 22-25, the following notation is used:
Р2 = ±(т2-1), P3=t3-3t + C, P4 = ±(t4-6t2 + 4CV-3).
22. yy'x-y= -^x
Solution in the parametric form:
з ^(Р2-тРз), where А = ±^Ъа.
23.
Solution in the parametric form:
= 10аР33/2р-9/8,
= 9аРз-1/2р-9/8(Рз2-Р2Р4), where A = ±9а2A0аJ/3.
24. yy'x-y= -—x + -—Ea;1/2 + 34A +
49 49
Solution in the parametric form:
x = аР2A4тРз - 9P2J, у = 28аР2-4Р3Dт2Р3 - ЗтР22
where A = — Ъ^/а.
12 А
25. уу'х-у= х Н B5ж1/2 + 41А + 10А2Ж-1/2).
4У У о
Solution in the parametric form:
x = аРзB1Р2Р4 - 16P|J, у = 21аРзР4(9Р22Р4
where A = —&\/а.
> In the solutions of equations 26-29, the following notation is used:
Si =exp(v/3r) +CsinT, S2 = 2exp(v/3r) - С sin т + v^ С cos r,
^з = 2ехр(у/3т) - Csinr - v^Ccost, S4 = 4Sr1Sr3 - 5|.
26. yy'x-y = -fx
Solution in the parametric form:
= 3aS-2S$, y = 2aS-2(S$-2S1S3), where A = 16CaK/2.
27. yy'x-y= -JL.X
Solution in the parametric form:
x = A8aS51/2S~5/'i, у = 5aS;1/2S~5/4(8Sl -
where A = D8аJ/5а2.
© 1995 by CRC Press, Inc.

28. уу' -у = ж H (-Зж1/2 + 23А + 12А2х~1/2).
49 49
Solution in the parametric form:
x = aS^GS1S3 - 2Slf, у = -7aS'iS'2DS'12S'2 - 4
where A = y/a/2.
29. yy'x-y= ~^-x + -|—-B1Ж1/2 + 35A + 6A2x
Solution in the parametric form:
x = a51-6A15254 - 64S?J, у = -Ila5f 654E| - 55|54 + 325?52),
where A = -32^.
> /n ifte solutions of equations 30-31, the following notation is used:
Xi = tanh(r + C) + tanr, T2 = tanh(r + C) — tanr,
6*i = coshr — sin(r + C), 02 = sinhr + cos(t + C), #3 = sinhr — cos(t + C).
30. yy'x-y = —j^x + Ax-5/3.
Solution in the parametric form with A < 0:
x = 8aT~3/2, y = 3aT1/2B-TiT2), where A = -12a8/3.
Solution in the parametric form with A > 0:
x = 4a<93/2<9~3/2, y = 3a<9~1/2<9~3/2(<92-<92<93), where A = 3a2DaJ/3.
31. yy' - у = ж Н (-10ж1/2 + 27А + 10А2х~1/2).
49 49
Solution in the parametric form with A < 0:
x = aA0 - 7TiT2J, y = 7aTi(T13 + 3TiT22-4T2), where A = -2-s/a.
Solution in the parametric form with A > 0:
ж = a<9-4G<92<93 - 5(92), у = -7а6»6»2(^ - 36»26»| + 2(92<93), where A = v^-
> /n the solutions of equations 32-43, the following notation is used:
C\Jv{t) + CiYv(j} for the upper sign (Bessel functions),
Z =
\ C\Iv{t) + C2Kv(t) for the lower sign (modified Bessel functions),
Z = Z1/3,
Ux = tZ't + \Z, U2 = U\ ± t2Z2, U3 = ±\t2Z3 - 2UXU2.
Remark. The solutions of equations 32-43 contain only the ratio Z'T/Z. There-
Therefore, function Z is defined in terms of two "arbitray" constants C\ and C2 (instead,
we may set, for instance, C\ = \, C2 = C).
© 1995 by CRC Press, Inc.

32. yy'x-y = Ax'1!2.
Solution in the parametric form:
x = aT-^3Z~2U2, y = aT-^3Z-2U2, where A = т\а3'2.
33. yy'x-y = Ax~2.
Solution in the parametric form:
x = 2ат4/3Z2UZX, y = ±3aT-2/3Z-1Ur1U3, where A = -36a3.
34. yy'x-y = A(n + 2)[хг/2 + 2(n + 2) A + (n + l)(n + 3)A2x~1/2}.
Solution in the parametric form:
x = aZ~2[fv -(u + 1)ZV]2, у = aZ-2(f2 - 2vZvfv ± t2Z2),
where A = ил/а, и = -^.
35. yy'x-y = A(n + 2)[x1/2 + 2(n + 2) A + Bn + З)^2»-1/2].
Solution in the parametric form:
x = af-2\r2Zv ± B - ^)/,]2, у = ±ат2/[/2 + 2A - u)Zvfv ± r
where A = T^Va, v = -^.
36. yy'x-y = Ax1/2 + 2A2 +
Solution in the parametric form:
x = A2Z-2{tZ'v - Zv)\ у = A2Z-2[t2{Z'vJ - (и2 т r2)Z2},
where В = A — u2)A3, prime denotes differentiation with respect to т.
37. yy'x-y = 2A2 - Ax1/2.
Solution in the parametric form:
x = a{Z'0)-2{TZ0 ± 2Z'0)\ у = ±aT(Z'o)-2[T(Z'oJ + 2ZOZ'O ± tZ2},
where A = л/а, prime denotes differentiation with respect to т.
38. yy'x-y= --^x + ^-(x1/2 + 8A + ЪА2х-1'2).
Solution in the parametric form:
x = ?,aU-A(Wi - 7t2Z2J, у = 28ат2Z2U~4(Зт2Z2 - ZUX - З^2),
where A = 2\/3a; Z and U\ are expressed in term of modified Bessel functions.
© 1995 by CRC Press, Inc.

fi fi \
39. yy'x - у = ж H Bж1/2 + 7А + 4A2x~1/2).
25 25
Solution in the parametric form:
x = aT-iZ-6(U1U2-2U3J, y = 5aT-iZ-6U2(Ul-U1U3), where A = -y/a/2.
40. yy^ -y= -^-ж + ЗАж-1/3 -12А2х~5/3.
Solution in the parametric form:
3ft —3/2/7—З/2 j —1/2/j2 dv ^ 2 v2 \
У = —T Z3/2 /3/2 (/з/2 " 2^3/2/3/2 " T Z3/2),
where ^3/2 and /3/2 are expressed interms of modified Bessel functions, and A
41. yy'x — у = —x + —\/x2 ± b2 ±
3b2
8 8 16л/ж2 ± b2
Solution in the parametric form:
where
vv'x-
b2
У
x =
У =
= fa2.
9
~~ ~32X
T"8~T
15
Ii2
^im2 - nu2u2 ±
3b2
42. ' " ¦"
64л/ж2 =F b2
Solution in the parametric form:
x = т Z ' U^ U^ Bt Z U3 i Зи2),
у = ±jT-1z-3/2u-3/2u-1/2(m3 т t2z3u3 - m2),
where b2 = 6a2.
15o2
43- уу'х -У = -—x- —\/x2~+~a
2
32 32 64л/ж2 + а2
Solution in the parametric form:
x = ^u-^u-^u2 - и3), у = ^u-3/2u3-\m2 - nu3 ±
© 1995 by CRC Press, Inc.

> In the solutions of equations 44~52, the following notation is used:
Function p = р(т) is given implicitly as follows:
dp
т =
\/±Dp3 -
-Co.
The upper sign in this formula corresponds to the classical elliptic Weierstrass func-
function p= р(т + С2, 0, 1).
44. yy'x-y = Ax2 - ——A \
625
Solution in the parametric form:
x = 5а(т рт у), у = ат
45. yy'x -y = —^-x + Ax2.
Solution in the parametric form:
x = bar p, у = ат
where A = ±—— a 1.
±Zd
where A = ±——— a 1.
±Zd
46- уу'х-у = ~kx
Solution in the parametric form:
x =
2
у = ат
^ —i
where A = ±——— a
rm:
47- УУ'Х — У = 12ж + Аж~5/2.
Solution in the parametric fo:
x = ap~&'7 E~i/7, y = ap~
48. yy'x - у = -f-ж + Аж-5/3.
Solution in the parametric form
where A = T147a7/2.
у =
49. уу^ - У = 2х + ZAilOx1/2 + 31A + 30А2х~1/2).
Solution in the parametric form:
ж = ар-2[тч/±Dр3-1)-Зр] , у = -2атр [р\/±Dр3 - 1) ± 2тр3 ± т],
where A = ^Ja.
© 1995 by CRC Press, Inc.

50. УУХ -y = 2x + 2A(-10x1/2 + 19A + 30A2x~1/2).
Solution in the parametric form:
x = aE~2(E1 -6E2J, y = -2aE~2(±6E3,-E2 + 7E1E2), where A=-y/a.
28 2A
51. yy'x-y= x H Eж1/2 + 106А + 15A2x~1/2).
1^1 1^1
Solution in the parametric form:
x = aB2p2E4 - 5J, у = ±Uap2E3GpE3 т 2т), where A = ±2y/a.
52. yy'-y= x H Eж1/2 + 262А + 65A2x~1/2).
49 49
Solution in the parametric form:
ж = а^зB8р?;| т 15-Б|J, у = 56aE^4ElFpE2 + EA), where A = тЗу^-
> In the solutions of equations 53-60, the following notation is used:
I = I — — incomplete elliptic integral of the second kind
J ^l^T ~ 1) in the form of Weierstrass
R = V±Dr3 - 1),
12
53. yy'x-y=-—x + Ax1/2.
Solution in the parametric form:
x = 7ат2/, у = -2aI-A(RI - 2т2), where A = ±^Vfa.
54. yy'x — у = 6ж + Ax~4.
Solution in the parametric form:
x = ат-3/51~2/5, y = aT-3/5I-2/5ERI1-2), where A = т150а5.
55. уу'х-у = 20ж -
Solution in the parametric form:
x = а/73/,2, у = -4а/-4/3(/22т9т/2), where A = ±108a3/2.
56. yy'x-y= Ig-x + Ax-7.
Solution in the parametric form:
ri/2 r-3/8 а r-3/2r-3/8/r т от2\ , ,, i За
x = аи J, , у =—I, J, и2-/з — 3ii , where Л = ±—--
© 1995 by CRC Press, Inc.

57. уу' -у = x H Dж1/2 + 61А + 12А2х~1/2).
49 49
Solution in the parametric form:
x = аGШг - ЗJ, у = Uah[±A0т3 - l)h - R], where A = y/a.
58. yy' - у = x H (x1/2 + 166A + 55A2x~1/2).
49 49
Solution in the parametric form:
x = а/2-4D2т/12 т 5/|J, у = т84а/2/2-4(Зт/| +I2T V2r2ll), where A = ±y/a.
4 A
59. yy'x-y= x H Gx1/2 + 49A + 6А2х~1/2).
2i%i 5U
Solution in the parametric form:
ж = а/-4E/2/з-16/2J, у=-5а/-4/з(±3/2-/2/з + 8/2/з), where А = 8уЪ.
60. yy'x-y= ^-x
Solution in the parametric form:
x = 2ат^2131/21-3/\ у = ат-1/2/Г1/2/2-3/4Bт/2 +12 - Зт2/2),
where A=-\а{2а)г'г.
61. УУ'Х-У = -^ж + Аж-^
The substitution x = т~3/2 leads to the equation
* = -f-"»» + ^--|^-»-|B
coincident with equation 1.3.3.13 when n = -1/2, с = 0, b = 3/4, d = 3,4/2, a2 = -3B.
f?o tin' ¦?# — 5 „ I 4™— 3/5 Rt»— ^/5 R >> П
The transformation
A 1 ,-\-5/4 5 /5 \i/2
У = -ж+ —В,
leads to an equation of the form 1.3.1.3:
2 2A / 5 \i/2 1
2 M / 5 у
9 3B V 27B J
63. уу4-у
The transformation
_ 4Q2w2 + hw + 6p) _ 4Q2w2 + hw + b0)
X~ Ы2 ' y-«+ Ы2
where parameters 62, 61, and 60 are found from the relations В = 4,Ab2 — &o and
С = b\ — 46062, leads to the Riccati equation:
'^ = (--^? + b2)w2
For С > 0, we may set b2 = 0, 61 = v7^, 60 = --B.
In books by Zaitsev & Polyanin A993, 1994) it is shown that the original equation
is reducible to the degenerate hypergeometric equation.
© 1995 by CRC Press, Inc.

64.
The substitution x = (?2 + \A) leads to an equation of the form 1.3.3.13 with n = 3,
a = 4/7, с = 0, b = A, d = \2A{\ - B):
yy't = D?2
+ -4гВ2\B - А)х^3 - \ВBА + 1) + В2A - ЗА)»-1/3 - АВ3х~2/3].
The transformation
-з L BC-2Bw) 1 / 1 \2 _2
x = w й, y= \i+——„ 1—-\[w + — )w
I 5{wz + -§-«;) J v ts J
leads to the Riccati equation:
УУа; У — 4 Ж 2
The transformation
. ,- / Я 1 1 ч _
where f =
leads to an equation of the form 1.3.1.46: wwi — w = -|g-? —
67. yy'x-y = --^x + l ^ f
Denote A = -j^-a and perform the transformation
,,/2 If 8^3 7a2\/- , , 3
X = ^ ' » = -2оГ + У^-Та-5оТ)^' where C = 2"To
As a result we obtain an equation of the form 1.3.4.30 with n = у, с = — -jg-
3\ 829 1 21 / 1 2
-—ajw+yz - yaz+ya Jw;0 = -—w +2zw.
68 wu' - v — —жх + ^-Ax
oo* У Ух У — 49 ^5 20
Denote ^4 = 8a~2. The transformation
V112 16У У' ад-7т + ж 42а
leads to an equation of the form 1.3.1.64 with В = —1/49:
12 39 2 15
^r + —а -ж
© 1995 by CRC Press, Inc.

fid nin/ ». — 33 I 286
196
Denote A = \a~2¦ The transformation
leads to an equation of the form 1.3.1.64 with В = —1/49:
—а -Ж
70. yy'x-y= —2&x + ^¦A
Denote A = I/a. The transformation
„7 3 a2 \, oM , „ 21
leads to an equation of the form 1.3.4.30 with n = 3, с = — -§g-a:
К 21 \ 2 ,i 3 ,
z——-a\w + 4:Z —7az + 3a \wz = — w +2zw.
71. yy'x — y = ax + bxm.
1°. For m^3, the transformation
2 2
, w = 2(m-3)B2(bxm-1-^T + — +a
leads to the equation
\TTl—о)
2°. Let тф\ and а > —1/4. Assume
(n + 2)(n + m + l) 1 / rn-1 \
a = —^—-—— r:——, whence щ о = — ±—, — m — 3 .
B 3J ' 1>2 2 V 'ii
Bn + m + 3J '
Then, the transformations
n+2_ m-l
x = ?m-lw y=
2n + m + 3 Vs m — 1 /
ation wL
/2n + m + 3\
where A = ) b, which is discussed below in Section 2.3.
V m — 1 /
reduce the original equation to the classical Emden—Fowler equation w'L = A^nwm,
7^. yy — у ^ Ж -
о
(ХТТЬ ТТЬ
Assume A = ; . .,. , В = ; . „,. . The transformation
2(т + 2уЬ2 2(т + 2NЬ2
г- (m + 2J m I m + 2, m 2(m +1) m m + 2
v/r = — —byx~m~1 H 6rr"m, u; = — -r + rrm H a
m m m + 2 m
leads to the equation
2m(m+l) _1/2
ww' -w=
(m + 2J
(see Table 1.2 with a = 0 in Subsection 1.3.1).
© 1995 by CRC Press, Inc.

73. yy'x-y = a2Xe2Xx - а(ЬХ + l)eXx + Ь.
Particular solution: yo = o,eXx — b.
74. yy'x-y = a2Xe2Xx + a\xeXx + beXx.
Particular solution: yo = cieXx + x -\ —.
aA
75. yy'x — у = 2o2AsinBAa;) + 2osin(Aa;).
Particular solution: yg = — 2asin(Aa?).
76. yy'x-y = a2fj'Jx - (/* ^ /;;, f = f(x).
\J x )
Particular solutions: y\ = afx -\ ——, %}2 = —a>fx H 7}
J x J x
1.3.2. Equations of the Form yyfx = f(x)y + 1
The substitution ? = у —\ax2 — bx leads to the Riccati equation with respect to
x = x(?): x'^ = \ax2 + bx + ?.
The substitution a!; = —(ax + 6) leads to an equation of the form 1.3.1.33: yyi =
3- УУ'х=(а )
V ax'
The substitution ^ = у — ax leads to the Bernoulli equation with respect to x = ж(?):
?a^ + а^ж + a2x2 = 0.
4- УУ'Х = (ax + Ь)-г/2у + 1.
2
The substitution z = —(ax + bI'2 leads to an equation of the form 1.3.1.2: yy'z =
Oj
y+\az.
5. yy'x = 3(ож3/2 + Sx)-1/^ + 1.
1 2
The substitution z = —(ax1'2 + 8I'2 leads to the equation of the form 1.3.1.10 with
a
r 2 а2 о
m = 3:yyz=y--z+—z.
6. yy'x = (ax'2/3 - J-a-1*-1/3^ + 1.
The transformation x = a3^2w3, у = ? — w2 leads to the Riccati equation: 3a3/2?w'e =
2
© 1995 by CRC Press, Inc.

7. yy'x = aeXxy + 1.
The substitution ?= -г-еЛа: leads to an equation of the form 1.3.1.16: yyi =
8. yy'x = (aeXx + be~Xx)y + 1.
The transformation
? = у+Ъ.е-**-±е**, w = ex*
leads to the Riccati equation: w'^ = aw2 + A?,w — b.
9- УУ'Х = °У cosh x + 1.
This is a special case of equation 1.3.3.20 with b = 0, с = 1.
Ю- УУ'х = oysinha; + 1.
This is a special case of equation 1.3.3.21 with b = 0, с = 1.
11. yy^, = acos(wx)j/ + 1.
The transformation
2 4u 8au
x = arctan , у = т-
a» ui
' & л г* 9 i 9
U> 16и2 + L0Z
leads to the Riccati equation: u'T = —2ти2 + аи — \ш2т.
12. yy'x = asin(u>x)y + 1.
The substitution x = ? + leads to an equation of the form 1.3.2.11: yyi =
acos(tt>?) у +1.
1.3.3. Equations of the Form yy'x = fi(x)y + /o(a?)
Preliminary comments. With the aid of the substitution С = / /i (ж) dx, these equations
are reducible to the form
t where №=Ш/Ш, A)
and by means of the substitution z = //о(ж) dx they can be reduced to the form
yy'z=g(z)y+l, where g(z) = Л(ж)//0(ж). B)
Concrete equations of the form A) and B) are outlined in Subsection 1.3.1 and 1.3.2,
respectively.
© 1995 by CRC Press, Inc.

1. yy'x = (ax + 3b)y + ex3 — abx2 — 2b2x.
The substitution у = x2t + bx leads to the linear equation with respect to x = x(t):
(-2t2 +at + c)x't = tx + b.
2. yy'x = Cax + b)y — а2ж3 — abx2 + ex.
The substitution у = xw+ax2 leads to the Bernoulli equation with respect to x = x(w):
(—w2 + bw + c)x'w = wx + ax2.
3. 2yy'x = Gax + 5b)y — За2ж3 - lex2 — 3b2x.
This is a special case of equation 1.3.3.11 with m = 3/2, к = 1/2.
4. yy'x = [C — m)x — l]y + (m — 1)(ж3 — ж2 — ax).
The transformation
x = w/z, y=-zm~x+x2-x-a
leads to the equation ш^ = w-\-az-\-zm whose solvable cases are outlined in Subsection
1.3.1 (see Table 1.1).
5- УУ'Х + x(ax2 + b)y + x = 0.
The substitution z = —\x2 leads to an equation of the form 1.3.2.1: yy'z =
(-2az + b)y + 1.
6. yy'x = 3(ож + Ъ)-г1ъх-ь1ъу + 3(ож + Ь)-2/3ж-7/3.
1 1 / ах + b \ !/з
The substitution w = 1 ( ) leads to an equation with separation of
xy 3 V x J
variables: w'x = x~1/3(ax + 6)/3(-i-a - 3w3).
7. 3yy'x = (-7Ax + 6s- 2Л)ж-1/3у
+ 2(Ax + 5X)(-Ax + 3s + 4Л)ж1/3 + 6(Xsx - 1)ж~2/3,
where A = As Cs + 4A).
The transformation
rr=(^ + As), у = (w + 4A + 3s - Ax)x2/3
leads to an equation of the form 1.3.4.12 with a = 1/3:
[(? + As)w + DA + 3s)?]«4 = \w2 + 2CA + s)w + 2?.
8. yy'x = ж" [A + 2п)ж + ап]у - nx2n(x + a).
The transformation
w 1 ,¦¦
x=—, y= -+xn+1+axn
leads to an equation with separation of variables: w'z = w~n — a.
9. yy'x = a(x - nb)xn~1y + c[x2 - Bn + l)bx + n(n + 1)Ь2]ж2"-1.
/IT \
The substitution ? = axn ( b) leads to an equation of the form 1.3.1.2: yyi
V n +1 / 4
у + (n + l)ca?.
© 1995 by CRC Press, Inc.

10. yy'x = [aBn + k)xk + b\xn~xy + (-a2nx2k - abxk + c)x2n~x.
The substitution у = xn(w + axk) leads to the Bernoulli equation with respect to
x = x(w): (—nw2 + bw + c)x'w = wx + axk+1.
11. yy'x = [oBm + k)x2k + b{2m - k)]xm-k-1y
- (a2mx4k + cx2k + b2m)x2m-2k-1.
The transformation z = xk, у = xm(t + axk + bx~k) leads to the Riccati equation
with respect to z = z(t):
(-mt2 + 2abm - c)z't = akz2 + ktz + bk. A)
The substitution
mt2 +c0 w't
z = ; -, where cq = с — 2abm,
ak w
reduces equation A) to a second order linear equation:
(mt2 + coJw't't + Bm + k)t(mt2 + co)w't + abk2w = 0. B)
The substitution
P= l , u=(l-er/2w, where ц=-И1±*
, u=(lerw, where ц=
t2 + (co/m) V ^ ' ^ 2m
brings equation B) to the Legendre equation:
A - ?>« " 2^4 + [1/A/ + 1) - M2(l " C2)"> = 0,
.2 m2 — k2 abk2
where v is a root of the quadratic equation v -\-v-\ -z = 0.
vmz mc
12. yy'x = [(га + 21 — 3)x + П — 21 + 3]x~ly
+ [(га + I - l)x2 + (n - ra - 21 + 3)x - n + I - 2]x1~21.
The transformation
w ^'
leads to the generalized Emden—Fowler equation: wi = A?nwm(w'cI, which is dis-
discussed in Section 2.5.
13. yy'x = [aBn + l)x2 +cx + bBn - l)]xn~2y
- (na2x4 + acx3 + dx2 + bcx + nb2)x2n~3,
where a, b, c, d, and n are arbitrary numbers.
The substitution у = xnt + axn+1 + bxn~1 leads to the Riccati equation with respect
to x = x(t):
(—nt2 + ct — d+ 2nab)x't = ax2 + tx + b.
© 1995 by CRC Press, Inc.

. yy'x = [a(n - l)x + bBX + n^x^iax + b)-x~2y
- [anx + b(X +
The substitution
= \— + —A—rrl xx+n(ax + b)~x
+ A
w xn(ax
leads to an equation of the form 1.3.4.5:
О + axn+1 + bxn)w'x = [anxn + b(X + п)ж"->.
15. yy'x = (aex + b)y + ce2x - abex - b2.
The transformation x = In w, у = tw + b leads to a linear equation: (—t2 + at + c)w't =
tw + b.
16. yy'x = [oB/it + X)eXx + Ъ\е»ху + (-а2це2Хх - abeXx + c)e2llx.
The substitution ? = ex leads to an equation of the form 1.3.3.10:
17. yy'x = (aeXx + b)y + c[a2e2Xx + ab(Xx + l)eXx + Ь2Хх].
The substitution ? = — eXx + bx leads to an equation of the form 1.3.1.2: yyi =
18. yy'x = eXx{2aXx + a + b)y - e2Xx(a2Xx2 + abx + c).
The substitution у = eXx(? + ax) leads to a linear equation with respect to x = ж(?
(-AC2 + &? - c)x'^ = ax + ?.
19. yy'x = eaxBax2 + 2x + b)y + e2ax(-ax4 - bx2 + c).
The substitution у = eax (?+x2) leads to the Riccati equation with respect to x = x (?
(-<2 + &? + c)x'^ = x2 + ?.
20. yy'x = (a cosh ж + b)y — obsinha; + с
The transformation t = y — a sinh x, ? = ex leads to the Riccati equation: 2(bt + c)?'t
a?2 + 2t? - a.
21. yy'x = (o sinh x + b)y — ab cosh ж + с.
The transformation t = y — acoshx, ? = ex leads to the Riccati equation: 2(bt + c)?'t
a?2 + 2t? + a.
22. yy'x = B In ж + a + l)y + x(— In2 ж — о In ж + b).
The transformation x = ew, y= (?,+w)ew leads to a linear equation: (—?2+a?+&)«;l
23. yy'x = B1п2ж + 21пж + о)у + ж(-1п4ж - о1п2ж + Ь).
Performing the transformation x = ew, у = (^+w2)ew, we obtain the Riccati equation:
© 1995 by CRC Press, Inc.

24. yy'x = ax соэ(и>ж2) у + x.
The substitution ? = -^-ж2 leads to the Abel equation of the form 1.3.2.11: yyl =
acosBtt>?) у + 1.
25. yy^, = ож эт(и>ж2) у + ж.
The substitution ? = -i-ж2 leads to the Abel equation of the form 1.3.2.12: yy'? =
asinBtt>?) у + 1.
1.3.4. Equations of the Form
Preliminary comments. With the aid of the substitution
= (у+—)е, where E = exp(- [ — dx), A)
^ Pi ^ ^ J Pi ^
w ¦
these equations are reducible to a simpler form:
ww'x=F1(x)w + F0(x), B)
where
A ?
Concrete Abel equations of the form B) are outlined in 1.3.1-1.3.3. In the degenerate cases
with Fq = 0 or Fi = 0, the variables in equation B) are separable.
1. (Ay + Bx + a)y'x + By + kx + b=0.
Solution: Ay2 + kx2 + 2(Bxy + ay + bx) = C.
2. (y + ax + b)y'x = ay +Cx + f.
The substitution у = и — ax — b leads to the equation
uu'x = (a + a)u + (j3 — aa)x + 7 — ba
which is separable with a = —a.
For а ф —a, the substitution и = (a + a)w yields an equation of the form 1.3.1.1
or 1.3.1.2:
ww'x = w + A~2(/3 — aa)x + A~2G — ba), where A = a + a.
3. (y + akx2 + bx + c)y'x = —ay2 + lakxy + my + k(k + b — m)x + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
\—az2 + (m — k)z + s — ck]x'z = akx2 + (b + k)x + z + c.
© 1995 by CRC Press, Inc.

4. (у + Ахп + а)у'х + пАхп~1у + кхт + Ь = 0.
2/с
Solution: у2 Н хт+1 + 2(Ахпу + ау + Ьх) = С.
т+ 1
5. (у + ахп+1 + Ъхп)у'х = (апхп + сх^^у.
The substitution y = xn(w—b) leads to the Bernoulli equation with respect to x = x(w):
[—nw2 + (bn + c)w — bc]x'w = wx + ax2.
6. жуу^, = ay2 +by + cxn + s.
The transformation ? = x~a, w = ——x~ay leads to the equation ww'? =
where A = —ab~2s, В =—ab~2c, m=(a — n)/a (see Subsection 1.3.1).
7. xyy'x = —ny2 + aBn + l)xy + by — a2nx2 — abx + с
The substitution у = w + ax leads to the Bernoulli equation with respect to x = x(w):
(—nw2 + bw + c)x'w = wx + ax2.
8. 2жуу^, = A — n)y2 + [aBn + l)x + In — l]y — a2nx2 — bx — n.
The transformation x = ?2, у = ?t + a?2 + 1 leads to the Riccati equation:
(-nt2 + 2an - b)?'t = a?2 + t? + 1.
9. (Axy — Aky + Bx — Bk)y'x = Cy2 + Dxy + (B — Dk)y.
The transformation x = w + к, у = ^w leads to a linear equation with respect to
w = w(x):
\{C - A)!? + D?]w'z = A?w + B.
10. [(Зож + Xs)y + DЛ + 3s)x]y'x = lay2 + 2(ЗЛ + s)y + 2x.
The substitution w = ay2 + (ЗА + s)y + x leads to an equation of the form 1.3.3.3:
2ww'y = {lay + 5b)w - 3a2y3 - 2cy2 - 3b2y, where b = s + 2X, с = ^-aA3A + 6s).
11. [Dax + Xs)y + DЛ + 3s)x]y'x = \ay2 + 2(ЗЛ + s)y + 2x.
The substitution w = \ay2 + (ЗА + s)y + x leads to an equation of the form 1.3.3.3:
2ww'y = {lay + 5b)w - 3aV - 2cy2 - 3b2y, where b = s + 2X, c= -|-aF0A + 25s).
12. BAxy + ay + bx + c)y'x = Ay2 + Ak2x2 + my + k(ak + b — m)x + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[Az2 + (m - ak)z + s - ck)x'z = 2Akx2 + BAz + ak + b)x + az + с
© 1995 by CRC Press, Inc.

Г 2(га + 1I, 1 - га , га-1
13. \2ху + A-т)Ау -—Lx\y'x = у2 + -—у + ж.
I га + 3 J 2 га + 3
The substitution w = у -\ у -\-х leads to an equation of the form 1.3.3.4:
ww'y = [C-m)y-l}w+(m-l)(y3-y2-ay), where a = A--^——^-
14. xBay + bx)y'x = oB — ra)y2 + b(l — m)xy + ex2 + ia;m+2.
The transformation z = y/x, w = —Axm + amz2 + bmz — с leads to the equation with
separated variables: w!, = mBaz + b)(amz2 + bmz — c).
15. (xy + x2 + a)y'x = y2 + xy + b.
Solution: (x+ yJ + a + b = C(bx — ayJ.
16. BAxy + Bx2 + b)y^ = Ay2 + k(Ak + В)ж2 + с.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
(Az2 + с - bk)x'z = BAk + B)x2 + 2Azx + b.
17. (Axy + Bx2 + kx)y'x = Dy2 + Exy + Fx2 + fey.
The substitution у = xz leads to a linear equation with respect to x = x(z):
[(D - A)z2 + (E- B)z + F]x'z = (Az + B)x + к
18. (Axy + Bx2 + kx)y'x = Ay2 + Bxy + (Ab + fe)y + Bbx + bk.
This is a special case of equation 1.3.4.22.
Solutions: у = Cx — b and Ay + Bx + к = 0.
19. BAxy + Bx2 + fca;)y^ = Ay2 + Сжу + Г>ж2 + fey - С/Зж - A/32 - k/3.
The substitution у = ?x + /3 leads to a linear equation with respect to x = x(?):
[-A(? + (C - B)S + D]x'^ = BA? + B)x + 2AC + k.
20. (Axy + Bx2 + kx)y'x = Ay2 + Сжу + Г>ж2 + (fe - AC)y - CCx - kC.
The substitution у = ?ж + C leads to a linear equation with respect to x = x(?):
[(C - B)S + D]x'^ = (AS + B)x + Af3 + k.
21. (Axy + Aky + Bx2 + Bkx)y'x = Су2 + Г>жу + k(D - B)y.
The transformation x = w — к, у = ?w leads to a linear equation:
[(C - A)S2 + (D- B)S}w'^ = (AS + B)w - kB.
© 1995 by CRC Press, Inc.

22. (Axy + Bx2 + сцх + Ъгу + ci)y'x = Ay2 + Вху + а2х + Ь2у + с2.
Jacobi equation.
1°. With the help of the transformation x = x + а, у = y + C, where a and C are the
parameters which are determined by solving the algebraic system
AaC + Ba2 + сца + hC + cx = О, A02 + Bap + a2a + b2p + c2 = 0,
we obtain the equation
(Axy + Bx2 + aix + hy)y's = Ay2 + Bxy + a2x + b2y,
where
A/3 + a1, b1=Aa + b1,
The transformation z = y/x, С = 1/ж leads to a linear equation:
[biz2 + Ei - b2)z - a2]Cz = (hz + 5i)C + Az + B.
2°. The original equation can be also rewritten in the form
{xy'x - y)(n3x + m3y + k3) - y'x(nix + mxy + h) + n2x + m2y + k2 = 0.
The solution of this equation in the parametric form can be obtained from the solution
of the following system of the constant-coefficient linear differential equations:
(x2)'t = n2x\ + m2x2 + k2xs,
{Xz)'t = П3Ж1 + ГП3Ж2 + /С3Ж3,
using the formulae x(t) = Xx/x3, y(t) = x2/x3.
23. (Axy + Bx2 + ay + bx + c)y'x = kAxy + kBx2 + my + k(ak + b — m)x + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(m - ak)z + s - ck]x'z = (Ak + B)x2 + (Az + ak + b)x + az + с
24. BAxy + Bx2 + ay + bx + c)y'x = Ay2 + k(Ak + B)x2 + aky + bkx + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
(Az2 + s- ck)x'z = BAk + B)x2 + BAz + ak + b)x + az + с
25. BAxy — Akx2 + ay + bx + c)y'x = Ay2 + my + k(ak + b — m)x + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[Az2 + (m - ak)z + s - ck]x'z = Akx2 + BAz + ak + b)x + az + с
26. BAxy + Bx2 + ay — akx + b)y'x = Ay2 + k(Ak + B)x2 + my — mkx + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[Az2 + (m - ak)z + s- bk]x'z = BAk + B)x2 + 2Azx + az + c.
© 1995 by CRC Press, Inc.

27. BAxy + Bx2 + ay + bx + c)y'x = Ay2 + k(Ak + B)x2 + by + ak2x + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[Az2 + (b- ak)z + s- ck]x'z = {2Ak + B)x2 + BAz + ak + b)x + az + c.
28. [Axy + Bx2 + (m — l)Aay — (Abm + Ba)x]y'x
= Ay2 + Bxy - (Ab + Bam)y + (m - l)Bbx.
This is a special case of equation 1.3.4.22.
Solution in the parametric form:
at + ACtm bt - BCtm
x =
t + C ' * t + C
The solution can be presented in an implicit form as well:
Cm(Ay + Bx)m + [A(b -y)+ B(a - rr)]m(ay - bx) = 0.
29. [(ож + c)y + A - ri)x2 + Bn - l)x — n]y'x = lay2 + 2xy.
The substitution w = ay + x leads to an equation of the form 1.3.4.8:
2yww'y = A — n)w2 + [aBn + l)y + 2n — l]w — a2ny2 — by — n,
where b = Bn — l)a — с
In
30. [(ж + c)y + (n + l)x2 - aBn + l)x + a2n]y'x = -y2 + 2xy.
The transformation
3n — 11 3n — 1 x
—, w =
n — 1 у n — 1 у n — 1
leads to an equation of the form 1.3.4.8:
2zww'z = A — n)w2 + [aBn + l)z + 2n — l]w — a2nz2 — bz — n,
Cn-l)c + anBn + l)
where b =
n-l
31. xBaxy + b)y'x = —a(m + 3)xy2 — b(m + 2)y + cxm.
The transformation
z = xy, w = —cxm+1 + a(m + l)x2y2 + b(m + l)xy
leads to the equation with separated variables: ш!, = (то + 1) Baz + b)(az + bz).
32. [@2Ж2 + 01Ж + Oo)y + b2x2 + bix + bo]y'x = c2y2 + C\y + cq.
This is the Riccati equation with respect to x = x(y).
The substitution x = ; — yields a second order linear equation:
О-2У + 02 W
hw'yy - \(h)'y + /1/2K + /o/22w = o,
where /j = ^ ; i = 1, 2, 3.
С2У + ciy + c0
© 1995 by CRC Press, Inc.

33. [A2а2ж2 - 7ож + 1)у + 4сж2 - 5Ьх]у'х = -2хCа2у2 + Icy + ЗЬ2).
The substitution w = хCа2у2 + 2cy + 3b2) leads to an equation of the form 1.3.3.3:
2ww'y = Gay + 5b)w - 3a2y3 - Icy2 - ЪЬ2у.
34. ж[(га - l)(Ax + B)y + m(Dx2 + Ex + F)]y'x
= [A(l - n)x - Bn]y2 + [DB - n)x2 + E(l - n)x - Fn]y.
Solution: Axy + Dx2 + Ex + By + F = Cxnym.
35. x2Baxy + b)y'x = —4ax2y2 — 3bxy + ex2 + k.
The transformation z = xy, w = 2ax2y2 + 2bxy — ex2 — к leads to an equation with
separated variables: w!, = 2{2az + b)Baz2 + 2bz — k).
36. (xy + axn + Ъх2)у'х =y2 + cxn + bxy.
The transformation t = y/x, z = xn~2 leads to a linear equation: (c — at)z't =
(n-2)(az + t + b).
37. жBож"у + Ъ)у'х = -aCn + m)xny2 - bBn + m)y + Axm + cx~n.
The transformation
z = xny, w = -Axn+m + (n + m) (az2 + bz)-c
leads to an equation with separated variables:
ww' =(n + mJBaz + b) (az2 + bz —V
V n + m)
38. yy'x = -ny2 + aBn + l)exy + by - a2ne2x - abex + с
Performing the transformation x = In ?, у = w + a?, we obtain the Bernoulli equation:
(-nw2 + bw + c)?'w = w? + a?2.
1.3.5. Some Types of First and Second Order Equations Reducible to
Abel Equations of the Second Kind
> Notation: f, g, h, p, ip, ip, Ф, F, and G are arbitrary functions of their arguments.
1. Quasi-homogeneous equation
f(xvy)xv+1y'x + g(xvy) + Axx = 0.
In the particular case when A = 0 this equation is homogeneous. The transformation
z = xvy, v = Axx + g(z) - vzf{z)
leads to the Abel equation:
vv'z = [-(A + v)f + g'z- vzfz\v + Xf(g - vzf).
© 1995 by CRC Press, Inc.

2. Quasi-homogeneous equation
f(xvy)xv+1yx + g(xvy) + xx[h(xvy)xv+1yx +р(х"у)} = О.
The transformation z = xvy, Q = x~x leads to the Abel equation:
{[g{z) - vzf{z)](,+p{z) - vzh(z)}? = Xf(zX2 + Xh(z)C
3. Equations of the theory of chemical reactors and the combustion theory
y'L - ay'x = /(»).
The substitution w(y) = y'x/a leads to the Abel equation:
ww'y -w = a~2f(y),
whose solvable cases are given in Subsection 1.3.1.
4. Equations of the theory of nonlinear oscillations
y'L + (р(у)у'х + y = o.
The substitution z(y) = y'x leads to the Abel equation:
zz'y + ip{y)z + у = 0, A)
which is reduced, with the aid of the substitution т = -j(a — у2), to the following form:
wfzb-v/tt — 2т)
zz'T= g(r)z + l, where gM = + v '- B)
Va — 2t
Concrete cases of equation B) are outlined in Subsection 1.3.2.
5. Equations of the theory nonlinear oscillations
I/»»+ *(»»)+1/= 0.
The transformation z = y'x, w = —y — Ф(у'х) leads to the Abel equation of the form A):
»!, + <&'z(z)w + 2 = 0.
6. Homogeneous equation with respect to the independent variable
x*y'L = x9(y)y'x + f(y)-
The substitution w(y) = xy'x leads to the Abel equation:
ww'y = \g{y) + l\w + f(y)-
7. Homogeneous equation in the extended sense
© 1995 by CRC Press, Inc.

The transformation t = yxk, и = xk(xy'x + ky) leads to the Abel equation:
uu't = [g(t) + 2/c + l]u + f(t) - ktg(t) - k(k + l)t.
To the Emden—Fowler equation, discussed in Section 2.3, correspond g(t) = —a, f(t) =Atm,
K — m-1 '
8. Homogeneous equation in the extended sense
The transformation
X i a+2 /3—1
V-jVx, v-x у
leads to the Abel equation:
[F(ri)v + G(ri) +T}- V2WV = [(/? - l)r? + a + 2}v.
To the generalized Emden—Fowler equation, discussed in Section 2.5, correspond ot = n — /,
/3 = m + /, ^G7) = ^V, G(t7) = 0.
9. Exponential-homogeneous equation
// ol Qii л/ /\i —2/ /\
Ухх — » \ Ух/ *^ cf\ Ух/*
The substitution С = жу^,, и = жа+2е/3?/ leads to the Abel equation:
10. Exponential-homogeneous equation
У
The substitution ? = y'x/y, w = eaxy^~1 leads to the Abel equation:
+ 9@ ~ C2]4 = К/? " 1)C + a]w.
1.4. Equations Containing Polinomial Functions of у
1.4.1. Abel Equations of the First Kind
y'x = h{x)y* + f2(x)y2 + fi(x)y + fo(x)
Preliminary comments.
1. If yo = yo{x) is a particular solution of the equation in question, the substitution
у — yo = l/w reduces it to the Abel equation of the second kind:
ww'x = -C/з2/о + 2/2Уо + h)w2 - C/зуо + /2)^ - /з,
which is discussed in Section 1.3. For fo{x) = 0, we may choose yo = 0 as a particular
solution.
2. The transformation
ti=fhE2dx, u=(y+-?-)E-1, where E = exp\ f (fr--gA dx],
J v ^/з' U \ 6J3) J
brings the original equation to the normal form:
where
1/2 . z/i
3 dx \ h ' 3/
© 1995 by CRC Press, Inc.

1. y'x = ay3 + bx~3!2.
This is a special case of equation 1.4.1.9 with n = —1/2.
2. y'x = —y3 + 3a2x2y — 2a3x3 + a.
The substitution у = 1/w + ax leads to the Abel equation of the form 1.3.2.1: ww'x =
iaxw + 1.
3- V'x = -V3 + («ж + Ь)у2.
The substitution у = —1/w leads to the Abel equation of the form 1.3.2.1: ww'x =
(ax + b)w + 1.
4- y'x = -y3 + (ax + b)-2y2.
The substitution у = —1/w leads to the Abel equation of the form 1.3.2.2: ш^, =
(ax + b)~2w + l.
5. у4 = -уЗ + (ож + Ь)-1/2у2
The substitution у = —1/w leads to the Abel equation of the form 1.3.2.4:
6. y'x = ay3 + ЗаЪху2 —Ъ — 2аЪ3х3.
This is a special case of equation 1.4.1.10 with n = 0, m = 1.
7. y'x = axy3 + by2.
The substitution и = xy leads to an equation with separation of variables: xux =
аи3 + bu2 + u.
8. y'x = axy3 + 3abx2y2 — b — 2ab3x4.
This is a special case of equation 1.4.1.10 with n = m = 1.
9. y'x = ax2n+1y3 + bx~n-2.
The substitution w = yxn+1 leads to an equation with separation of variables: xw'x =
aw3 + (n + l)w + b.
n+1
tor a = —
as
77 I 1
For a= 2 and b= -|-^4(n + l), the solution in the parametric form is written
where F = т - \\п\т + \\ +С.
10. y'x = axny3 + 3abxn+my2 - bmxm~1 - 2ab3xn+3m.
The substitution w = у + bxm leads to the Bernoulli equation: w'x = axnw3 —
3ab2xn+2mw.
11. y'x = axny3 + 3abxn+my2 + cxky - 2ab3xn+3m + bcxm+k - bmx™-1.
The substitution w = у + bxm leads to the Bernoulli equation: w'x = axnw3 +
(cxk - 3ab2xn+2m)w.
© 1995 by CRC Press, Inc.

12. 9у'х = -хт(ах1-т + ЬJХ+1у3 -х-2т(9а + 2 + 9Ьтхт-1)(ах1-т + Ь)
With A = — -, the substitution
3a(l — m)
\—A—2
У = (— + ^t ) (ax1- + b)~X
\w ax + bxm )v '
leads to the equation ww'x = w + ax + bxm outlined in Subsection 1.3.1.
13. xy'x = ax4y3 + (bx2 — l)y + ex.
The substitution w = xy leads to an equation with separation of variables: w'x =
x(aw3 + bw + c).
14. xy'x = ay3 + 3abxny2 - bnxn - 2ab3x3n.
The substitution w = у + bxn leads to the Beroulli equation: w'x = ax~1w2' —
Заб2^2™-1^.
15. xy'x = ax2n+1y3 + (bx - n)y + ex1-™.
The substitution w = yxn leads to an equation with separated variables: w'x = aw3 +
bw + с
16. xy'x = axn+2y3 + (bxn - l)y + ex"-1.
The substitution w = xy leads to an equation with separation of variables: w'x =
xn~1(aw3 + bw + c).
17. x2y'x = y3 - 3a2x4y + 2a3xe + 2ax3.
The transformation x = —, у = \- ax leads to an equation of the form 1.3.2.2:
? w
ww'e = 3a!;~2w + 1.
18. y'x = -(ax + bxm)y3 + y2.
The substitution у = —1/w leads to the equation ww'x = w + ax + bxm outlined in
Subsection 1.3.1.
19. у'х = (Ах2 + + )у + у
The substitution у = —1/w leads to the Abel equation of the form 1.3.1.63: ш^. =
2 l2
' = -y3 + aeXxy2
20. y'x = -y3 + aeXxy
The substitution у = —1/w leads to the Abel equation of the form 1.3.2.7: ww'x =
aeXxw
21. y'x = -y3 + 3a2e2Xxy - 2a3e3Xx + a\eXx.
The substitution у = \- aeXx leads to the Abel equation of the form 1.3.2.7:
w
ww'x = iaeXxw + 1.
© 1995 by CRC Press, Inc.

22. y'x = —|.A-1e2AV + f\2e'Xx.
The solution in the parametric form is written as
x(l ^y-F where F = т - \\п\т + \\+C.
23. y'x = ae2Xxy3 + beXxy2 + cy + de~Xx.
The substitution у = we~Xx leads to an equation with separated variables: w'x =
aw3 + bw2 + (c + X)w + d.
24. y'x = aeXxy3 + 3abeXxy2 + cy - 2ab3eXx + be.
The substitution w = y+b yields the Bernoulli equation: w'x = aeXxw3 + (c—3ab2eXx)w.
25. y'x = aeXxy3
The substitution w = у + be]xx leads to the Bernoulli equation: w'x = aeXxw3 —
26. y'x = aeXxy3
The substitution w = у + be^x leads to the Bernoulli equation: w'x = aeXxw3
()
27. y'x = aeXxy3
The substitution w = у + be]xx leads to the Bernoulli equation: w'x = aeXxw3
28. y'x = aeXxy3 + 3abe^x+^xy2 + [Cab2 + c)e^x+2^x + s]y
+ b(ab2 + c)e<-x+3^x + b(s -
The substitution w = у + be^x leads to the Bernoulli equation: w'x = aeXxw3 +
[ce(A+2M)a: + s}w^
29. y'x = [a + Ь ехрBж/о)]у3 + у2.
The substitution у = —1/w leads to an equation of the form 1.3.1.9: ww'x = w — a —
Ьещ>Bх/а).
30. y'x = -J-аж ехрBож2)у3 + A - -§-аж2) ехр(-ож2).
The substitution у = ( \-x ) exp(—ax2) leads to an equation of the form 1.3.1.16:
V 2aw J
ww'x = w + Fax)~1.
31. y'x = —оехрBож3)у3 + A — 2ож3) ехр(—ож3).
The transformation ? = x2, у = [
V 3aw
form 1.3.1.32: юи)[ = w± 2(9a)-1C/2.
The transformation ? = x2, у = [ \- x) exp(—ax3) leads to an equation of the
V 3aw I
/
© 1995 by CRC Press, Inc.

32. y'x = -ож ехрBож3)у3 + 2жA — ож3) ехр(—ож3).
The substitution у = ( \-х2 ) ехр(—ах3) leads to an equation of the form 1.3.1.33:
V 3aw J
ww'x = w + (9a)~1 x~2.
33. y'x = ay3 + b cosh(\x)y2.
The transformation t = 1 sinh(Aa;), ? = eXx leads to the Riccati equation:
У А
2a?'t = b?2 - 2Щ - b.
34. y'x = ay3 + bsinh(Xx)y2.
The transformation t = 1—— cosh(Arr), ? = eXx leads to the Riccati equation:
2a?'t = Ъ?2 - 2Щ + b.
35. y'x = —y3 + 3o2 cosh2 ж у — 2o3 cosh3 ж + о sinh ж.
The substitution у = \- a cosh ж leads to the Abel equation of the form 1.3.2.9:
w
ww'x = 3a cosh x w + 1.
36. y'x = —y3 + 3o2 sinh2 ж у — 2o3 sinh3 ж + о cosh ж.
The substitution у = \- a sinh ж leads to the Abel equation of the form 1.3.2.10:
w
ww'x = 3a sinh ж w + 1.
37. y'x = —у3 + осоэ(а;ж)у2.
The substitution у = —1/w leads to the Abel equation of the form 1.3.2.11: ш^. =
acos(uix)w + 1.
38. y'x = —у3 + оэт(а;ж)у2.
The substitution у = —1/w leads to the Abel equation of the form 1.3.2.12: ww'x =
asin(uix)w + 1.
39. y'x = —y3 + Зо2 соэ2(Лж)у + оЛэт(Лж) + 2o3 соэ3(Лж).
The substitution у = асов(Аж) leads to the Abel equation of the form 1.3.2.11:
w
ww'x = —3acos(Xx)w + 1.
40. y'x = —y3 + За2 эт2(Лж)у + оЛсоэ(Лж) - 2o3 эт3(Лж).
The substitution у = \-a вш(Аж) leads to an equation of the form the Abel equation
w
of the form 1.3.2.12: ww'x = 3asin(Xx)w + 1.
bfq2 -\ — )y + cfq3, f = fix), a = д(ж).
The substitution у = gw leads to an equation with separation of variables: w'x =
fg2(aw3 + bw + c).
© 1995 by CRC Press, Inc.

42. y'x = fy3 + 3fhy2 + (g + 3fh2)y + fh3 + gh - h'x,
where / = f(x), g = g(x), h = h(x).
The substitution w = у + h(x) leads to the Bernoulli equation: w'x = g(x)w + f(x)w3.
43-
2(од + ЬK /
—5 — + С =—In lap+ 61, where и = —-,
ий — аи +1 a j (ад -\
лл i ( t\i \( af + b9\u , У-9 f, , У-f ,
44. y; = (y -f)(y-g) (y - lK ' * '
a+b J f-g g-f
where / = /(ж), # = g(x), h = h(x).
Solution:
1.4.2. Equations of the Form
2
Preliminary comments.
1. For A22 = 0, this is the Abel equation (see Subsection 1.3.4). For Bu = 0, this is
the Abel equation with respect to x = x(y).
2. The transformation z = y/x, ? = x~2, leads to the Abel equation of the second kind:
QZ — -DOjs + ^22^ + \A\2 — -022^ -+- ^^1Ц — D\2)Z — -"lljsx
= 2^oC2 + 2(^22Z2 + Al2Z + Ац)(.
3. The transformation x = x + а, у = у + C, where a and C are the parameters which
are determined by solving the second order algebraic system
A22/32 + A12a/3 + Alia2 + Ao = О, В221З2 + B12a/3 + Blia2 + Bo = 0,
leads to the equation
[А22У2 + A12xy + Ацх2 + BA22C + A12a)y + BAxla + A120)x)]y'^.
= В22У2 + B12xy + Вцх2 + BB22[3 + B12a)y + BВца + B12[3)x,
The transformation ? = y/x, w = 1/x reduces this equation to the Abel equation of the
second kind:
= («гС + a>i)w + (^-ггС + -^-12^ + An)w.
where
a2 = 2A22C + A12a, b2 = 2B22C + B12a, аг = 2А1га + A12C, Ъг = 2В1га + B12C.
© 1995 by CRC Press, Inc.

4. The substitution у = t + ex, where parameter e is determined by solving the cubic
equation
(A22e2 + A12e + Ац)е - B22e2 - B12e - Вц = О,
leads to the Abel equation of the second kind with recpect to x = x(t):
where Q = 2B22s + B12 - sBA22s + A12).
1. (Ay2 + x2)y'x = -2xy + Bx2 + a.
Solution: Ay3 - Bx3 + 3(x2y - ax) = C.
2. (Ay2 + Bx2 - a2B)y'x = Cy2 + IBxy.
The transformation x = w + а, у = ?w leads to the linear equation:
+ B)w + 2aB.
3. (Ay2 + Bxy + Cx2)y'x = Dy2 + Exy + Fx2.
Homogeneous equation. The substitution z = y/x leads to an equation with
separated variables:
xz'x = (Az2 + Bz + Cy^-Az3 + (D- B)z2 + (E - C)z + F].
4. (Ay2 — lAkxy + Bkx2)y'x = -By2 + IBkxy — Ak3x2 + a.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[-(Ah + B)z2 + a]x'z = k(B - Ak)x2 + Az2.
5. (Ay2 + IBxy + Ak2x2)y'x = By2 + 1Ак2ху + Bk2x2 + a.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(B - Ak)z2 + a]x'z = 2k(Ak + B)x2 + 2(Ak + B)zx + Az2.
6. (Ay2 + Bxy + Cx2 + a)y'x = Aky2 + Bkxy + Ckx2 + b.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
(b - ak)x'z = (Ak2 + Bk + C)x2 + BAk + B)zx + Az2 + a.
7. (Ay2 + IBxy + Dx2 + a)y'x = -By2 - 2Dxy + Ex2 + b.
Solution: Ay3 - Ex3 + 3(Bxy2 + Dx2y + ay - bx) = C.
8. (Ay2 - lAxy + Bx2 + A- B)y'x = -Ay2 + IBxy - Bx2 + A-B.
This is a special case of equation 1.4.2.21 with a = 1, b = 1.
© 1995 by CRC Press, Inc.

9. (Ay2 + 2Axy + Bx2 + A - B)y'x = Ay2 + 2Bxy + Bx2 - A + B.
This is a special case of equation 1.4.2.21 with a = 1, b = —1.
10. (Ay2 - AAxy + Bx2 + AA- B)y'x = -2Ay2 + 2Bxy - 2Bx2 + 8A - 2B.
This is a special case of equation 1.4.2.21 with a = 1, b = 2.
11. (Ay2 + AAxy + Bx2 + AA- B)y'x = 2Ay2 + 2Bxy + ABx2 -8A + 2B.
This is a special case of equation 1.4.2.21 with a = 1, b = —2.
12. (Ay2 - 6Axy + Bx2 + 9A- B)y'x = -3Ay2 + 2Bxy - 3Bx2 + 27A - 3B.
This is a special case of equation 1.4.2.21 with a = 1, b = 3.
13. (Ay2 + 6Axy + Bx2 + 9A- B)y'x = 3Ay2 + 2Bxy + 3Bx2 - 27A + 3B.
This is a special case of equation 1.4.2.21 with a = 1, b = —3.
14. 2(Ay2 - Axy + Bx2 + A- AB)y'x = -Ay2 + ABxy - Bx2 + A- AB.
This is a special case of equation 1.4.2.21 with a = 2, b = 1.
15. 2(Ay2 + Axy + Bx2 + A - AB)y'x = Ay2 + ABxy + Bx2 - A + AB.
This is a special case of equation 1.4.2.21 with a = 2, b = —1.
16. (ay2 — 2bxy + ax2 + ab2 — a3)y'x = —by2 + 2axy — bx2 + b3 — a2b.
This is a special case of equation 1.4.2.21 with A = 1, В = 1.
17. (ay2 - 2bxy - ax2 + ab2 + a3)y'x = -by2 - 2axy + bx2 + b3 + a2b.
This is a special case of equation 1.4.2.21 with A = 1, В = —1.
18. (ay2 - 2bxy + 2ax2 + ab2A - 2a3)y'x = -by2 + Aaxy - 2bx2 + b3 - 2a2b.
This is a special case of equation 1.4.2.21 with A = 1, В = 2.
19. (ay2 - 2bxy - 2ax2 + ab2 + 2a3)y'x = -by2 - Aaxy + 2bx2 + b3 + 2a2b.
This is a special case of equation 1.4.2.21 with A = 1, В = —2.
20. (Ay2 + Bxy + Cx2 + a)y'x
= Dy2 + kBAk + B- 2D)xy + k(-Ak2 + Dk + C)x2 + b.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(D - Ak)z2 + b - ak]x'z = (Ak2 + Bk + C)x2 + BAk + B)zx + Az2 + a.
21. (aAy2 - 2bAxy + aBx2 + ab2A - a3B)y'x
= —ЪАу2 + 2аВху — ЪВх2 + Ъ3А — а2ЪВ.
The transformtion x = w + а, у = ^w + b leads to the linear equation:
(-aA?3 + bA?2 + aB? - ЬВЩ = (aA?2 - 2bA? + aB)w + 2a2В - 2b2A.
© 1995 by CRC Press, Inc.

1.4.3. Equations of the Form
(А22У2 + A12xy + Ацсс2 + A2y +
= В22У2 + B12xy + Вцж2 + B2y
Preliminary comments.
1. For A22 = 0, this is the Abel equation (see Subsection 1.3.4). For Вц = О this is the
Abel equation with respect to x = x(y).
2. The transformation ? = y/x, w = 1/x leads to the Abel equation of the second kind:
{[A2e + (Аг - B2)i - Bx}w + A22e + (A12 - B22)i2 + (Au - ??12)? - Bu}^
= (A2? + Ax)w2 + (A22?2 + A12? + Au)w.
3. In Paragraph 3 of Subsection 1.4.4, another thransformation is given which reduces
the original equation to the Abel equation of the second kind.
4. Dynamical systems of the second order
^r = P(x,y), ^- = Q{x,y) A)
at at
which describe the behavour of the simplest Lagrangian and Hamiltonian systems in me-
mechanics are often reduced to equations of the considered type when
P(x, y) = f(x, y)(A22y2 + A12xy + Ацх2 + A2y +
Q(x, y) = f(x,y)(B22y2 + B12xy + BllX2 + B2y + Birr),
B)
where / = f(x,y) is an arbitrary function.
In particular, dynamical systems A) with functions B) and / = 1 are met with in
analyzing complex equillibrium states. In this case, functions P and Q are substituted by
their Taylor-series expansions in the vicinity of the equallibrium state x = у = 0 with the
first and second order terms retained.
When obtained the solution of the ordinary differential equation
(A22y2 + A12xy + AllX2 + A2y + Axx)y'x = B22y2 + B12xy + Bxlx2 + B2y + Bxx
in the parametric form x = x(u,C\), у = y{u,C\), the solution of the system A), B) is
determined by the formulae
/ПГ A11
P(x(u,d), 2/0,Ci))
The latter relation defines the implicit dependence of parameter uoni: и = u(t, C\, C2),
and makes it possible to find, with the aid of two former formulae, the dependence of x and
у on t.
1. (y2 - x2 + ay)y'x = y2 - x2 + ax.
Solution in the parametric form:
1eAt, y = -at + C\t\-1eAt.
* This section was written with A.I. Zhurov
© 1995 by CRC Press, Inc.

2. (у2 - ж2 + ау)у'х = 2у2 - 2ху + ау.
Solution in the parametric form:
x = t + Ct2ea/t, y = Ct2ea/t.
3. (y2 — ж2 + ay — ax)y'x = y2 — x2 — ay + ax.
Solution in the parametric form:
x = at + Ce2t, y = -at + Ce2t.
4. (y2 - x2 + ay + 2ax)y'x = y2 - x2 + lay + ax.
Solution in the parametric form:
5. (y2 — x2 + ay + 2ax)y'x = 2xy — 2x2 + ay + 2ax.
Solution in the parametric form:
6. (y2 — x2 + ay — 2ax)y'x = 4y2 — бжу + 2ж2 + ay — 2ax.
Solution in the parametric form:
x=±t + C\t\2/3ea'\ y=\t + СЩ2/3еа'
7. {у2 - x2 + ay + 3ax)y'x = -y2 + 4xy - 3x2 + ay + Зож.
Solution in the parametric form:
x=\t + C\t\-1e~a/t,
у = -\t
8. (у2 - xy + ay + ax)y'x = xy — x2 + ay + ax.
Solution in the parametric form:
x = -t + C\t\-1ea/t, y = t + C\t\-1ea/t.
9. (y2 — xy + ay + ax)y'x = y2 — xy + 2ay.
Solution in the parametric form:
x = -at + Ct2e\ y = Ct2e\
10. (y2 — xy + ay — 2ax)y'x = Зу2 — 5жу + 2ж2 + ay — 2ax.
Solution in the parametric form:
x=\t + C\t\1/2ea/t, y = t + C\t\1/2ea/t.
© 1995 by CRC Press, Inc.

11. (у2 + ху — 2х2 + ау + ах)у'х = у2 + ху — 2х2 + lax.
Solution in the parametric form:
12. (y2 + xy — 2x2 + ay + ax)y'x = 2y2 - xy — x2 + ay + ax.
Solution in the parametric form:
x = t + C\t\3ea/t, y = -t + C\t\3ea/t.
13. (y2 + xy — 2x2 + ay — ax)y'x = y2 + xy — 2x2 — 2ay + 2ax.
Solution in the parametric form:
x = at + Ce3t, у = -2at + Ce3t.
14. (y2 + xy — 2x2 + ay — 2ax)y'x = by2 — Ixy + 2ж2 + ay — 2ax.
Solution in the parametric form:
x=\t + C\t\3/iea/t, y=\t + C\t\3/iea/t.
15. (у2 - 2жу + ж2 + ау)у'х = ay.
Solution: x = у +
С - In \y\
16. (у2 - 2жу + ж2 + ay + ax)y'x = —y2 + 2xy — x2 + ay + ax.
Solution in the parametric form:
+ Ct ^+Ct
17. (у2 - 2жу + ж2 + ay + 2ax)y'x = -2(y2 - 2жу + ж2) + ay + 2ax.
Solution in the parametric form:
+ct +CL
18. (у2 — 2жу + х2 + ay — 2ах)у'х = 2(у2 — 2жу + ж2) + ау — 2ож.
Solution in the parametric form:
a -, 2a „
X = -rr—r-r + Ct, у= —ГТ + Ct.
\n\t\ ' У ln\t\
19. (у2 + 2жу + x2 + ay + 2ax)y'x = —y2 — 2xy — x2 + 2ay + ax.
Solution in the parametric form:
x = C2(t1'3 + — )+Ct, y=-C2(t1/3 + —)+Ct, a^O.
V 5a J ' y V 5a J ' ^
© 1995 by CRC Press, Inc.

20. (у2 + 2жу + ж2 + ay — ах)у'х = —у2 — 2ху — х2 + оу — ож.
Solution in the parametric form:
¦""'¦
21. (у2 + 2жу + х2 + ay — 2ах)у'х = —у2 — 2ху — ж2 — 2оу + ож.
Solution in the parametric form:
x = C2(t3 + — )+Ct, y=-C2(t3 + —)+Ct, а
V a J V a J
22. (у2 + 2жу - Зж2 + ay + ax)y'x = 3y2 - 2жу - 1ж2 + ay + ax.
Solution in the parametric form:
x = \t + Ct2ea/t, x = -\t + Ct2ea/t.
23. (у2 + 2жу - Зж2 + ay + ax)y'x = y2 + 2xy — Зж2 - ay + Зож.
Solution in the parametric form:
x = at + ClipV*, у = -Sat
24. (у2 + 2жу - Зж2 + ay + 2ах)у'х = у2 + 2жу - Зж2 + Зож.
Solution in the parametric form:
x = at + C\t\~3e16t, у = -Sat + C\t\~3e16t.
25. (у2 - ж2 + оу + Ьж)у^, = у2 - ж2 + by + ax.
Solution in the parametric form:
a+b a+b
x = (a - b)t + C\t\ «-f-e4*, у = (b - a)t + C\t\ «-*> e4*, а ф b.
26. (y2 — xy + ay + bx)y'x =y2 —xy+ (a + b)y.
Solution in the parametric form:
a+b a+b
x = -bt + C\t\ ь e\ y = C\t\ ь e\ b ф 0.
27. (у2 + жу - 2ж2 + оу + Ъх)у'х = у2 + ху - 2х2 + F - а)у + 2ож.
Solution in the parametric form:
x= {2a-b)t + C\t\~^a-=bem, у = 2(b-2a)t +C\t\~^b e9t, Ьф2а.
28. (у2 - 2жу + ж2 + оу - аЪх)у'х = Ъ(у2 — 2ху + ж2) + оу - оЬж.
Solution in the parametric form:
a 1 _, ab 1 _, , . „
x= . л . ... +Ct, y= +Ct, Ьф\.
b — 1 In |t| b — 1 In \t\
© 1995 by CRC Press, Inc.

29. (у2 + 2жу — Зх2 + ау + Ъх)у'х = у2 + 1ху — Зх2 + F — 1а)у + Зах.
Solution in the parametric form:
^rbe16t, у = Щ - ia)t + C\t\~^ь еш, Ь ф За.
30. (у2 — Зху + 2х2 + ау + Ъх)у'х = у2 — Зху + 2х2 + (За + Ъ)у — lax.
Solution in the parametric form:
+Ъе-\ у = 2Bа + b)t
31. (у2 + Зжу - 4ж2 + ау + Ъх)у'х = у2 + Зху — 4ж2 + F — Зо)у + 4ож.
Solution in the parametric form:
x={^a-b)t + C\t\~^=be2bt, у = 4F - 4a)i + C\t\~~t=b e25t, Ьф
32. [у2 2 ^ 2 2
Solution in the parametric form:
33. (у2 - 2Ажу + А2ж2 + by - Ъх)у'х = Ау2 - 1А2ху + А3х2 + by - Ъх.
Solution in the parametric form:
x =
34. [у2 - 2Ажу + BA - 1)ж2 + by - АЬх]у'х = B - А)у2 - 2жу + Аж2 + by - АЬх.
Solution in the parametric form:
35. (y2-2Axy+A2x2+ay+bx)y'x =
Solution in the parametric form:
where а + Ь ф 0 and B - А)а + ЬфО.
36. [у2 - (А + 2)жу + (А + 1)ж2 + by - АЪх]у'х = -Аху + Ах2 + by - АЬх.
Solution in the parametric form:
x=
+C\t\e, y
© 1995 by CRC Press, Inc.

37. [Ay2 + xy - (A + l)x2 + Ьу + bx]y'x = (A + l)y2 - xy - Ax2 + by + bx.
Solution in the parametric form:
x = t + C\t\2A+1eb^, y = -t + СЩ2А+1еь'К
38. {Ay2 + Bxy + Cx2 + kx)y'x - Dy2 + Exy + Fx2 + ky.
The substitution у = xz leads to a linear equation with respect to x = x(z):
[-Az3 + (D- B)z2 + (E- C)z + F]x'z = (Az2 + Bz + C)x + k.
39. (Ay2 + Bxy + Cx2 - a.By - a.Cx)y'x = Dy2 + Exy + <x(C - E)y.
The transformation x = w + а, у = ?,w leads to a linear equation:
4 = [A?2 + B? + C)w + aC.
40. (Ay2 + IBxy + Ak2x2 + ay + bx)y'x = By2 + 1Ак2ху + Bk2x2 + by + ak2x.
This is a special case of equation 1.4.3.57 with С = Ak2.
41. (Ay2 + IBxy + Ak2x2 +ay + bx)y'x = By2 + 1Ак2ху + Bk2x2 + aky + bkx.
This is a special case of equation 1.4.3.62 with С = Ak2.
42. (Ay2 + IBxy + Ak2x2 +ay — akx)y'x = By2 + 1Ак2ху + Bk2x2 + my — mkx.
This is a special case of equation 1.4.3.61 with С = Ak .
43. (Ay2 + IBxy - Bkx2 + ay + bx)y'x = By2 + 1Ак2ху - Ak3x2 + by + ak2x.
This is a special case of equation 1.4.3.58 with m = b.
44. (Ay2 + IBxy - Bkx2 + ay + bx)y'x = By2 + 1Ак2ху — Ak3x2 + aky + bkx.
This is a special case of equation 1.4.3.62 with С = —Bk.
45. (Ay2 + IBxy — Bkx2 + ay — akx)y'x = By2 + 1Ак2ху — Ak3x2 + my — mkx.
This is a special case of equation 1.4.3.61 with С = —Bk.
46. (Ay2 + lAkxy + Cx2 + ay + bx)y'x = Aky2 + 2Ak2xy + Ckx2 + by + ak2x.
This is a special case of equation 1.4.3.57 with В = Ak.
47. (Ay2 + lAkxy + Cx2 + ay + bx)y'x = Aky2 + 1Ак2ху + Ckx2 + aky + bkx.
This is a special case of equation 1.4.3.62 with В = Ak.
48. (Ay2 + lAkxy + Cx2 -\-ay — akx)y'x = Aky2 + 2Ak2xy + Ckx2 + my — mkx.
This is a special case of equation 1.4.3.61 with В = Ak.
49. (Ay2 - lAkxy + Bkx2 + ay + bx)y'x = -By2 + IBkxy - Ak3x2 + by + ak2x.
This is a special case of equation 1.4.3.59 with m = b.
© 1995 by CRC Press, Inc.

50. (Ay2 — 2Akxy + Bkx2 + ay + bx)y'x = -By2 + 2Bkxy - Ak3x2 + aky + bkx.
This is a special case of equation 1.4.3.59 with m = ak.
51. [y2 + lAxy + A2x2 + (A- l)By - 2ABx]y'x
= -A(y2 + 2Axy + A2x2) - (A2 + l)By + A(A - l)Bx.
Solution in the parametric form:
52. [у2 - 2Axy + A2x2 + (В - l)ky + (A - B)kx]y'x
= A(y2 - 2Axy + A2x2) + (AB - l)ky - A(B - l)kx.
Solution in the parametric form:
53. [2y2 -(A + 3)xy + (A + l)x2 + By - ABx]y'x
= (A + l)y2 - (ЗА + l)xy + 2Ax2 + By - ABx.
Solution in the parametric form:
54. [2y2 - (ЗА + l)xy + (ЗА - l)x2 + By - ABx]y'x
= C - A)y2 - (A + 3)xy + 2Ax2 + By - ABx.
Solution in the parametric form:
55. [A(y2 - 2xy + x2) - A(A - B)y + B(A - B)x]y'x
= B(y2 - 2xy + x2) - A(A - B)y + B(A - B)x.
Solution in the parametric form:
А В
x = —- + Ct, y = -j—т + Ct.
In \t\ In \t\
56.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
(n - ak)zx'z = (Ak2 + Bk + C)x2 + [BAk + B)z + ak + b]x + Az2 + az.
57. (Ay2 + 2Bxy + Cx2 + ay + bx)y'x
= By2 + 2Ak2xy + k(-Ak2 + Bk + C)x2 + by + ak2x.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(B - Ak)z + b-ak]zx'z = (Ak2 + 2Bk + C)x2 + [2(Ak + B)z + ak + b]x + Az2 + az.
© 1995 by CRC Press, Inc.

58. (Ay2 + 2Bxy - Bkx2 + ay + Ъх)у'х
= By2 + 2Ак2ху — Ак3х2 + ту + к(ак + Ь — т)х.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(B - Ak)z + m - ak]zx'z = (Ak2 + Bk)x2 + [2(Ak + B)z + ak + b]x + Az2 + az.
59. (Ay2 — lAkxy + Bkx2 + ay + bx)y'x
= —By2 + IBkxy — Ak3x2 + my + k(ak + b — m)x.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[-(Ak + B)z + m- ak]zx'z = k(B - Ak)x2 + (ak + b)x + Az2 + az.
60. (Ay2 + IBxy + Ak2x2 + ay + bx)y'x
= By2 + 2Ak2xy + Bk2x2 + my + k(ak + b- m)x.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(B - Ak)z + m - ak]zx'z = 2k(Ak + B)x2 + [2(Ak + B)z + ak + b]x + Az2 + az.
61. (Ay2 + 2Bxy + Cx2 + ay - akx)y'x
= By2 + 2Ak2xy + k(—Ak2 + Bk + C)x2 + my — mkx.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(B - Ak)z2 + m - ak]zx'z = (Ak2 + 2Bk + C)x2 + 2(Ak + B)zx + Az2 + az.
62. (Ay2 + 2Bxy + Cx2 + ay + bx)y'x
= By2 + 2Ak2xy + k(-Ak2 + Bk + C)x2 + aky + bkx.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
(B - Ak)z2x'z = (Ak2 + 2Bk + C)x2 + [2(Ak + B)z + ak + b]x + Az2 + az.
63. {(A - l)y2 + [2 - A(k + l)]xy + (Ak - l)x2 + By- Bkx}y'x
= (A- k)y2 + [2k - A(k + l)]xy + (A - l)kx2 + By - Bkx.
Solution in the parametric form:
64. [A(ay2 + Cxy + -yx2) + Ba - A2cr)y + (/3 - ABcr)x]y'x
+ B(ay2 + (Зху + 7Ж2) + (/3 - ABcr)y + B7 - B2cr)x = 0.
Solution: ay2 + /Зху + jx2 — Aery — Bax + a = С exp(—Ay — Bx).
65. (A22y2 + A12xy + Anx2 + A2y + Агх)у'х
= B22y2 + kBA22k + A12 - 2B22)xy + k(-A22k2 + B22k + А1г)х2
+ B2y + k(A2k + Ai - B2)x.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(B22 - A22k)z + B2- A2k)zx'z
= (A22k2 + A12k + Ац)х2 + [2(A22k + A12)z + A2k + Аг]х + A22z2 + A2z.
© 1995 by CRC Press, Inc.

> In equations 66-70, the following notation is used:
A = Ab - aB, 6 = Ab + aB.
66. (Aa2y2 — 2Aabxy + Ab2x2 — AAay + AaBx)y'x
= a2By2 - 2aBbxy + Bb2x2 - AAby + ABbx.
Solution in the parametric form:
67. [kAa2y2 — кёаху + kaBbx2 — mAAay + (maB — A) Ax]y'x
= kAaby2 — kebxy + kBb2x2 — (mAb + A) Ay + mABbx.
Solution in the parametric form:
x = At + aC\t\m+1eM, y = Bt + bC\t\m+1eM.
68. [mAa2y2 — а(тё — A)xy + b(maB — A)x2 + kAAay — kAaBx]y'x
= a(mBb + A)y2 — b(m6 + A)xy + тВЪ2х2 + к AAby — к ABbx.
Solution in the parametric form:
x = At + aC\t\m+1ek/t, y = Bt + bC\t\m+1ek/t.
69. (kA3y2 - 2kA2Bxy + kAB2x2 - 2Aa2y + 2Aabx)y'x
= kA2By2 - 2kAB2xy + kB3x2 - 2Aaby + 2Ab2x.
Solution in the parametric form:
x = ACy^kt3 + 1 + aC2t, у = BC3^±kt3 + 1 + ЪСН.
70. [kA3y2 - 2kA2Bxy + kAB2x2 + mAAay - (mAb + A) Ax]y'x
= kA2By2 — 2kAB2xy + kB3x2 + (maB — A) Ay — mABbx.
Solution in the parametric form:
x = AC2 (tm+1 + ——t2) + aCt, у = ВС2 (tm+1 + ——t2) + bCt, тф\.
V m — 1 / V m — 1 /
1.4.4. Equations of the Form
(А22У2 + A12xy + Ацсс2 + A2y + Axx + A0)y'x
= B22y2 + B12xy + Вцсс2 + B2y + Bxx + Bo
Preliminary comments.
1. With A22 = 0, this is the Abel equation (see Subsection 1.3.4). With Вц = О, this
is the Abel equation with respect to x = x(y).
See Subsection 1.4.2 for the case A2 = Ax = B2 = B\ = 0.
See Subsection 1.4.3 for the case Ao = Bo = 0.
© 1995 by CRC Press, Inc.

2. The transformation x = x + а, у = у + C, where a and C are the parameters which
are determined by solving the second order algebraic system
A22/32 + A12a/3 + Аца2 + A2/3 + Ага + Ao = 0,
B22/32 + B12a/3 + Blia2 + B2/3
leads to the equation
(A22f + A12xy + Axlx2 + a2y + сцх)^ = B22f + B12xy + Bxlx2 + b2y + hx, A)
where
a2 = 2A22/3 + A12a + A2, сц = 2Аца + A12j3 + Аъ
b2 = 2B22/3 + B12a + B2, 61 = 2Blia + B12/3 + Вг.
The transformation ? = y/x, w = 1/x reduces equation A) to the Abel equation of the
second kind:
{Ы2 + («i - Ы? " h]w + A22?3 + (A12 - B22)i2 + (Ац - B12)i - ВИЦ
= (a2C + cn)w2 + (A22?2 + A12? + An)w.
3. The substitution у = z + ex, where parameter e is determined by solving the cubic
equation
(A22e2 + A12e + Ац)е - B22e2 - B12e - Bn = 0,
leads to the Abel equation of the second kind with respect to x = x(z):
[(Qz + R)x + (B22 - A22e)z2 + (B2 - A2e)z + Bo - Aoe]x'z
= {A22e2 + A12e + Аг1)х2 + [BA22e + A12)z + A2e + Аг]х + A22z2 + A2z + Ao,
where
Q = 2B22e + B12 - eBA22e + A12), R = B2e + B1- e(A2e + Аг).
This is a special case of equation 1.7.1.6 with f(z) = z~2.
2. (Ay2 + Bxy — aBy + kx — cxk)y'x = Cy2 + Dxy + (k — oiD)y.
The transformation x = w + а, у = w^ leads to a linear equation with respect to
3 2 2 + k.
3. (Ay2 + lAxy + Bx2 + A-B)y'x = Ay2 + IBxy + Dx2 + 2(B- D)x + D-A.
The transformtion x = w + 1, у = ^w — 1 leads to a linear equation:
(-Af - Af + B? + D)w't = {Af + 2A? + B)w + 2(B - A).
4.
The transformtion x = w — 1, у = ^w — 1 leads to a linear equation:
(-A?3 + A?2 + B? + C)w'z = [A?2 - 2A? + B)w + 2(A - B).
© 1995 by CRC Press, Inc.

5. (Ay2 + lAxy + Bx2 + A- B)y'x = Ay2 + IBxy + Cx2 + 2(C- B)x -A + C.
The transformtion x = w — 1, у = ?w + 1 leads to a linear equation:
(-A?3 - A? + B? + СЩ = [A? + 2A? + B)w + 2(A - B).
6.
The transformtion x = w + 1, у = ^w + 1 leads to a linear equation:
(-A?3 + A?2 + B? + C)«4 = [A?2 - 2A? + B)w + 2(B - A).
7. (Ay2 - 2Axy + Bx2 + A- B)y'x
= Cy2 + 2Bxy + Dx2 - 2(A + C)y - 2(B + D)x + 2A + C + D.
The transformtion x = w + 1, у = ?w + 1 leads to a linear equation:
{-Af + BA + C)C2 + ВС + #K = (^ " 2AZ + B)w + 2(B " A)-
8. BAy2 - 2Axy + Bx2 + 2A- AB)y'x
= -Ay2 + 2Bxy + Dx2 - 2(B + 2D)x + A + AD.
This is a special case of equation 1.4.4.34 with a = 2, C = 1, С = —А.
9. (Ay2+4Axy+Bx2+4A-B)y'x = 2Ay2+2Bxy+Cx2-2(C-2B)x+C-8A.
The transformtion x = w + 1, у = ^w — 2 leads to a linear equation:
3 2 + B? + C)«4 = (A?2 + AA? + B)w + 2B- 8A.
10. (Ay2-4Axy+Bx2+4A-B)yfx = -2Ay2+2Bxy+Cx2-2BB+C)x+8A+C.
The transformtion x = w + 1, у = ?w + 2 leads to a linear equation:
3 2 2 B)w + 2B- 8Л
11. (Ay2 + 4Axy + Bx2 + 4A- B)y'x
= Cy2 + 2Bxy + 2Bx2 + 4(C - 2A)y + 2B + 4C - 16A.
The transformtion x = w + 1, у = ^w — 2 leads to a linear equation:
4 = (Af + AA? + B)w + 2B- 8A.
12. BAy2+2Axy+Bx2+2A-4B)y'x = Ay2+2Bxy+Dx2+2(B-2D)x+4D-A.
This is a special case of equation 1.4.4.34 with a = 2, C = —1, С = A.
13. BAy2+2Axy-Bx2+2A+4B)y'x = Ay2-2Bxy-Dx2+2(B-2D)x-A-4D.
This is a special case of equation 1.4.4.34 with a = —2, C = 1, С = —А.
14. (Ay2 + 2Bxy + Ak2x2 + ay + bx + m)y'x
= By2 + 2Ak2xy + Bk2x2 + by + ak2x + s.
This is a special case of equation 1.4.4.27 with С = Ak2.
© 1995 by CRC Press, Inc.

15. (Ay2 + 2Bxy + Ak2x2 + ay + bx + m)y'x
= By2 + 2Ak2xy + Bk2x2 + aky + bkx + s.
This is a special case of equation 1.4.4.32 with С = Ak2.
16. (Ay2 + 2Bxy + Ak2x2 + ay - akx + b)y'x
= By2 + 2Ak2xy + Bk2x2 + my — mkx + s.
This is a special case of equation 1.4.4.31 with С = Ak2.
17. (Ay2+2Bxy-Bkx2+ay+bx+c)y'x = By2+2Ak2xy — Ak3x2+by+ak2x+s.
This is a special case of equation 1.4.4.28 with m = b.
18. (Ау2+2Вху-Вкх2+ау+Ьх+тп)у'х = By2+2Ak2xy-Ak3x2+aky+bkx+s.
This is a special case of equation 1.4.4.32 with С = —Bk.
19. (Ay2 + 2Bxy — Bkx2 + ay — akx + b)y'x
= By2 + 2Ak2xy — Ak3x2 + my — mkx + s.
This is a special case of equation 1.4.4.31 with С = —Bk.
20. (Ay2+2Akxy+Cx2+ay+bx+m)y'x = Aky2+2Ak2xy+Ckx2+by+ak2x+s.
This is a special case of equation 1.4.4.27 with В = Ak.
21. (Ay2+2Akxy+Cx2+ay+bx+m)y'x = Aky2+2Ak2xy+Ckx2+aky+bkx+s.
This is a special case of equation 1.4.4.32 with В = Ak.
22. (Ay2 + 2Akxy + Cx2 + ay - akx + b)y'x
= Aky2 + 2Ak2xy + Ckx2 + my — mkx + s.
This is a special case of equation 1.4.4.31 with В = Ak.
23. (Ay2 — 2Akxy + Bkx2 + ay + bx + c)y'x
= -By2 + 2Bkxy — Ak3x2 + by + ak2x + s.
This is a special case of equation 1.4.4.29 with m = b.
24. (Ay2 - 2Akxy + Bkx2 + ay + bx + c)y'x
= —By2 + 2Bkxy — Ak3x2 + aky + bkx + s.
This is a special case of equation 1.4.4.29 with m = ak.
25. (Ay2 + 2Bxy + Cx2 - 2ACy + kx + AC2)y'x
= By2 + Exy + Fx2 + ky - ECx - B02 - kC.
The substitution w = у — C leads to an equation of the form 1.4.3.38:
(Aw2 + 2Bxw + Cx2 + kx)w'x = Bw2 + Exw + Fx2 + kw,
where к = к + 2В/3.
© 1995 by CRC Press, Inc.

26. (Ay2 + Вху + Cx2 + ay + Ьх + m)y'x
= Aky2 + Bkxy + Ckx2 + ny + (ak + b — n)x + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(n - ak)z + s- mk]x'z = (Ak2 + Bk + C)x2 + [BAk + B)z + ak + b]x + Az2 + az + m.
27. (Ay2 + IBxy + Cx2 + ay + bx + m)y'x
= By2 + 2Ak2xy + k(-Ak2 + Bk + C)x2 + by + ak2x + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(B - Ak)z2 + (b- ak)z + s- mk]x'z =
(Ak2 + 2Bk + C)x2 + [2(Ak + B)z + ak + b]x + Az2 + az + m.
28. (Ay2 + IBxy - Bkx2 + ay + bx + c)y'x
= By2 + 2Ak2xy — Ak3x2 + my + k(ak + b — m)x + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(B-Ak)z2 + (m-ak)z+s-ck}x'z = (Ak2+Bk)x2 + [2(Ak+B)z+ak+b]x+Az2+az+c.
29. (Ay2 — lAkxy + Bkx2 + ay + bx + c)y'x
= —By2 + 2Bkxy — Ak3x2 + my + k(ak + b — m)x + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[-(Ak + B)z2 + (m - ak)z + s - ck]x'z = k(B - Ak)x2 + (ak + b)x + Az2 + az + c.
30. (Ay2 + 2Bxy + Ak2x2 + ay + bx + c)y'x
= By2 + 2Ak2xy + Bk2x2 + my + k(ak + b- m)x + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
2 + (m-ak)z+s-ck}x'z = 2k(Ak+B)x2 + [2(Ak+B)z+ak+b]x+Az2+az+c.
31. (Ay2 + 2Bxy + Cx2 + ay - akx + b)y'x
= By2 + 2Ak2xy + k(-Ak2 + Bk + C)x2 + my - mkx + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
32. (Ay2 + 2Bxy + Cx2 + ay + bx + m)y'x
= By2 + 2Ak2xy + k(-Ak2 + Bk + C)x2 + aky + bkx + s.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(B - Ak)z2 + s -mk]x'z = (Ak2 + 2Bk + C)x2 + [2(Ak + B)z + ak + b]x + Az2 + az + m.
33. [A(ay2 + Cxy + 7Ж2) + (A6 + 2a)y + (As + C)x + Act + 6]y'x
+ B(oty2 + Cxy + fx2) + (B6 + /3)y + (Be + 21)x + Bcr + г = 0.
Solution: ay2 + /3xy + jx2 + 6y + ex + a = Сещ>(—Ау — Bx).
© 1995 by CRC Press, Inc.

34. (otAy2 - 2(ЗАху + Вх2 + <хC2А - а.2В)у'х
= Cy2+2Bxy + Dx2-2/3(/3A + C)y-2(
The substitution х = w + а, у = ^w + C leads to a linear equation:
[-aA?3 + B/3A + C)i2 + B? + D]w'i = (aA?2 - 2C A^ + B)w + 2(aB - C2A).
35. (A22y2 + A12xy + Ацх2 + A2y + Агх + A0)y'x
= B22y2 + kBA22k + A12 - 2B22)xy + k(-A22k2 + B22k + Аг1)х2
+ B2y + k(A2k + Ai - B2)x + Bo.
The substitution у = z + kx leads to the Riccati equation with respect to x = x(z):
[(B22 - A22k)z2 + (B2 - A2k)z + B0- Aok]x'z =
(A22k2 + A12k + Ац)х2 + [2(A22k + A12)z + A2k + Ax]x + A22z2 + A2z + Ao.
36. (A22y2 + A12xy + Anx2 + A2y + Агх + A0)y'x
= B22y2 + B12xy + BllX2 + B2y + BlX + Bo.
Here A^j, Вц, and A\ are arbitrary parameters, and the other parametes are defined
by the equations
A2 = -A12a - 2A22C,
Ao = -А1Хо? + A22p2 - Aia,
B2 = BAn - B12)a + (A12 - 2B22)/3 + Аъ
Вг = -2Вца - B12[3,
Во = В1га2 + (В12 - 2А11)а0 + (В22 - А12)C2 - АгC,
(а, C are arbitrary parameters).
The transformation x = w + а, у = ^w + C leads to a linear equation:
+ (B22 - A12)i2 + (B12 - AU)Z + Вц]«4 = (A22?2 + A12? + A^w + k,
where к = 2Аца + А12C + Ax.
1.5. Nonlinear Equations of the Form f(x,y)yx = g(x,y)
Containing Arbitrary Parameters
1.5.1. Equations Containing Power Functions
1. y^ 1/2
2.
The substitution w = B/A) ^fy leads to the Abel equation of the form 1.3.1.32: ww'x
w + 2BA-2x~1'2.
Let A = ±2a~1Vb, В = ^46 (b > 0).
The solution in the parametric form is written as
x = af(r), y = b[2T±f(T)}2,
where /(т) = exp(=Fr2) / exp(=Fr2) dr + С
-i
© 1995 by CRC Press, Inc.

3. y'x
The substitution w = 2A~1y/y leads to the Abel equation of the form 1.3.1.33: ww'x =
22
4. y'x = a^/y + bx + cx
The substitution w = 2а~г^/у leads to the Abel equation of the second kind: ww'x
w + 2a~2(bx + cxm), whose special cases are outlined in Subsection 1.3.1.
5. y'x = ayn + bx !-« .
Solution:
dw
= In |ж| + С, where w = yx n~1 .
awn + -jb^w + b
6. y'x = Ays - Bxk.
The transformation x = (w'z) k , у = A(—j , where A = (—r) S , leads to the
generalized Emden—Fowler equation:
which is discussed in Section 2.5 (in the classification table, one should search for the
equations satisfying the condition n + m + 1 = 0).
7- y'x = (ax + by + c)n.
This is a special case of equation 1.7.1.1 with /(?) = ?™'.
8. y'x = axm-n-nmyn + bxm.
Solution:
dw _, , fa , .
+ C = b\ — In \x
, „Га _m_i л
where w = * —yx , A=
V b
wn -Xw + 1
m +
yx , A=;
b b
9. y'x = axn-1ym+1 + bxnk-1ymk+1.
This is the homogeneous equation in the extended sense of the form 1.7.1.3 with
bxmy + c
This is a special case of equation 1.7.1.4 with f(x) = axk, g(x) = bxm, h(x) = cxs,
n = 1/2.
11. y'x = axky1+n + bxmy + cxsyl~n.
This is a special case of equation 1.7.1.4 with f(x) = axk, g(x) = bxm, h(x) = cxs.
© 1995 by CRC Press, Inc.

12. y'x = xn-1y1-m(axn + Ьут)к.
This is a special case of equation 1.7.1.7 with /(?) = ?k.
13. xy'x = у + axn-mym + bxn~kyk.
The substitution у = xw leads to an equation with separation of variables: w'x =
14. (ayn + bx)y'x = 1.
Solution: x = eby(c + a f yne~bv dyj.
15. x(xyn + a)y'x + by = O.
Solution: nb-a = x (Суа1ъ + уп).
16. x(aym + m)yx = у [bx71^-^ymX - n].
This is a special case of equation 1.7.1.15 with /(?) = a?, g(?) = 1, /i(?) = 6?л, к = п.
17. (ож" + Ьж2 + сху)у'х = kxn + Ьжу + су2.
The transformation t = y/x, z = xn~2 leads to a linear equation: (k — at)z't =
(n-2)(az + b + ct).
18. (ayn + bx2 + cxy)y'x = kyn + Ьжу + су2.
The transformation t = y/x, z = xn~2 leads to a linear equation: tn(k — at)z't =
(n-2)(atnz + b + ct).
19. (axn + byn + x)y'x = axkyn~k + /3xmyn-m + y.
The transformation t = y/x, z = xn~x leads to a linear equation:
(atn~k + f3tn~m - btn+1 - at)z't = (n - l)(btn + a)z + n-l.
20. (axn + byn + Ax2 + Bxy)y'x = axkyn~k + /3xmyn-m + Axy + By2.
The transformation t = y/x, z = xn~2 leads to a linear equation:
(atn~k + f3tn~m - btn+1 - at)z't = (n - 2){btn + a)z + (n - 2){Bt + A).
21. [(ax + by)n + bx]y'x = c(ax + by)m - ax.
This is a special case of equation 1.7.1.13 with /(?) = ?n, g(?) = 1, /i(?) = c?m.
22. [(ож + by)n + by]y'x = c(ax + by)m - ay.
This is a special case of equation 1.7.1.14 with /(?) = ?"¦, p(^) = 1,
23. (аж + /3y + 7)"y^ = (ож + by + c)n.
This is a special case of equation 1.7.1.6 with /(?) = ^n.
© 1995 by CRC Press, Inc.

24. (axn + Ъут)у'х = xn-1y1-m.
This is a special case of equation 1.7.1.7 with /(?) = l/?.
25. (oym + bxn + s)y'x + otxk + bnxn~1y + /3 = 0.
Solution:
aip(y) + аф(х) + bxny + sy + Cx = C,
where
{уГП+1 ( fe+l
Итф-l, ., л I if к ф -1,
m + 1 ^ V>0) = < к + 1
ln|y| ifm = -l, [ln|rr| if fe = —1.
26. (aa^y™ + Ьжу + cyk)y'x = ayP + (ЗуЧ + f.
This is the Riccati equation with respect to x = x(y).
27. (axnym + x)y'x = bxkyn+m-k + y.
The transformation t = y/x, z = xn+m~1 leads to a linear equation: tm(btn~k — at)z't
(n + m-l)(atmz + l).
28. x(axnym + ot)y'x + y(bxnym + C) = 0.
Solution:
(yaxb)A (yaxld)B „ , . mC-na „ mb-na
л 1 ^ = ^> where A = —-—-—, В = —-—-—.
А В a/3-ba a/3-ba
29. x(anxkyn+k + s)y'x + y(bmxm+kyk + s) = 0.
Solution: akyn + bkxm - s(xy)~k = С
30. (axnym + Ax2 + Bxy)y'x = bxkyn+m-k + Axy + By2.
The transformation t = y/x, z = xn+m~2 leads to a linear equation: tm{btn~k — at)z't
(n + m-2) (atmz + Bt + A).
31. (amxnym-1 + byk)y'x + anxn-1ym + cxs = 0.
This is a special case of equation 1.7.1.18 with f(y) = byk, g(x) = cxs.
32. (axnym + bxyk)y'x = ays + C.
This is the Bernoulli equation with respect to x = x(y) (see Subsection 1.1.5).
33. x(axn-kym + m)y'x = y{bxXn~kyXm - n).
This is a special case of equation 1.7.1.15 with /(?) = a?, g(?) = 1,
34. x(axnym-k + m)y'x = y{bxXnyXm~k - n).
This is a special case of equation 1.7.1.16 with /(?) = a?, g(?) = 1,
© 1995 by CRC Press, Inc.

35. (axn+1ym-1 + Ьхпк+1утк-1)у'х = cxnsyms.
This is a special case of equation 1.7.1.3 with /(?) = c^s(a^
36. (axn + bym)ky'x = cxn-1y1-m.
This is a special case of equation 1.7.1.7 with /(?) = c?~k.
37. xy'x = у + a\Jy2 + bx2.
Solution: у + а/у2 + bx2 = Cxa+1.
ж + о ж — о \ . /ei e2,
38- \ei—— + е2~— )vL-vi — + — )=0'
where r ^ = (ж + оJ + у2, Гз = (ж — оJ + у
This is the equation of force lines corresponding to the Coulomb law.
x + a x — a
Solution: ei \- e<i = С
Т\ Г2
1.5.2. Equations Containing Exponential Functions
1. y'x = aev + bex.
Solution in the parametric form:
( f е6т
x = In т, у = Ът — In ( С — a I dr
\ J т
2. y'x = aev + bxn.
Solution in the parametric form:
with n ф —1,
b , |\ a f 22_ / Ът
I СI +1
-J— b , |\ a f 22_ / Ът . , .
x = т «+1 , у = т -In С It «+1 exp ) ат ;
n+1 [ n+lJ Vn+1
with n = —1, b ф —1,
with n = —1, b = —1,
ж = ет, у = — т — ln(C — ат).
3. y^ = ay-1 + bex.
Solution in the parametric form:
x = iniAE-1) tt2, y = B[2 ± exp(T
where a = T2-B2, b = ±A~1B, E = /ехр(тт2) dr + С
© 1995 by CRC Press, Inc.

4. y'x = Aey+ax - a.
This is a special case of equation 1.7.1.2 with /(?) = Ae^, n = 1, b = 0.
5. y'x = aeux+Xy + be»*.
This is a special case of equation 1.7.2.5 with f(x) = aevx, g(x) = be]xx.
6. y'x = aeux+Xy + bxn.
This is a special case of equation 1.7.2.5 with f(x) = aevx, g{x) = bxn.
7. y'x = axneXy + bevx.
This is a special case of equation 1.7.2.5 with f(x) = ax", g(x) = bevx.
8. y'x = axneXy + bxm.
This is a special case of equation 1.7.2.5 with f(x) = axn, g(x) = bxm.
9. y'x = aevx+Xy + be»x~Xy.
This is a special case of equation 1.7.2.8 with f(x) = aevx, g(x) = 0, h(x) = belxx.
10. y'x = axneXy + bxme~Xy.
This is a special case of equation 1.7.2.8 with f(x) = axn, g(x) = 0, h(x) = bxm.
11. y'x = (y + aeXx)n - a\eXx.
This is a special case of equation 1.7.2.10 with /(?) = ?n.
12. y'x = (aey +bx
Solution in the parametric form:
( 1 -. f
= ехр{ [/(t)+C]}, y=f{T)+C, where /(r)=/
к (It
efeT) '
13. y'x =
Solution in the parametric form:
kdr
x =
r 1 ч
eXP{r+T[f(r)+C]},
14 ?/ — n/rn—lp\ny i u^m — lpXmy
This is a special case of equation 1.7.2.2 with /(?) = a^
15. y'x = axn~1eay + bxnm~1eamy.
This is a special case of equation 1.7.2.4 with /(?) = a^
© 1995 by CRC Press, Inc.

16. y'x = aeXnxyn+1 + be~Xx.
This is a special case of equation 1.7.2.1 with /(?) = a?n+1 + b.
17. y'x = aeaxym+1 + beanxynm+1.
This is a special case of equation 1.7.2.3 with /(?) = a? + 6?n.
18. y^ = aeXnxyn+1 + beXmxym+1.
This is a special case of equation 1.7.2.1 with /(?) = a?n+1 +
19. y^ = ae2ax~fiv + beaa: + ceaa:~/31'.
This is a special case of equation 1.7.2.9 with /(?) = ? + с
20. y^ = axnyk + bxneaxyk+1 - ay.
This is a special case of equation 1.7.2.7 with /(?) = ?n, g(?) = a + b?, m = 1.
21. y'x = aeXxy1+n + be»xy + cevxy1~n.
This is a special case of equation 1.7.1.4 with f(x) = aeXx, g(x) = be^x, h(x) = cevx.
22. y'x = aeXxy1+n + be»xy + cxmy1~n.
This is a special case of equation 1.7.1.4 with f(x) = aeXx, g(x) = be^x, h(x) = cxm.
23. y'x = axky1+n + beXxy + cxmy1-n.
This is a special case of equation 1.7.1.4 with f(x) = axk, g(x) = beXx, h(x) = cxm.
24. y'x = aeXxy1+n + bxmy
This is a special case of equation 1.7.1.4 with f(x) = aeXx, g(x) = bxm, h(x) = celxx.
25. y'x = aeXxy1+n + bxmy + cxky1~n.
This is a special case of equation 1.7.1.4 with f(x) = aeXx, g(x) = bxm, h(x) = cxk.
26. xy'x = axn+key + bxnm+kemy - n.
This is a special case of equation 1.7.2.6 with f(x) = xk~1, g(?) = a^ + b^m.
27. (by + X)y'x = ceax+by - ay.
This is a special case of equation 1.7.1.14 with /(?) = A, g(?) = 1, h((,) = ce^.
28. xyy'x = axney — ny.
This is a special case of equation 1.7.2.11 with /(?) = a?, a = 1.
29. жу2у^, = axnev — ny2.
This is a special case of equation 1.7.2.12 with /(?) = a?, a = 1.
© 1995 by CRC Press, Inc.

30. (aey + Ьех)у'х = 1.
Solution in the parametric form:
e
x = ат — In (С — b I dr ), у = In т.
31.
Solution in the parametric form:
with n ф —1:
n+1 | n+
with n = —1, а ф —1:
/ Ц-Г ( aT \ j 1
— It n+1 exp <ir
1У РКп + 1У J
= eT
with n = —1, a = —1:
x = — т — ln(C — Ьт), у = ет.
32. (aev + Ъх)у'х = 1.
Solution in the implicit form:
e ifb^l,
ж = < l-o
U^C + ay) if 6 = 1.
33. (оеа + Ьх2)у'х = 1.
Solutions in the parametric form:
^ (^) Z = СгМт) + C2Y0(t)
and
where Jo and lo are Bessel functions, Iq and ATo are modified Bessel functions.
34. (aey + bx~x)y'x = 1.
Let a = ±A/B, b = ^p2A2. The solution in the parametric form is written as
here /(t) = / ехр(тт2) dr + С
35. (Ьеа« + c)y^ = eax+by - aeay.
This is a special case of equation 1.7.2.13 with /(?) = c, g(?) = 1,
© 1995 by CRC Press, Inc.

36. (aeax + bePy)yx = e™"*.
This is a special case of equation 1.7.2.9 with /(?) = ?-1.
37. (eax+-fy + a/3)y'x + bevx+Py + aa = 0.
This is a special case of equation 1.7.2.15 with f(y) = elv, g(x) = I
38. (eax+by + bx)y'x = ceax+by - ax.
This is a special case of equation 1.7.1.13 with /(?) = e^, д(?) = 1,
39. (еах+ьУ + by)y'x = ceax+by - ay.
This is a special case of equation 1.7.1.14 with /(?) = e^, g(?) = 1, h(?) =
40. (oeaa;ym + b)y'x = y.
This is a special case of equation 1.7.2.3 with /(?) = (a? + 6).
41. (eaa:y" + a/3)y'x + bevx+t3y + oa = 0.
This is a special case of equation 1.7.2.15 with f(y) = yn, g(x) = bevx.
42. (eaxyn + a/3)y'x + Ьхте^ + aa = 0.
This is a special case of equation 1.7.2.15 with f(y) = yn, g(x) = bxm.
43. (eaxym + mx)y'x = y(beanxynm - ax).
This is a special case of equation 1.7.2.17 with /(?) = ?, g(?) = 1, h(?,) = b?n.
44. x(xneay + ay)y'x = bxnmeamy - ny.
This is a special case of equation 1.7.2.16 with /(?) = ?, g(?) = 1,
45. (ож^е^ + bxe»y)y'x = evy.
This is the Bernoulli equation with respect to x = x(y) (see 1.1.5).
46. (axneXy + bxym)y'x =
This is the Bernoulli equation with respect to x = x(y).
47. (axnym + bxeXy)y'x = yk.
This is the Bernoulli equation with respect to x = x(y).
48. (axnym + bxyk)y'x = eXy.
This is the Bernoulli equation with respect to x = x(y).
49. {amxnym-1 + b)y'x + anxn~1ym + ceXx = 0.
This is a special case of equation 1.7.1.18 with f(y) = b, g(x) = ceXx.
© 1995 by CRC Press, Inc.

50. (amxnym-1 + ЬеХу)у'х + anxn-1ym + с = 0.
This is a special case of equation 1.7.1.18 with f(y) = beXy, g(x) = с
51. (amxnym-1 + byk)y'x + anxn-1ym + ceXx = 0.
This is a special case of equation 1.7.1.18 with f(y) = byk, g(x) = ceXx.
52. [(ож + by)n + beax]y'x = c(ax + by)m - aeax.
This is a special case of equation 1.7.2.14 with /(?) = f\ g(x) = 1, /i(?) = c?m.
53. [(ож + by)n + beay]y'x = c(ax + by)m - aeay.
This is a special case of equation 1.7.2.13 with /(?) = ?n, g(x) = 1, h(?) = c?m.
1.5.3. Equations Containing Hyperbolic Functions
1. y'x = ocosh(Ay) + bcosh.(vx).
This is a special case of equation 1.7.2.18 with f(x) = 0, g(x) = a, h{x) = bcosh(vx).
2. y'x = osinh(Ay) + bsinhfra).
This is a special case of equation 1.7.2.18 with f(x) = a, g(x) = 0, h(x) = bsmh(yx).
3. y'x = axn cosh(Ay) + bxm.
This is a special case of equation 1.7.2.18 with f(x) = 0, g(x) = axn, h(x) = bxm.
4. y'x = axn sinh(Ay) + bxm.
This is a special case of equation 1.7.2.18 with f(x) = ax", g(x) = 0, h{x) = bxm.
5- y'x = ауг+п + by + c sinh( Аж)у1-".
This is a special case of equation 1.7.1.4 with f(x) = a, g(x) = b, h(x) = csinh(Aa;).
6. y'x = ay1+n + b зтЬ(Аж)у + су1-"-.
This is a special case of equation 1.7.1.4 with f(x) = a, g(x) = 6sinh(Aa;), h{x) = с
7. y'x =ycosha;(aj/r'msinh"a; + bym).
This is a special case of equation 1.7.2.20 with /(?) = a?n + b?.
8. y'x =ysinhs(oj/nmcosh"a; + bym).
This is a special case of equation 1.7.2.22 with /(?) = a(,n + b?.
9. жу^, = (ож" cosh у + b) cothy.
This is a special case of equation 1.7.2.23 with /(?) = a? + b.
© 1995 by CRC Press, Inc.

10. xy'x = (axn sinh у + b) tanh y.
This is a special case of equation 1.7.2.21 with /(?) = a? + b.
11. (aym cosh ж + b)y^, = ym+1 sinh ж.
This is a special case of equation 1.7.2.22 with /(?) = ?(a? + 6).
12. (oym sinh x + b)y'x = ym+1 cosh ж.
This is a special case of equation 1.7.2.20 with /(?) = ?(a? + 6).
13. (axn + bx cosh™ y)y4 = yk.
This is the Bernoulli equation with respect to x = x(y) (see Subsection 1.1.5).
14. (ox" + Ьж tanh™ y)y'x = yk.
This is the Bernoulli equation with respect to x = x(y).
15. (axn + bx cosh y)y'x = coshfc(Ay).
This is the Bernoulli equation with respect to x = x(y).
16. (ax11 + bx tanh y)y'x = tanhfc(Ay).
This is the Bernoulli equation with respect to x = x(y).
17. {amxnym-1 + b)y'x + anxn~1ym + csinhk(\x) = 0.
This is a special case of equation 1.7.1.18 with f(y) = b, g(x) = csinh (Xx).
18. {amxnym-1 + b)y'x + anxn~1ym + ctanhfc^) = 0.
This is a special case of equation 1.7.1.18 with f(y) = b, g(x) = ctanh (Xx).
19. (axnym + bx)y'x = coshfc (Ay).
This is the Bernoulli equation with respect to x = x(y).
20. (axnym
This is the Bernoulli equation with respect to x = x(y).
21. (ож" cosh у + bx)y'x = sinhfc(Ay).
This is the Bernoulli equation with respect to x = x(y).
22. (ож" tanh у + bx)y'x = yk.
This is the Bernoulli equation with respect to x = x(y).
23. (amxnym-1 + bsinhfc y)y'x + anxn~1yrn + с = 0.
This is a special case of equation 1.7.1.18 with f(y) = b sinh y, g(x) = с
24. (amxnym-1 + btanhfc y)y'x + anxn~1ym + с = 0.
This is a special case of equation 1.7.1.18 with f(y) = b tanh y, g(x) = c.
© 1995 by CRC Press, Inc.

1.5.4. Equations Containing Logarithmic Functions
This is a special case of equation 1.7.2.3 with /(?) = ln? + /?.
2. y^ = axkn-1ykm+1(n\nx +
This is a special case of equation 1.7.1.3 with /(?) = a?fe
3. y'x = axny In2 у + Ьж^у In у + cxky.
This is a special case of equation 1.7.3.1 with f(x) = axn, g(x) = bxm, h(x) = cxk
4. xy'x = (ay + n In x)m +/3.
This is a special case of equation 1.7.2.4 with /(?) = lnm ? + /3.
5. xy'x = y(n In x + m In y).
This is a special case of equation 1.7.1.3 with /(?) = ln?.
6. mxy'x = axsyk(n In ж + m In y) — ny.
This is a special case of equation 1.7.1.5 with f(x) = J^xs~1, g(?) = ln^.
7. (ха + Ь)у'х =ужа-1 + сAпу-1пж).
This is a special case of equation 1.7.1.12 with /(?) = 6, p(^) = cln?, /i(^) = 1.
8. ж(ау + f3)y'x = n In ж + (а — n)y.
This is a special case of equation 1.7.2.16 with /(?) = /3, p(?) = 1, h((,) =
9. ж(о + mxk)y'x = y(bn In ж + bmlny — rafe).
This is a special case of equation 1.7.1.15 with /(?) = a, #(?) = 1, h((,) = bln
10. ж(о + myk)y'x = y(bnlnx + brainy — nyk).
This is a special case of equation 1.7.1.16 with /(?) = a, #(?) = 1,
11. (огаж^у-1 + b)y4 + anxn-1ym + c\nk(Xx) = 0.
This is a special case of equation 1.7.1.18 with f(y) = b, g(x) = clnfe(Aa;).
12.
Solution in the parametric form:
„Ьт
ОТ II 7 i SI\ 1
x = e I — / (XT + С¦ 1 —— шт, у = т.
13. жAп у)у^, = у(ахпкук + Ъхпу) — ny In у.
This is a special case of equation 1.7.3.7 with /(?) = a?fe + 6?, m = 1.
© 1995 by CRC Press, Inc.

14. x(a + m In y)y'x = у(bxnym — n In у + с).
This is a special case of equation 1.7.3.9 with /(?) = a, #(?) = 1, /i(?) = 6? + a
15. (аж™ + Ьж In™ y)y^ = lnfc(Ay).
This is the Bernoulli equation with respect to x = x(y).
16. x(axnym + m\nx)y'x = y(bxnkymk -nlnx).
This is a special case of equation 1.7.3.10 with /(?) = a?, g(?) = 1,
17. x(axnym + mlnt/)t? = y(bxnkymk - nIny).
This is a special case of equation 1.7.3.9 with /(?) = a?, p(^) = 1,
18. (om"|/ra-1 + b \nk y)y'x + anxn~1ym + с = 0.
This is a special case of equation 1.7.1.18 with f(y) = blnk y, g(x) = с
19. (axn lnm у + bx)y'x = lnfc(Ay).
This is the Bernoulli equation with respect to x = x(y).
20. (axn In™ у + Ьж lnfc y)y'x = ys.
This is the Bernoulli equation with respect to x = x(y).
1.5.5. Equations Containing Trigonometric Functions
1- y'x = a cos(oy) + /3 cos(bx).
This is a special case of equation 1.7.4.11 with f(x) = a, g(x) = 0, h(x) = Ccos(bx).
2. y'x = эт(аж) cos(by) + соэ(ож) sin(by).
This is a special case of equation 1.7.1.1 with /(?) = sin?, с = 0.
3. y'x = atan(focy).
The solution is given by the relation
I exp(-ji2) cos{yabxt) dt = сещ>[-^аЬх2), where
Jo
b
= y\l—.
4. y'x = bxn cos(oy) + cxm.
This is a special case of equation 1.7.4.11 with f(x) = bxn, g(x) = 0, h{x) = cxm.
5. yfx = bxnsin(ay) + cxm.
This is a special case of equation 1.7.4.11 with f(x) = 0, g(x) = bxn, h(x) = cxm.
© 1995 by CRC Press, Inc.

6. y'x = у cos x(aynm sin™-1 х + Ьут).
This is a special case of equation 1.7.4.4 with /(?) = a?n + &?.
7. y^ = у sin жСау"™ cos"-1 ж + bym).
This is a special case of equation 1.7.4.3 with /(?) = a?n +
„ , sin2 у cos2 у
8- Ух = a , +b
cos2 ж sin2 ж
This is a special case of equation 1.7.4.14 with /(?) = a? + &;.
9. y^ = оу1+" + Ьу + с5т(Лж)у1-".
This is a special case of equation 1.7.1.4 with f(x) = a, g(x) = b, h(x) = csin(Aa;).
10. y'x = ay1+n + b sin(Xx)y + cyx~n.
This is a special case of equation 1.7.1.4 with f(x) = a, g(x) = 6sin(Aa;), h(x) = с
11. жу^, + о эт(Ьж + су) = 0.
bx ~\~ cxi
The substitution w = x tan leads to the Riccati equation of the form 1.2.2.22:
2xw'x - bw2 + 2(ac - l)w - bx2 = 0.
12. жу^, = ож2 tan(by) + y.
The substitution у = xw leads to an equation of the form 1.5.5.3: w'x = ata,n(bxw).
13. жу^, = ож" cos2 у + b cos у sin у.
This is a special case of equation 1.7.4.8 with /(?) = -j(aC + &)•
14. жу^, = ож" sin2 у + b cos у sin y.
This is a special case of equation 1.7.4.7 with /(?) = -j(aC + &)•
15. жу^, = ож sinfc у cos2"k у — n sin 2y.
This is a special case of equation 1.7.4.18 with f(x) = axm~2nk~1, g(?) = ?k.
16. A + tan2 y)y'x = a tan у + b tan у + еж" tan1"™ у.
This is a special case of equation 1.7.4.19 with f(x) = a, g(x) = b, h(x) = cxn.
17. (amxnym-1 + b)y'x + anxn-1ym + csink(\x) = 0.
This is a special case of equation 1.7.1.18 with f(y) = b, g(x) = csinfe(Aa;).
18. (amxnym-1 + b)y'x + anxn~1yrn + ctanfc^) = 0.
This is a special case of equation 1.7.1.18 with f(y) = b, g(x) = ctanfe(Aa;).
© 1995 by CRC Press, Inc.

19. (axnym + Ьх)у'х = cosk(\y).
This is the Bernoulli equation with respect to x = x(y) (see 1.1.5).
20. (axnym + bx)y'x = tanfc(Ay).
This is the Bernoulli equation with respect to x = x(y).
21. (aym cos x+ b)yfx=ym+1 sin x.
This is a special case of equation 1.7.4.3 with /(?) = ?(a? + Ь)~г.
22. (aym sin x + b)y'x = ym+1 cos x.
This is a special case of equation 1.7.4.4 with /(?) = ?(a? +
23. (axn + bx cos™ y)y^ = yk.
This is the Bernoulli equation with respect to x = x(y).
24. (axn + bx cosm y)y'x = cosk(Xy).
This is the Bernoulli equation with respect to x = x(y).
25. (amxnym-1 + bcosk y)y'x + anxn-1ym + с = 0.
This is a special case of equation 1.7.1.18 with f(y) = bcosky, g(x) = с
26. (axn cosm у + bx)y'x = cosk(Xy).
This is the Bernoulli equation with respect to x = x(y).
27. (axn + bx tan y)y'x = yk.
This is the Bernoulli equation with respect to x = x(y).
28. (axn + bx tan y)y'x = tanfc (Ay).
This is the Bernoulli equation with respect to x = x(y).
29. (amxnym-1 + btank y)y'x + апхп-1ут + с = 0.
This is a special case of equation 1.7.1.18 with f(y) = 6tanfe y, g(x) = с
30. (axn tan у + bx)y'x = tanfc (Ay).
This is the Bernoulli equation with respect to x = x(y).
1.5.6. Equations Containing Combinations of Exponential, Hyperbolic,
Logarithmic, and Trigonometric Functions
1. y'x = axneXy +b\nmx.
This is a special case of equation 1.7.2.5 with f(x) = axn, g(x) = blnm x.
© 1995 by CRC Press, Inc.

2. y'x = a lnn(vx)eXy + bxm.
This is a special case of equation 1.7.2.5 with f(x) = a\nn(yx), g(x) = bxm.
3. у'х =
This is a special case of equation 1.7.2.2 with /(?) = aln
4. y'x = ae-Xx(Xx
This is a special case of equation 1.7.2.1 with /(?) = alnm?.
5. y'x = ay In2 у + by In у + ceXxy.
This is a special case of equation 1.7.3.1 with f(x) = a, g(x) = b, h(x) = ceXx.
6. y'x = ay In2 у + beXxy In у + су.
This is a special case of equation 1.7.3.1 with f(x) = a, g(x) = beXx, h(x) = с
7. y'x = aev sin ж + btana;.
This is a special case of equation 1.7.5.6 with /(?) = a? + b.
8. y^, = (oe1 sin у + b) tany.
This is a special case of equation 1.7.5.4 with /(?) = a? + 6.
9. y^ = aex sin2 у + be~x cos2 y.
This is a special case of equation 1.7.5.8 with /(?) = -j(aC + &/?)•
10. y^ = a cos"(/ita;)eAj' + bxm.
This is a special case of equation 1.7.2.5 with f(x) = acosn(/ix), g(x) = bxm.
11. y'x = axneXy +bcosm(ixx).
This is a special case of equation 1.7.2.5 with f(x) = axn, g(x) = bcosm(iix).
12. y'x = axneXy +btanm(txx).
This is a special case of equation 1.7.2.5 with f(x) = ax", g(x) = bta,nm(iix).
13. y'x = a tan"(/ita;)eAj' + bxm.
This is a special case of equation 1.7.2.5 with f(x) = ata,nn(/ix), g(x) = bx
14. y'x = AeXx cos(oy) + Be** sin(oy) + AeXx.
The substitution w = tan(-jay) leads to a linear equation: w'x = aBe^xw + aAeXx.
15. y'x = osin(/ita;) sinh(Ay) + bcos(/ita;) cosh(Ay).
This is a special case of equation 1.7.2.18 with f(x) = asm(/ix), g(x) = bcos(/ix),
h(x) = 0.
© 1995 by CRC Press, Inc.

16- Угх = ay \п2 у + by \ny + сsinn(Xx)у.
This is a special case of equation 1.7.3.1 with f(x) = a, g(x) = b, h(x) = csinn(Aa;).
17. A + tan2 y)y'x = a tan1+r™ у + b tan у + ceXx tan1"™ y.
This is a special case of equation 1.7.4.19 with f(x) = a, g(x) = b, h(x) = ceXx.
18. (aex cos у + b)y'x = cot y.
This is a special case of equation 1.7.5.5 with /(?) = (a? + b)~1.
19. (aex sin у + b)y'x = tan у.
This is a special case of equation 1.7.5.4 with /(?) = (a? + b)~1.
20. (aey cos x + b)y'x = tan x.
This is a special case of equation 1.7.5.6 with /(?) = (a? + b)~1.
21. (aea sin ж + b)y'x = cot ж.
This is a special case of equation 1.7.5.7 with /(?) = (a? + 6).
22. (eaxyn + a/3)y'x + be^ In™ ж + oa = 0.
This is a special case of equation 1.7.2.15 with f(y) = yn, g(x) = blnm x.
23. (eaxyn + a/3)y'x + be^ cosm ж + oa = 0.
This is a special case of equation 1.7.2.15 with f(y) = yn, g(x) = bcosm x.
24. (eax cos" у + aC)y'x + be^ сов^Лж) + aa. = 0.
This is a special case of equation 1.7.2.15 with f(y) = cos™ y, g(x) = bcosm(\x).
1.6. Equations Not Solved for Derivative
1.6.1. Equations of the Second Degree in y'x
1. (y'J2 = ay + bx2.
See equation 1.6.3.43.
2. (y'xJ =y + ax2 + bx + c.
The substitution w = 2л/у + ax2 + bx + с leads to an equation of the form 1.3.1.2:
ww'x — w = Aax + 2b.
3- (y'J2 = ay3 + by + c.
dy
Solution: x = С ±
\/ay3 + by
© 1995 by CRC Press, Inc.

See equation 1.6.3.26.
5- (У'х? = ау-
The substitution aw = 2\/ay-\-b^/x + с leads to the Abel equation of the form
1.3.1.32: ww'x -w = ba^x'1/2.
6- (y'xf + ay'x + by = O.
Solution in the parametric form:
bx = -2t-alnt + C, by=-t2-at.
7- (v'xJ + аУУх = bx + c.
Differentiate the equation with respect to x, take у as the independent variable, and
assume ? = y'x. As a result we obtain a linear equation with respect to у = y(?):
«2 - ЪЦ + a?y + 2?2 = 0.
8- Ю2 + axy'x + by + ex2 = 0.
The transformation x = e*, у = х2и leads to an autonomous equation:
/ / a \2 a2
[u't + 2u+-j = — -c-bu.
Having extracted the root and carried over the terms 2u+ ^a from the left-hand side
to the right-hand side, we obtain an equation of the form 1.1.2.
9- У = xy'x + ax2 + b(y'xJ + cy'x + d, аф О.
Differentiating with respect to x and changing to new variables t = y'x and w(t) = —2ax,
we arrive at the Abel equation of the form 1.3.1.2: ww't — w = —Aabt — 2ac.
Ю- (У'хJ + (ax + b)y'x - ay + с = 0, а ф 0.
Solutions:
у = (ax + b)C + aC2 + ca~x and Aay = 4c- (ax + bJ.
!!• (УхJ + (ay + bx)y'x + abxy = 0.
This equation can be factorized:
Therefore, the solutions are
у = Ce~ax and у = -\bx2 + С
© 1995 by CRC Press, Inc.

12. (у'ху + ах2у'х + Ьху = 0.
The transformation z = In ж, и = ух~3 leads to an equation not depending implicitly
on z:
(u'zJ + (a + 6u)u'z + Ca + b + 9u)u = 0.
Rewriting the latter equation to solve for u'z, we obtain an equation of the form 1.1.2.
13. Ю- = у + ая
The substitution
leads to the Abel equation of the form 1.3.1.10:
ww'x -w = — m '2x + 2a(m + l)xm.
14. (y'xJ = Xy + ож2 + bxm+1 + с
With A^O, the substitution Aw; = 2(Ay + ax2 + bxm+1 + cI'2 leads to the Abel
equation:
2x + 2b\-2(m + l)xm,
which is outlined in Subsection 1.3.1.
The special cases of the original equation are equations 1.6.1.1, 1.6.1.2, 1.6.1.4,
1.6.1.5, and 1.6.1.13.
Solution in the parametric form:
— (J I /7 lyi ( -f- I \ / 4-Z I 1 I ^i — fi-f-
16. x(y'xJ = axy + b.
See equation 1.6.3.32.
17. x{y'x) = axy + bx + с, о ф 0.
The substitution aw = 2\/ay + b + ex'1 leads to the Abel equation of the form
1.3.1.33: ww'x -w = -2ca~2x-2.
18. x(y'xf - ayy'x + b = 0.
With а ф 1, the solution in the parametric form is written as
x = Ctk H 12, aty = xt2 + b, where к = .
2a — 1 a—I
With a = 1, the solution is C(y — Cx) = b.
© 1995 by CRC Press, Inc.

19. x(y'xJ + ayy'x + bx = O.
With а ф — 1, the solution in the parametric form is written as
eve
In addition, there is the solution у = ±x\
V
V a+1
With a = — 1, the solution in the parametric form is written as
= С*ехР(-|-), y = x(t+±).
20. х(у'хJ - уу'х + ау = 0.
Solution in the parametric form:
x = C(t- a) exp(-i/a), у = Ct2 exp(-i/a),
In addition, there is the solution у = 0.
21- x(y'xJ - yy'x + ax2y'x + Ьу'х + с=0, а ф 0.
Divide the both sides by y'x and differentiate with respect to x. Changing to new
variables t = y'x and w(t) = —2ax, we arrive at the Abel equation of the form 1.3.1.33:
ww't — w = act~2.
22. y(y'xJ + axy'x + by = 0.
Solution in the parametric form is defined by the relations
a + 26
axt + у (b + t2) = 0, Cy (t2 + a + b)m = tb/(a+b), where m =
2(a + b) '
In addition, there is the solution у = ±x\f—a — b corresponding to the limit С —>• oo.
23. x(y'xJ + (a - y)y'x + b = 0.
Solutions: C(Cx — у + a) +b = 0 and (y — aJ = Abx.
24. ax(y'xJ + (bx — ay + k)y'x — by = 0.
kC
Solution: у = Cx -\ ———.
aC + b
In addition, there is the exceptional solution which may be written in the para-
parametric form as
_ bk _ kt
X~~ {at + bJ ' V~Xt+laTb'
25. ax(y'xJ - (ay + bx-a- b)y'x + by = 0.
DifFeretiating with respect to x and factorizing, we obtain
{2axy'x -ay-bx + a + b)yxx = 0.
Equating both factors to zero and integrating, we arrive at the solutions:
у = Cx -\ ^r—p- and (ay + bx — a — bJ — ^abxy = 0.
aC — b
© 1995 by CRC Press, Inc.

26. x(y'xJ + ayy'x + bxnym = 0.
The substitution x = e* leads to an equation of the form 1.6.1.56: (y't) + ayy't +
(i)
27. x2(y'xJ - Bxy + a)y'x + y2 = 0.
Solutions:
= aC2x + aC and y = .
Ax
28. ax2(y'xJ - 2axyy'x +y2 - 0@ - l)x2 = 0.
Solutions:
у ± уУ + ax2 = Cx1+k, where к = у/(a - I)/a.
29. (a2 - I)x2(y'xJ + 2xyy'x - y2 + a2x2 = 0.
Solution in the parametric form:
x = C(t2+ l)~1'2(t + yt2+ l) °, у = xt + axyt2
The equation can be factorized:
(y'x+ay3)(x2y'x+b)=0.
Equating each of the factors to zero, we obtain the solutions:
y2 = 2ax + С and у = b/x + C.
31. axy(y'xJ - (ay2 + bx2 + k)y'x + bxy = 0.
This differential equation presents an equation of curvature lines of a surface defined
by the relation
Ax2 + By2 + Cz2 = 1,
where
a = AB(C-B), b = AB(A-C), k = C(B-A).
Solutions: (aC - b)y2 = C(aC - b)x2 - kC and ay2 = bx2 ± 2xy^bk - k.
32. y2(y'xJ = ax2y2 + b.
See equation 1.6.3.34.
33. у2(ухJ = ах-2/5у2+Ь.
See equation 1.6.3.28.
34. y2(y'xJ + 2axyy'x + A - a)y2 + ax2 + (a - l)b = 0.
Solutions:
y2+ax2-b = (a-l)(x + CJ and y2 +ax2 -b = 0.
© 1995 by CRC Press, Inc.

35. (о - Ь)у2(у'хJ - 2Ъхуу'х + ау2 - Ьх2 - аЬ = 0.
Solutions:
х2 + у2 = Сх + Ь- — с2, and (а - Ь)у2 - Ьх2 = (а - b)b.
4а
36. (х2-а)(у'хJ-2хуу'х-х2 = 0.
Solving for у, differentiating with respect to x, and setting w(x) = y'x, we obtain a
factorized equation:
(xw'x — w)(x2w2 + x2 — aw2) = 0.
Equating each of the factors to zero, we arrive at the solutions:
у = —— (ж2 - а- С2) and у2 + х2 = a (y^O).
37. (x*-a2)(yxJ + 2xyyx + y2 = 0.
The equation can be factorized:
{xy'x + ay'x + y){xy'x - ay'x + y) = 0.
Equating each of the factors to zero, we obtain the solutions:
(x + a)y = С and (x — a)y = C.
38. (x2 + a)(y'xJ-2xyy'x + y2 + b = 0.
Differentiating with respect to x, we obtain a factorized equation:
[(x2 + a)y'x - xy]y%x = 0.
Therefore, the solutions of the original equation are
y = Cix + C2, where aC2 + C\ + b = 0; bx2 + ay2 + ab = 0.
39. (ay-x2)(yxJ + 2xyyx-y2 = 0.
Solution: (Cy + xJ = Aay.
40. (ay2 + bx)(y'xJ = 1.
See equation 1.6.3.44.
41. (ax2 + by)(y'xJ = x2y.
See equation 1.6.3.46.
42. (axy + b)(y'xJ=y.
See equation 1.6.3.33.
© 1995 by CRC Press, Inc.

43. (у2 - а2х2)(у'хJ + 2хуу'х + A - а2)*2 = 0.
Solution in the parametric form:
Ct С
С
x=
i2 + 1 Vi2 + 1
44. (oy - bxJ[a2(y'xJ + b2] - k2(ay'x + bJ = 0.
Solve the equation for ay — bx and differentiate with respect to x. Setting w(x) = y'x,
we obtain a factorized equation with respect to w(x):
(aw - b)[(a2w2 + b2f/2 ± abkw'x] = 0.
Equating each of the factors to zero and integrating, we arrive at the solutions:
(bx - Cf + (ay - Cf = k2 and ay - bx = ±kV2.
45. x3(y'xJ +x2yy'x + a = 0.
Solutions:
Cxy = C2x + a and xy2 = 4a.
46. xy2(y'xJ = ay2 + bx.
See equation 1.6.3.45.
47. (ax2y2 + b)(y'xJ=x2.
See equation 1.6.3.35.
48. (xy'x + aJ — lay + x2 = 0, а ф 0.
The substitution 2ay — x2 = u2 leads to the equation xuu'x — a(u — a) + x2 = 0.
Next assuming и — a = xw(x), we obtain (xw + a)w'x + w2 + 1 = 0. Taking w as the
independent variable, we arrive at a linear equation whose solution is
x= (w2 + 1)/2 [C -a\n(w + yV + 1)].
49. (xy'x + y + laxJ = 4(xy + ax2 + b).
The substitution и = xy-\- ax2 + b leads to an equation of the form 1.1.2: u'x
50. (a^y + bx)(y'xJ = 1.
See equation 1.6.3.27.
51. (ож2у3/5 + by)(y'xJ = x2y.
See equation 1.6.3.29.
© 1995 by CRC Press, Inc.

52. (о2ж + b2y + с2)(у'хJ + (агх + Ьгу + сг)у'х + аох + Ьоу + с0 = 0.
The Legendre transformation x = u't, у = tu't — и (у'х = i) leads to a linear equation:
[f(t)+tg(t)}u't=g(t)u + h(t),
where
f(t) = a2t2 + cut + a0, g{t) = b2t2 + b1t + b0, h(t) = -c2t2 - at - c0.
53. (y'xJ = аеУ + b.
See equation 1.6.3.3 with к = 2.
54. (y'J2 = a + bex.
See equation 1.6.3.4 with к = 2.
55. (y'J2 = ay2 + bex.
See equation 1.6.3.8 with к = 2.
56. (y'xJ + ayy'x + beXxym = 0.
With m^2, solving for y'x and performing the substitution w = e^xym~2, we arrive
at an equation with separated variables of the form 1.1.2:
w'x = Xw H — (a ± у a2 — 4bw )w.
With m = 2, solving the original equation for y'x, we obtain an equation with
separation of variables:
2y'x =y(-a±\fa?
57. x2(y'xJ = ax2ey+b.
See equation 1.6.3.9 with к = 2.
58. (аеУ + Ьх2)(у'хJ = 1.
See equation 1.6.3.9 with к = —2.
59. (аеху2+Ь)(у'хJ=у2.
See equation 1.6.3.8 with к = —2.
60- (у'хJ = ау + Ыпх.
See equation 1.6.3.13.
61. (y'xJ = Xy + a\nx + b, Л фО.
The substitution Xw = 2\JXy + alnx + b leads to the Abel equation of the form
1.3.1.16: io< -w = 2aX~2x-1.
62. (y'xJ - xyy'x + y2 ln(oy) = 0.
Solutions:
ay = ехр(Сж — С2) and ay = ехр(^-ж2).
63. (о In» + Ъх){у'хJ = 1.
See equation 1.6.3.14.
© 1995 by CRC Press, Inc.

1.6.2. Equations of the Third Degree in y'x
This is a special case of equation 1.8.1.13 with f(w) = w3.
2- a(y'xf + by'x = x.
This is a special case of equation 1.8.1.1 with f(w) = aw3 + bw.
3. a(y'xf + Ьу'х = у.
This is a special case of equation 1.8.1.2 with f(w) = aw3 + bw.
4- a(v'xf + xv'x = v-
This is a special case of equation 1.8.1.6 with f(w) = aw3.
5- a(v'xK + bxv'x = у-
This is a special case of equation 1.8.1.7 with f(w) = bw, g(x) = aw3.
6- Ю3 - axy'x + x3 = 0, о ф 0.
Solution in the parametric form:
at „ a2 4i3 + l
с+
+ 1 ' У ' 6 (*з + 1J '
~ axyy' + lay1 =
7- (y'xf ~ axyy'x + lay1 = 0.
Differentiating with respect to x and eliminating y, we obtain a factorized equation
with respect to w(x) = y'x:
[2(«4J — axw'x + aw](9w — ax2) = 0.
Equating each of the factors to zero and integrating, we find the solutions:
y=±C(x-CJ and У=^х3.
8. a(y'xf + b(y'xf = x.
This is a special case of equation 1.8.1.1 with f(w) = aw3 + bw2.
9- a(y'xf + b(y'xf = y.
This is a special case of equation 1.8.1.2 with f(w) = aw3 + bw2.
Ю- Ю3 + a(y'xJ + by + abx + d = 0.
Solution in the parametric form:
2bx = -Ы2 + 2at - 2a2 ln(i + a)+C, by = -abx - t3 - at2 - d,
In addition, there is the solution by = —abx — d.
© 1995 by CRC Press, Inc.

11. a(y'xf + b(y'xf +Cyx=y + d.
Solution in the parametric form:
x = С + \at2 + 2Ы + cln \t\, у = at3 + bt2 +ct-d.
12. a(y'xf + Ьх(у'хJ = у.
This is a special case of equation 1.8.1.7 with f(w) = bw2, g(x) = aw3.
13. ax(y'xf + by'x = y.
This is a special case of equation 1.8.1.7 with f(w) = aw3, g(w) = bw.
14. ax(y'xf + Ь(у'хJ = у.
This is a special case of equation 1.8.1.7 with f(w) = aw3, g(w) = bw2.
15. (ax + by + c)(y'xK = ax + fly +7.
Dividing both sides by ax + by + с and raising to the power 1/3, we finally arrive at
an equation of the form 1.7.1.6 with f(w) = w~x/3.
16. ax3/2(y'xf +2xy'x=y.
Solution: (y - aC3'2f = 4Сж.
17. (x2 - а2)(у'хУ + bx(x2 - а2)(у'хУ + y'x + bx = 0.
The equation can be factorized:
(y'x + bx)[(y'xJ(x2-a2) + l}=0,
whence we find the solutions:
1 x
у = bx2 + С and у = ± arcsin \- C.
2 a
18. axn(y'xf + xy'x = y.
This is a special case of equation 1.8.1.8 with f(w) = awn.
19. (xy'x - yK + ay + bx = 0.
This is a special case of equation 1.8.1.10 with f(w) = 1, g(w) = a, h(w) = b, n = 3.
20. (xy'x - yK + ayy'x + bx = 0.
This is a special case of equation 1.8.1.10 with f(w) = 1, g(w) = aw, h(w) = b, n = 3.
21. (xy'x - yK + axy'x + by = 0.
This is a special case of equation 1.8.1.10 with f(w) = 1, g(w) = b, h(w) = aw, n = 3.
© 1995 by CRC Press, Inc.

1.6.3. Equations of the Form (yfx) = f(y) -\-g(x)
Preliminary comments.
1. In the general case, the equation
(y'x)k = f(y) + g(x) A)
is reduced, with the aid of the transformation
to the same form
(u')k = F(u) -
where functions F = F(u) and G = G(t) are defined parametrically by the following formulae:
u = Jlf(y)}-1'kdy,
2. Taking у as the independent variable, we obtain from equation A) an equation of
the same class for x = x(y):
(x'y)-1/k = g(x) + f(y)-
3. The equation
y'x = aVy + 9(x) (k = l, f = ay/y)
is reduced, with the aid of the substitution w(x) = 2а~1л/у, to the Abel equation ww'x — w =
2a~2g(x) which is outlined in Subsection 1.3.1.
4. The equation
y'x = y-1 + g(x) (k = l, f = y~1)
is an alternative form of writing the Abel equation yy'x = g(x)y + 1 which is outlined in
Subsection 1.3.2.
5. The equation
y'x = ays+g{x) (k = l, f = ays)
is reduced, with the aid of the substitution w = у — f g(x) dx followed by raising both sides
of the equation to the power of 1/s, to an equation of the class in question:
p
(w'x) = w + / g(x) dx.
J
6. The equation
{y'xf = ay + g{x) (/c = 2, / = ay, афО)
is reduced, with the aid of the substitution aw = 2\Jay + g(x), to the Abel equation of the
second kind:
ww'x = w + <p(x), where ip = 2a~ g'x(x),
© 1995 by CRC Press, Inc.

which is outlined in Subsection 1.3.1.
7. The equation
(y'xI/2 = ay + g(x) (A; = 1/2, f = ay)
is reduced, by squaring both sides and performing the substitution z = ay + g(x), to the
Riccati equation
z'x = az2 + g'x.
For some specific functions g = g(x), the solutions of the latter equation are given in
Section 1.2.
8. The equation
{y'xI/2 = ay1'2 + g{x) (к = 1/2, f = ay1'2)
can be reduced, by squaring both sides and performing the substitution у = ехр(а x)?, to
the Abel equation of the second kind:
а2х)д? + \ ещ>(-а2х)д2
(see Subsection 1.3.3).
9. The equation
1/2 ax {к = -1/2, g = ax)
can be reduced, by squaring both sides and performing the substitution v = f(y) + ax, to
the Riccati equation:
v'y = av2 + f'y.
For some specific functions g = g(x), the solutions of the latter equation are given in
Section 1.2.
10. For the sake of convenience, in Tables 1.5-1.9 are listed all the equations outlined
in Subsection 1.6.3. Five classification tables are given below which classfy the equations
wherein functions / and g are of the same form. The rightmost columns of the tables
present the numbers of equations where the corresponding solutions are given. After the
tables follow the equations—they are combined into groups so that the solutions of the
equations within each group are expressed in terms of the functions indicated before the
groups as a notation list.
1. (y'x)k = Ay* + B.
Solution
ion: x= f(Ays + B)~1/kdy
J
2.
Solution: y= f(A + BxrI/kdx
3. (y'x)k = АеУ + В.
Solution: x = f(Aev + В)~г'к dy + С
© 1995 by CRC Press, Inc.

TABLE 1.5
Solvable equations of the form (y'x) = Ays + Bxr
к
arbitrary
arbitrary
arbitrary
(fc^-1, 1)
arbitrary
arbitrary
-2
-2
-2
-2
-2
-2
-1
-1
-1
-1
-1
-1
-1
-1
s
arbitrary
(s^k)
arbitrary
к
1-k
0
1
-1
-1
-2/5
1/2
2
2
arbitrary
(s^O)
arbitrary
(s^-2,0)
-2
-2
-2
-1
-1/2
-1/2
r
fcs
/c — s
0
к
1 + k
arbitrary
1
-2
1
-2
1
-2
1
1
2
-1
1/2
2
1/2
-1
1/2
Equation
1.6.3.7
1.6.3.1
1.6.3.6
1.6.3.2
1.6.3.5
1.6.3.46
1.6.3.33
1.6.3.29
1.6.3.27
1.6.3.35
1.6.3.44
1.6.3.10
1.6.3.15
1.6.3.21
1.6.3.31
1.6.3.36
1.6.3.12
1.6.3.40
1.6.3.25
к
1
-1
-1/2
-1/2
1/2
1/2
1
1
1
1
1
1
1
2
2
2
2
2
2
s
1
1
arbitrary
(s^-1,0)
-1
1
1
-1
-1
-1
1/2
1/2
1/2
1/2
-2
-2
-2
1
1
1
r
1
1/2
1
1
arbitrary
(r^-1,0)
-1
-2
-1/2
1
-2
-1
-1/2
1
-1
-2/5
2
-1
1/2
2
Equation
1.6.3.23
1.6.3.42
1.6.3.17
1.6.3.38
1.6.3.16
1.6.3.37
1.6.3.20
1.6.3.39
1.6.3.22
1.6.3.30
1.6.3.11
1.6.3.24
1.6.3.41
1.6.3.45
1.6.3.28
1.6.3.34
1.6.3.32
1.6.3.26
1.6.3.43
4.
Solution: y=
f(A
BexI/kdx
5. (y'x)k = Ay + Bx.
Solution in the parametric form:
= [(Ат1/к + В)-1 dr + C, у = -|- [т - В [(Ат1/к + В)-1 dr - ВС1].
© 1995 by CRC Press, Inc.

TABLE 1.6
Solvable equations of the form
(y'x)k = АеУ + Bxr
к
arbitrary
arbitrary
-1
-1
-1
-1/2
l—i
Г
-к
0
-1
1
2
1
arbitrary
Equation
1.6.3.9
1.6.3.3
1.5.2.34
1.5.2.32
1.5.2.33
1.6.3.19
1.5.2.2
TABLE 1.8
Solvable equations of the form
(y'x)k = АеУ + Bex
к
-1
Equation
1.5.2.30
1.5.2.1
TABLE 1.7
Solvable equations of the form
(y'x)k = Ays + Bex
к
arbitrary
arbitrary
-1
1/2
1
s
к
0
arbitrary
1
-1
Equation
1.6.3.8
1.6.3.4
1.5.2.31
1.6.3.18
1.5.2.3
TABLE 1.9
Solvable equations containing
logarithmic function
Form of equation
{y'x)~2 = A\ny + Bx
(y',)-1 = Alny + Bx
{y'xf = Ay + В In x
Equation
1.6.3.14
1.5.4.12
1.6.3.13
6. (y'x)K = Ay i-«. + Bx ,
Solution in the parametric form:
X = <
where
А = а
fe+i
b(k - 1)
a(k +1)
fe-i
ks
7. (y'xy = Ays + Bx «•— , к ф s.
Solution in the parametric form:
У =
s-*)k -т\ dr,
k-s
© 1995 by CRC Press, Inc.

8. (y'x)k = Ayk + Be^.
Solution in the parametric form:
x = / [(A + Be-kTI/k - -i] dr + C
9. (y'x)k = Aey + Bx~k.
Solution in the parametric form:
З + Ае^у1
Solution: x = eBy (a f yse~By
> In the solutions of equations 11-14, the following notation is used:
F= [/2 l
11. y'x = Ay1/2 +
Solution in the parametric form:
x = a.Fexp(=FT2), у = Ь[2т ±
where A = ±2a~1b1/2, В =
12. (y'x)-1 = Ay-1+
Solution in the parametric form:
x = а[2т ± i^exp(TT2)]2, у =
where^ = T4a, В = ±2а1/2Ь~1.
13. Ю2 = Ay + B\nx.
Solution in the parametric form:
x = ai^exp(=FT2), у = Ь{[2т ± i^exp(=FT2)]2 ± 41n(aF) - 4т2},
where A = Aa~2b, В =
(У'х)~2 = Alny
Solution in the parametric form:
x = а{[2т ± i^exp(=FT2)]2 ± 41n(bF) - 4т2}, у = bFe
where A = ^fAaB, В =
© 1995 by CRC Press, Inc.

> In the solutions of equations 15-19, the following notation is used:
Jv{t) + C?Yv{t) for the upper sign,
Z =
' C\Iv{t) + C<iKv(j) for the lower sign,
where Jv and Yv are Bessel functions, Iv and Kv are modified Bessel functions.
Remark. The solutions of equations 15-19 contain only the ratio Z'T/Z = (lnZyT.
Therefore, for the sake of symmetric appearance, two "arbitrary" constants C\ and C%
are indicated in the definition of function Z (instead, we may set, for instance, C\ = 1
and C2 = C).
(y'J'1 = Ays + Bx2, зф-2,зфО.
Solution in the parametric form:
where v =^y, A = T^— ab'1'8, B =
2
1 fi 111 I Atl I TRt1^* T* ~^~ 1 С ~T~ fl
\&X/ if • 7 / 7/
Solution in the parametric form:
ж = ат2!У, у = 6т2!У [тAп Z); + rv ± ^-т2],
1 , , if (r + lNl!/2 r+1 r, .
where jy = , A = 6 — , B = + a"r6A.
r+1 L 2a J 2r
17- (У'ХГ1/2 = Ays + Bx, зф -1, в ^ 0.
Solution in the parametric form:
x = ar^[r(lnZyT + v± ^-r2], у = Ът2\
1 , s + 1 , sn n гг ( + l)V2
where v = -TT, A = T—ab-°B, В = a [--
Ю1/2 = Ау + Ве*.
Solution in the parametric form:
х = Ы(ат2),
where гу = 0, A = b~1(-^-I 2
19. (y'x)~1/2 = Aey + Bx.
Solution in the parametric form:
±] = Ып(т2),
where v = Q, А = т~ab^B, B = a~1(-—V .
Z\ V Zi /
© 1995 by CRC Press, Inc.

> In the solutions of equations 20-35, the following notation is used:
( C1J1/3(t) + C2Y1/3(t) for the upper sign,
I CiIi/3(t) + C2K1/z{t) for the lower sign,
where J1/3 andY1/% are Bessel functions, I1/3 andK1/3 are modified Bessel functions;
f/i = tZ't + \Z, U2 = U2 ± t2Z2, U3 = ±\t2Z3 - 2VXV2.
Remark. The solutions of equations 15-19 contain only the ratio Z'T/Z = (h\Z)'T.
Therefore, for the sake of symmetric appearance, two "arbitrary" constants C\ and C2
are indicated in the definition of function Z (instead, we may set, for instance, C\ = 1
and C2 = C).
20. y'x = Ay-1 + Bx~2.
Solution in the parametric form:
x =
у =
where A = 2a~1b2, В = т4
where A = Тт*-
21. (у'х) = Ay -
Solution in the parametric form:
22. y'x = Ay-1 + Bx.
Solution in the parametric form:
x = aT-2l3Z~1U1, y = bT-Al3Z~2U2, where A = ^fa
23. (y^,) = Ay + Bx—1.
Solution in the parametric form:
x = ат-^3г-2и2, y = bT~2'3Z-1U1, where A = 2a~4
24. y'x = Ay1'2 + Bx-1/2.
Solution in the parametric form:
x = aT-^3Z-2U2, y = bT-s'3Z-AU2, where A = 2a~1
25- «Г* = Ay-1/2 + Bx1/2.
Solution in the parametric form:
x = ат~8/3.?~4[/|, у = bT~4/3Z~2U2, where А = =р-^с
26. (y'xJ = Ay + Bx1/2.
Solution in the parametric form:
x = aT-^3Z-2U2, у = bT~s/3Z-4(U2 ± |т
where A = 4a~2b, В = т4а~
В = 2a2
= 2a~2b.
~1b1/2,
В =
© 1995 by CRC Press, Inc.

27. (ух)~2 = Ay1/2 + Bx.
Solution in the parametric form:
x = aT-*l3Z-\Ul ± f т2^), у =
where A = T-f-a&~1/2-B, В = Aab~2.
28. (y'xJ = Ay-2 + Bx-2/5.
Solution in the parametric form:
x = ат-ь^г-ъ/2и1/2, у = bT-^3Z-2(Ul ± f
where A = т^а-2'ъЪ2В, В = ^a
29. (у'хГ2 = Ау-2/5+
Solution in the parametric form:
x = aT-^3Z-2(U2 ± f T2^)^, у = 1
where A = ^а2Ь~^5, В = т^а2Ъ-2'ъА.
30. y'x = Ay1/2 + Bx-2.
Solution in the parametric form:
х = ат^г2и-\ у = Ът-^г-2и-2и1 where A = ifa^2, В = -АаЬ.
31. (y'J-1 = Ay~2+Bx1/2.
Solution in the parametric form:
U-2Ul y = br^3Z2U-\ where A=~Aab,
32. (yxJ = Ay +
Solution in the parametric form:
x = aTA/3Z2U-1, у = Ьт-А/3г-2и-2(и2-Ш3), where A=^-a~2b, B = AabA.
33. (y'x)~2 = Ay-1 + Bx.
Solution in the parametric form:
х = ат-4/3г-2и-2(и2-Ш3), у = Ьт4/3г2и~\ where A = 4,abB, B=^-ab~2.
34. (y'xJ = Ay~2 + Bx2.
Solution in the parametric form:
x = aT2'3ZU-1/2, у = br-^Z^U-'iU2 ~ AU3I'2,
where A = Aa2b2B, В = ^-а~Ч2.
35. (ухГ2 =
Solution in the parametric form:
x = ar-^Z^U^iU2 - W3I'2, у = bT2'3ZU-1/2,
where A = ^§-а2Ь~4, В = Aa2b2A.
© 1995 by CRC Press, Inc.

> In the solutions of equations 36-46, the following notation is used:
{C\TV + C<it~v for the upper sign,
C\ ъш(у In t) + Ci cos(jv In t) for the lower sign,
{A + v)C\Tv + A — v)C<it~v for the upper sign,
{C\ — vCi) sin(jvlnT) + (Ci + vC\) сов(гЛпт) for the lower sign,
Remark. The expressions for R and Q contains two "arbitrary" constants C\
an C2- One of them may be fixed to set it equal to any nonzero number (for example,
we may set C2 = ±1Л while another constant remains arbitrary.
36. (у'хГ1 = Ау-2+
Solution in the parametric form:
x = ar-^R^Q, у = Ьт2, v = y/\l - AAB\,
where A = ab, B = a~1b~1.
37. (y'xI/2 = Ay +
Solution in the parametric form:
where A = b~1(—— V 2, В = 1 T V abA.
\ 2) 2
1(—— V 2, В = 1
\ 2a)
38. (у'хГ1/2 = Ay'1 +
Solution in the parametric form:
where A = lTV abB, Д = а~1(-^-
Z \ ZO
2
1/2
39. y'x = Ay-1 +
Solution in the parametric form:
= ar2R2, y = brQ, where A = (-1 ± v2)^—, В = a~1/2b.
2a
40. (у'хГ1 = Ау-1/2
Solution in the parametric form:
7 2
= arQ, y = bT2R2, where A = ab~1/2, В = {-l±v2) — .
ZQ
© 1995 by CRC Press, Inc.

41. y'x = Ay1/2 + Bx.
Solution in the parametric form:
x = arR, y = bT2Q2, where A = 2(-l±v2)a~1b1/2, В = 4а6.
42. (у'хГ1 = Ау +
Solution in the parametric form:
x = ar2Q2, y = brR, where A = 4ab~2, В = 2(-l ± i/2)a1/2b~1.
43. (y'xJ = Ay + Bx2.
Solution in the parametric form:
x = arR, y = bT2[Q2-{-l±v2)R2}, where A = 16a~2b, B = (-l±v2)a~2bA.
44. (y'x)~2 = Ay2 + Bx.
Solution in the parametric form:
x = aT2[Q2-(-l±v2)R2}, y = brR, where A= (-I±v2)ab~2 В, B = 16ab~2.
45. (y'xJ = Ay-2 +
Solution in the parametric form:
x = ut2R2, y = bT[Q2-{-l±v2)R2}112,
where A= {-l±v2)a~1b2B, B = a~1b2.
46. (у'хГ2 = Ау-12
Solution in the parametric form:
x = aT[Q2-{-l±v2)R2}112, y = br2R2,
where A = a2b~\ В = (-1 ± v2)a2b-xA.
1.6.4. Other equations
1. у = xy'x + ax2 + Ь^/у^. + с, а ф 0.
Differentiating the equation with respect to x and changing to new variables t = y'x and
w(t) = —2ax, we arrive at the Abel equation of the form 1.3.1.32: ww't—w = —abt~x/2.
2- V = xy'x + ax2 + b(y'xJ + c(y'x)m+1 + d, а ф 0.
Differentiating the equation with respect to x and changing to new variables t = y'x
and w(t) = —lax, we arrive at the Abel equation:
ww't -w = -4a6i - 2ac(m + l)tm,
whose solvable case are outlined in Subsection 1.3.1.
© 1995 by CRC Press, Inc.

3. а(у'х)п
Solution in the parametric form:
with n ф —1, m ф —1,
x = atn + btm, y = C
^t +
n+l m+1
with n = —1, m ф —1,
x=j+btm, y = C ^p
4. a(t?r
Solution in the parametric form:
with n/1, m/1,
n — 1 m — 1
with n = 1, m ф 1,
ГН1'1, y = atn + btm;
— Г7 .. &m _! _
m — 1
5- У = xy'x + a(y'x)n.
Solution: у = Cx + aCn. In addition, there is the solution у = Ах п~г , where
6- У = xy'x + axn(y'x)m.
This is a special case of equation 1.8.1.8 with f(w) = awm.
7. у = axn(y'xJn + 2xy'x.
This is a special case of equation 1.8.1.9 with f(w) = awn.
8. y'x = axn(xy'x - y)m.
The Legendre transformation x = w't, у = tw't — w (y'x = t) leads to the equation
t = awm(w't)n. By integrating, we obtain the solution in the parametric form:
with m ф —n, n ф —1,
1 г 1 ~~
1 + n
with m = —n, n ф —1,
,lr ,1-| r , 1 -, r
x = C[ — ) exp 1[ — ) , y = C\t[ — ) — 1 exp
Ve/ [n+1 \ a/ J |_ V a I \ [n+lVa
with m ф —n, n = —1,
a m i
x = — [a(l - m) In |i| + C] 1~m , у = tx - [a(l - m) In |i| + C] 1~m ;
with m = 1, n = —1,
© 1995 by CRC Press, Inc.

9. x = aexp(Ay^) + Ьехр(цу'х).
This is a special case of equation 1.8.1.1 with f(w) = aexp(Aw) + Ьехр(/ш).
10. у = оехр(Лу4) + Ьех.р(цу'х).
This is a special case of equation 1.8.1.2 with f(w) = aexp(Aw) + Ьехр(/ш).
11. у = xy'x + ax
This is a special case of equation 1.8.1.8 with f(w) = aexp(Aw).
12. у = ажехр(Лу^,) + Ъех.р(цу'х).
This is a special case of equation 1.8.1.7 with f(w) = aexp(Aw), g(w) = bex
13.
Solution in the parametric form:
with а ф § and ф —1,
1
ж= YCt «+1 , y = (xt + lnt
at a
with а = 0,
t '
with a = —1,
and у = ln(-
14. у = жу^ + ах2 + biny^, + с, а ф 0.
Differentiating the equation with respect to x and changing to new variables t = y'x and
«;(?) = — 2ax, we arrive at the Abel equation of the form 1.3.1.16: ww't — w = —2abt~1.
15. у = xy'x + axn In™ (Xy'x).
This is a special case of equation 1.8.1.8 with f(w) = alnm(Xw).
16. y = xy'x + axn sin™ (fct?).
This is a special case of equation 1.8.1.8 with f(w) = asinm(kw).
17. y = xy'x + axn cosm(kyfx).
This is a special case of equation 1.8.1.8 with f(w) = acosm(kw).
18. y = xy'x + axn tanm(kyfx).
This is a special case of equation 1.8.1.8 with f(w) = at&nm(kw).
© 1995 by CRC Press, Inc.

1.7. Equations of the Form F(x,y)yx = G(x,y)
Containing Arbitrary Functions
Notation: f, g, and h are arbitrary composite functions whose argument, indicated
after the function name, may depend on both x an y.
1.7.1. Equations Containing Power Functions
With b = 0, we have an equation of the form 1.1.1. With b ф 0, the substitution
u(x) = ax + by + с leads to an equation of the form 1.1.2: u'x = bf(u).
2- y'x
The substitution и = у + axn + b leads to an equation of the form 1.1.2: u'x = f(u).
3. y'a = ^f(xnym)-
Homogeneous equation in the extended sense.
The substitution z = xnym leads to an equation with separation of variables:
xz'x = nz + mzf(z).
4- y'x = f()y+g()y + ()y
The substitution w = yn leads to the Riccati equation: w'x = nf (x)w2 +ng(x)w+nh(x).
5- V'x = ~ — -+ykf(x)g(xnym)-
m x
m x
The substitution z = xnym leads to an equation with separation of variables: z'x =
n—nk k+m—1
rax m f(x)z m g{z).
6- y* =
67 — cC ca — 1
With A = ap — ba ф 0, the transformation x = и -\ — , у = v{u) -\ —
yields
, / au+bv
vu = f[
¦ аи + [3v
Dividing both the numerator and denominator of the fraction on the right-hand side
by u, we obtain a homogeneous equation of the form 1.1.6.
With A = 0, b ф 0, the substitution v(x) = ax + by + c leads to an equation of the
form 1.1.2:
With A = 0, /3 ф 0, the substitution v(x) = ax + /3y + 7 also leads to an equation
of the form 1.1.2:
, , a t ( bv + ф - 67
v a + f3f[
f3f
(iv
© 1995 by CRC Press, Inc.

7. y'x = xn-1y1-mf(axn + bym).
The substitution w = axn + bym leads to an equation with separation of variables:
8- VnVrx + axn + g(x)f(yn+1 + ахп+г) = О.
The substitution w = yn+1 + axn+1 leads to an equation with separation of variables:
w'x + (n+l)g(x)f(w)=0.
9. [xnf(y) + xg(y)]y'x = h(y).
This is the Bernoulli equation with respect to x = x(y) (see 1.1.5).
10. [x2+xf(y)+g(y)]yx =
This is the Riccati equation with respect to x = x(y) (see Section 1.2).
V'x = [f(x)y + 9(x)} y/(y-a)(y-b).
The substitution u2 = (y — a)/(y — b) leads to the Riccati equation:
±2u'x = [bf(x) + g(x)]u2 - af(x) - g(x).
"¦ Ю
The substitution у = xt leads to the Bernoulli equation with respect to x = x{t):
[g(t)-tf(t)}x't =
13. [f(ax + by) + bxg(ax + by)]y'x = h(ax + by) — axg(ax + by).
The substitution t = ax + by leads to a linear equation with respect to x = x(t):
[af(t)+bh(t)]x't =
14. [f(ax + by) + byg(ax + by)]y'x = h(ax + by) — ayg(ax + by).
The substitution t = ax + by leads to a linear equation with respect to у = y(t):
[af(t) + bh(t)]y't = -ag(t)y + h(t).
15. x[f(xnym) + mxkg(xnym)]yx = y[h(xnym) - nxkg(xnym)\.
The transformation t = xnym, z = x~k leads to a linear equation: t[nf(t) + mh(t)]z't =
-kf(t)z - kmg(t).
16. x[f(xnym) + mykg(xnym)]y'x = y[h(xnym) - nykg(xnym)].
The transformation t = xnym, z = y~k leads to a linear equation: t[nf(t) + mh(t)]z't =
-kh(t)z + kng(t).
17. x[sf(xnym) - mg(xkys)]y'x = y[ng(xkys) - kf(xnym)].
The transformation t = xnym, w = xkys leads to an equation with separated variables:
tf(t)w't = wg(w).
© 1995 by CRC Press, Inc.

18- [/(У) + атхпут-1]у?а + д(х) + апхп-1ут = 0.
Solution: / f(y) dy + f g(x) dx + axnym = C.
df dg
19. f(x,y)yx+g(x,y) = 0, where —— = ——.
ox ay
Total differential equation.
Solution:
rv
f(xo,t)dt+ / g(t,y)dt = C,
where xq and yo are arbitrary numbres.
1.7.2. Equations Containing Exponential and Hyperbolic Functions
The substitution и = eXxy leads to an equation of the form 1.1.2: u'x = f(u) + Xu.
2. y'x =
The substitution и = eXyx leads to an equation with separated variables: xu'x =
\u2f(u) + u.
3. У'х = Vf(eaxym).
Exponential homogeneous equation.
The substitution z = eaxym leads to an equation of the form 1.1.2: z'x = az +
mzf(z).
4- У'х = -f(xneay).
x
Exponential homogeneous equation.
The substitution z = xneay leads to an equation with separated variables: xz'x =
nz + azf(z).
5- y'x = f(x)exy+g(x).
The substitution и = e~Xy leads to a linear equation: u'x = —Xg(x)u — А/(ж).
6- y'x = -^+f(x)g(xney).
The substitution z = xnev leads to an equation with separation of variables: z'x =
f(x)zg(z).
7- y'x = -— У + ykf(x)g(eaxym).
m
The substitution z = eaxym leads to an equation with separation of variables:
fe+m-l
(l-k)x\f(x)z
. m
\ Ot 1 k+m — l
2X =mexp —A — k)x \f(x)z m g(z).
© 1995 by CRC Press, Inc.

8- У'х = f(x)exy + д(х) + Н(х)е~хУ.
The substitution u = eXy leads to the Riccati equation: u'x = Xf (x)u2+Xg(x)u+Xh(x).
9. y'x = e^
The substitution w = aeax + berv leads to an equation with separated variables:
w'x = eax[aa + bpf(w)}.
Ю- V'x = f(v + aeXx + b) - a\eXx.
The substitution w = у + aeXx + b leads to an equation of the form 1.1.2: w'x = f(w).
11. y'x = -—+ ПХ ' .
yx +
ax xy
The substitution t = xneay leads to a linear equation with respect to у = y(t):
aHf(t)y't = -ny + af(t).
12. y'x =+ '
ax xy2
The substitution t = xneay leads to the Riccati equation: a2tf(t)y't = —ny2 + af(t).
13. [f(ax + by) + ЬеаУд(ах + by)]y'x = h(ax + by) - аеаУд(ах + by).
The transformation t = ax + by, z = e~ay leads to a linear equation: [af(t) + bh(t)]z't =
—ah(t)z + aag(t).
14. [f(ax + by) + beaxg(ax + by)]y'x = h(ax + by) - aeaxg(ax + by).
The transformation t = ax + by, z = e~ax leads to a linear equation: [af(t) + bh(t)]z't =
-af(t)z - abg(t).
15. [eaxf(y) + aC]y'x + e^gix) + aa = 0.
Solution:
/ e~pvf{y) dy+ I e-axg(x) dx - ae"^"^ = С
16. x[f(xneay) + ауд(хпеаУ)]у'х = Н(хпеаУ) - пуд(хпеаУ).
The substitution t = xneay leads to a linear equation with respect to у = y(t): t[nf(t) +
ah(t)}y't = -
17. [f(eaxym) + mxg(eaxym)]y'x = y[h(eaxym) - axg(eaxym)}.
The substitution t = eaxym leads to a linear equation with respect to x = x(t):
t[af(t) + mh(t)]x't = mg(t)x + f(t).
- VL = f(x) sinh(Ay) + g(x) cosh(Ay) + h(x).
The substitution и = eXy leads to the Riccati equation: 2u'x = A(/ + g)u2 + 2Xhu
49 ~ /)•
© 1995 by CRC Press, Inc.

19. y'x = /(ж) sinh2(Ay) + g(x) cosh2(Ay) + h(x) sinhBAy) + s(x).
The substitution w = tanh(Ay) leads to the Riccati equation: w'x = A(/ + s)w2 +
2Xhw + X(g - s).
20. y'x = ycothxf(ymsinhx).
The transformation t = sinhrr, z = ym leads to an equation of the form 1.7.1.3:
tz't = mzf(tz).
21. y'x = ж tanh у f(xn sinh y).
The transformation t = xn, z = sinh у leads to an equation of the form 1.7.1.3:
ntz[ = zf{tz).
22. y'x = ytanhx f(ym cosh x).
The substitution t = cosh ж leads to an equation of the form 1.7.1.3: ty't = yf(tym).
23. y'x = ж cothy f(xn coshy).
The substitution z = coshy leads to an equation of the form 1.7.1.3: xz'x = zf(xnz).
1.7.3. Equations Containing Logarithmic Functions
!• y'x = f(x)yln2y +g(x)y\ny + h(x)y.
The substitution и = In у leads to the Riccati equation: u'x = f(x)u2 + g(x)u + h(x).
2- yfx=x-1ym+1f(ym\nx).
The substitution t = In x leads to an equation of the form 1.7.1.3: y't = — [tymf(tym)].
ъ
3. yfx=x-n-1yf(xn\ny).
z f(xnz)
The substitution z = In у leads to an equation of the form 1.7.1.3: z' = .
X XnZ
4. y'x =ж-1е«/(е«1пж).
The substitution t = In ж leads to an equation of the form 1.7.2.4: y't = —[tev f(tev)].
5- y'x=ye-xf(exlny).
f(exz)
The substitution z = In у leads to an equation of the form 1.7.2.3: z' = z .
exz
6. y'x = -пж~1у1пу + у/(ж)з(жпу).
The substitution w(x) = ж™ In у leads to an equation with separation of variables:
w'x = xnf(x)g(w).
© 1995 by CRC Press, Inc.

п у yf(xnym)
7. V = .
m x ж In у
The transformation t = xnym, z = \ny leads to a linear equation: m?tf(t)z't =
-nz + mf(t).
n у yf(xnym)
m x
The transformation t = xnym, z = \ny leads to the Riccati equation: m2tf(t)z't =
-nz2+mf(t).
9. x[f(xnym) + mlny g(xnym)]y'x = y[h(xnym) - nlny g(xnym)].
The transformation t = xnym, z = \ny leads to a linear equation: t[nf(t) +mh(t)]z't =
-ng(t)z + h(t).
10. x[f(xnym) + m\nxg(xnym)]y'x = y[h(xnym) - n\nxg(xnym)}.
The transformation t = xnym, z = lnx leads to a linear equation: t[nf(t) + mh(t)]z't =
mg(t)z + f(t).
1.7.4. Equations Containing Trigonometric Functions
1. y'x =y™+1 sin xF(ym cos ж).
This is an equation of the form 1.7.4.3 with /(?) =
2. y^ = у"г+1 cos xF(ym sin ж).
This is an equation of the form 1.7.4.4 with /(?) =
3. y^, = у tan ж/(у cos ж).
The substitution t = cos ж leads to an equation of the form 1.7.1.3: y't = f{tym)
ъ
4- y'x = vcotxf (vm sin x) ¦
The substitution t = sin ж leads to an equation of the form 1.7.1.3: y't = —f{tym).
5. y'x=x~1tanyf(xnsiny).
The transformation t = xn, z = sin у leads to an equation of the form 1.7.1.3: ntz't =
zf{tz).
6. y'x = ж cot yf(xn cos y).
The transformation t = xn, z = cosy leads to an equation of the form 1.7.1.3: ntz't =
-zf(tz).
7. y'x = x-1 sin 2yf(xn tan y).
The transformation t = xn, z = tan у leads to an equation of the form 1.7.1.3: ntz't =
2zf(tz).
© 1995 by CRC Press, Inc.

8. y'x = x г sin2yf(xn coty).
The transformation t = xn, z = coty leads to an equation of the form 1.7.1.3: ntz't =
-2zf(tz).
9- y'x = ^—f(y
эт2ж
The substitution t = tana; leads to an equation of the form 1.7.1.3: 2ty't = yf(tym).
10. y'x = ——/(ymcotx).
sin2a;
The substitution t = cot ж leads to an equation of the form 1.7.1.3: 2ty't = —yf(tym).
11. y'x = f(x) cos(oy) + g(x) sin(oy) + h(x).
The substitution и = tan(ay/2) leads to the Riccati equation: 2u'x = a(h — f)u +
2agu + a(f + h).
12. y'x = f(x) cos2(oy) + g(x) sin2(oy) + h(x) sinBoy) + s(x).
The substitution и = tan(ay) leads to the Riccati equation: u'x = a(g + s)u2 + 2ahu +
a(f + s).
13. y'x = f(y + a tan x) — a tan2 x.
The substitution и = y + atana; leads to an equation of the form 1.1.2: u'x = a + f(u).
sin 2y
14. yx= /(tan x tan y).
sin 2x
The transformation t = tana;, z = tan у leads to an equation of the form 1.7.1.3:
tz't = zf{tz).
15. y'x = cot x tan y/(sin x sin y).
The transformation t = sin a;, z = sin у leads to an equation of the form 1.7.1.3:
tzt = zf(tz)-
f(x)
16. y' =— cot ж tan у И«/(этжэ
/(у)
cosy
The substitution w(x) = sin a; sin у leads to an equation with separated variables:
w'x = sin xf(x)g(w).
17. y' = ^— |-cos2y/(ж)з(tanжtany).
Э1п2ж
The substitution w(x) = tan a; tan у leads to an equation with separated variables:
w'x = taxi xf(x)g(w).
18. y'x = — nx~1 sin2y + cos2 yf(x)g(x2n tany).
The substitution w(x) = a;2™ tan у leads to an equation with separated variables:
w'x = x2nf(x)g(w).
19. A + tan2 y)y'x = /(ж) tanm+1 у + g(x) tany + h(x) tan1" y.
The substitution и = tanm у leads to the Riccati equation: u'x = mf(x)u2 + mg(x)u +
mh(x).
© 1995 by CRC Press, Inc.

1.7.5. Equations Containing Combinations of Exponential, Logarithmic,
and Trigonometric Functions
1. y'x = -sin2y + cos2yf(x)g(e2xtany).
The substitution w(x) = e2xta,ny leads to the equation with separated variables:
w'x = e2xf(x)g(w).
2 у' =
ex sin у
This is an equation of the tipe 1.7.5.5 with /(?) =
3. y'x = ev cos ж F(ev sin ж).
This is an equation of the tipe 1.7.5.7 with /(?) =
4. y'x = tany f(ex siny).
The substitution z = sin у leads to an equation of the form 1.7.2.3: z'x = zf(exz).
5- Vx=coty f(excosy).
The substitution z = cosy leads to an equation of the form 1.7.2.3: z'x = —zf(exz).
6. y^, =tana;/(e!'cosa;).
The substitution t = cos a; leads to an equation of the form 1.7.2.4: ty't = —f(tev).
7. y'x=cotxf(evsinx).
The substitution t = sin a; leads to an equation of the form 1.7.2.4: ty't = f(tev).
8. y'x = sin 2yf(ex tany).
The substitution z = tana; leads to an equation of the form 1.7.2.3: z'x = 2zf(exz).
9. y'x = sin2y f(excoty).
The substitution z = cot a; leads to an equation of the form 1.7.2.3: z'x = —2zf(exz).
10. y' = — SlnV .
ex cos у
This is an equation of the form 1.7.5.4 with /(?) =
11. y'x = ey sin ж F(ey cos ж).
This is an equation of the from 1.7.5.6 with /(?) = t;F(!;).
/ _
Ух —
эт2ж
The substitution t = tana; leads to an equation of the form 1.7.2.4: 2ty't = f(tev)
© 1995 by CRC Press, Inc.

13. y/=
sin2a;
The substitution t = cot ж leads to an equation of the form 1.7.2.4: 2ty't = —f(tey).
14. y'x = e-
The substitution u = Xx+lny leads to an equation of the form 1.1.2: u'x =e~uf(u) + X.
15. y'x = exyf(Xy + \nx).
The substitution u = Xy + lnx leads to the equation with separated variables: xu'x =
Xeuf(u) + 1.
1.8. Equations of the Form F(x,y,y'x) = 0 Not Solved for
the Derivative and Containing Arbitrary Functions
1.8.1. Some Equations
1- x = f(y'x).
The solution is written in the parametric form:
f(t), y = Jtfl(t)
dt
2- У = f(y'x)-
The solution is written in the parametric form:
*= №)^г+С, у = /(*).
3.
у
This is an equation of the form 1.7.2.3. Change to a new variable w(x) = xy'x/y; divide
both sides of the equation by xnym and differentiate with respect to x. As a result
we arrive at an equation with separation of variables: xf'w(w)w'x = (mw + n)f(w).
The solution is written in the parametric form:
n)f(w)
In addition, there are solutions у = A^x~n^m', where A^ are roots of the equation
A™ - f{-n/m) = 0.
4. хпеаУ = f(xy'x).
The substitution у = Inu leads to an equation of the form 1.8.1.3: xnua = f(xu'x/u).
5. eaxyn = f{y'x/y).
The substitution x = lni leads to an equation of the form 1.8.1.3: tayn = f(ty't/y).
© 1995 by CRC Press, Inc.

У = ху'х + f(y'x).
The Clerot equation.
Solution: y = Cx + f(C).
In addition, there is a particular solution which may be written in the parametric
form as
* = -fl(t), У = -tfi{t) + f{t).
y = xf(yfx)+g(yx).
The Lagrange—d'Alembert equation.
With f(t) = t, see equation 1.8.1.6. Having differentiated with respect to x, we
arrive at a linear equation with respect to x = x(t), where t = y'x:
8. y = xnf(y'x)+xyx.
Differentiating with respect to x and denoting t = y'x, we obtain the Bernoulli equation
for x = x(t):
x —
n f(t) " nf(t)
x2~n = 0.
9. y = f(x(yxJ)+2xyx.
Solution: [y - f(C)}2 = ACx.
10. y = f(x(y'x)n)
n
n - 1
nC та-1
Solution: у = f(Cn) -\ -x n .
it -L
11. (xy'x - y)nf(y'x) + yg(y'x) + xh(y'x) = 0.
With the aid the Legendre transformation x = u't, у = tu't — и (у'х =t), we obtain the
Bernoulli equation:
h(t)}u't=g(t)u
12. Ю2 + [f(x) + g(x)]y'x + f(x)g(x) = 0.
The equation can be factorized: [y'x + f(x)] [y'x +g(x)] = 0, i.e., it falls into two simpler
equations y'x + f(x) = 0 and y'x + g(x) = 0. Therefore, the solutions are
f f
f(x) dx = С and у + / g(x) dx = C.
J J
f = f(x), g = g(x).
Solution:
( (x \ ( fx \
expl-/ f dx jsinl / \/g-f2dx + C\ iig>f2,
V Jo / \Ja J
У =
Cexpl — / f dx
exp — / / dx) cosh
if g<f2.
© 1995 by CRC Press, Inc.

f(y'x) +ax + by + s = O.
Solution in the parametric form:
bt
by=-ax-s-f(t).
In addition, there is a particular solution у = ax + /3, where a and /3 are determined
by solving the system of two algebraic equations a + ba = 0 and f(a) + bC + s = 0.
Setting u(x) = yy'x + x and differentiating with respect to x, we obtain
u'x[fu{u)-2u + 2x}=0. A)
Equating the first factor to zero, after integrating we find y2 = — (x — CJ + B.
Substituting the latter into the original equation, we have В = f(C). As a result we
obtain the solution: y2 = f(C) — (x — CJ.
There is also an exceptional solution that corresponds to equating the second
factor of A) to zero. The solution in the parametric form is written as
x = u-\f'u(u), y2=f(u)-i[fUu)}2-
16. у = a(y'xJ + f(x - 2ay'x).
This is a special case of equation 1.8.1.18 with n = 2.
Solution:
In addition, there is the following solution written in the parametric form:
x = t + 2afi(t), y 2
17. y = 2a(yxK + f(x-3a(yxJ).
This is a special case of equation 1.8.1.18 with n = 3.
Solution:
3a ) '
In addition, there is the following solution written in the parametric form:
18. y = a(n- l)(y'x)n + f(x - aniy'X-1).
Differentiating with respect to x, we obtain a factorized equation:
[1 - an(n - l)(yx)n-2yx'x] [y'x - fl{t)] = 0, A)
where t = x — an(y'x)n . Equate the first factor to zero and integrate the obtained
equation. Substituting the expression obtained into the original equation, we find the
solution:
Equating the second factor in A) to zero, we have another solution which can be
written in the parametric form as
]n-\
x = t + an
[f't{t)]n-\ y = f{t)+a{n-l)[fl{t)]n.
© 1995 by CRC Press, Inc.

19. f(x2 + y2)^(y'xJ + l=xy'x-y.
Setting x = r(t) cost, у = r(t) sini and integrating, we obtain the solution:
*
f
x/r2-/2(r2)
20. *(fx + fyyx, f-x(fx + fyy'x))=0,
Differentiating with respect to x, we obtain
дФ дФ
where Фи = —— and Фу = —— are partial derivatives of function Ф(и, v). Equating
аи av
the first factor to zero, we find the solution:
f(x,y) = Cx + A, where Ф(С,А)=0.
It remains to be checked whether the equation Фи — хФу = 0 possesses any solutions
and which of them satisfy the original equation.
1.8.2. Some Transformations
1. x = f(y,y'x).
Substituting t = y'x and differentiating both sides of the equation with respect to x,
we obtain an equation with respect to у = y(t):
[1 - tfy(y, t)]y't = tft(y, t), where ft = Ц-, /„ = Ц-.
If у = y(t) is the solution of the latter equation, the solution of the original equation
may be presented in the parametric form as
x = f(y(t), t), у = y(t).
2. y = f(x,y'x).
Differentiating with respect to x and setting t = y'x, we obtain an equation with respect
to x = x(t):
[t - fx(x, t)]x't = ft(x, t), where ft = -^-, fy = -^-.
If x = x(t) is the solution of the latter equation, the solution of the original equation
may be presented in the parametric form as
x = x{t), y = f(x{t),t).
© 1995 by CRC Press, Inc.

о гП11т — f( <rk1lS XVa
л. x у — j\x у ,
V у
I
Set z = xkys and w= ——. Divide both sides of the equation by xnym and differentiate
У
with respect to x. As a result we arrive at the following equation with respect to
w = w(z):
z(sw + k)(fz + fww'z) = (mw + n)f, where f = f(z, w),
which is usually is simpler than the the original equation, since it is readily solved for
the derivative. If w = w(z) is the solution of the equation obtained, the solution of
the original equation is written in the parametric form as
X у — Z, X у — J[Z, W(Z)j.
4. xneay = f(xmel3y, xy'x).
The substitution у = lnu leads to an equation of the form 1.8.2.3:
5. eaxyn = f(ePxym, y'Jy).
The substitution x = lni leads to an equation of the form 1.8.2.3:
6- f(x, xy'x - y, y'x) = 0.
The Legandre transformation x = u't, у = tu't — и (y'x = t), where и = u(t), leads to
the equation f(u't, u, t) = 0. The inverse transformation: t = y'x, u = xy'x —y,u't = x.
7- Ю = Ay + f(x).
With А ф 0, the substitution Xw = 2\J\y + f(x) leads to the Abel equation of the
second kind:
ww'x = w + <p(x), where ip = 2A~ f'x(x),
which is outlined in Subsection 1.3.1 for specific functions tp.
8- У = xy'x + ax2 + f(y'x), афО.
Differentiating the equation with respect to x and changing to new variables t = y'x
and w = —lax, we arrive at the Abel equation of the second kind:
ww't = w + <p(t), where ip = —2aft(t),
which is outlined in Subsection 1.3.1 for specific functions ip.
© 1995 by CRC Press, Inc.

Chapter 2
Second Order
Differential Equations
2.1. Linear Equations
2.1.1. Preliminary Comments
1. A homogeneous linear equation of the second order has the form
f2(x)y':x + h(x)y'x + fo(x)y = 0. A)
Let yo = Уо{х) be a nontrivial particular solution (у ф 0) of this equation. Then the
general solution of equation A) can be found from the formula
V = Vo(c1+C2 [ ^-z-dx), where F = [ A dx. B)
v J Уо ' J h
For specific equations described below in 2.1.2-2.1.8, often only particular solutions are
given, while the general solutions can be obtained with formula B).
2. The substitution и = y'x/y brings equation A) to the Riccati equation
f2(x)u'x + f2(x)u2 + h{x)u + fo(x) = 0
which is discussed in Section 1.2.
3. Assuming
y = u(rr)exp(-— / -j-dxj C)
v 2 J J2 '
yields from equation A) the canonical (or normal) form
u';x + f{x)u = 0, where / = A - U^ff - ± (AV . D)
Substitution C) is a special case of the more general transformation (ip is an arbitrary
function):
which reduces the original equation to the canonical form.
© 1995 by CRC Press, Inc.

4. A nonhomogeneous linear equation of the second order has the form
ЬШхх + /iWx + ШУ = 9(x). E)
Let 2/1 = 2/1 (ж) and 2/2 = 2/2 (ж) be two nontrivial linearly-independent B/1/2/2 ^ const)
solutions of the corresponding homogeneous equation with g = 0. Then the general solution
of equation E) can be found from the formula
„ „ f g dx f g dx
y = Ciyi+C2y2+y2J yi — w-yiJ y2 — w, F)
where W = 2/1B/2)* - 2/2B/1)^-
Given a nontrivial particular solution уг = у\(х) of the homogeneous equation with
g = 0, formula F) can be used for the construction of the general solution of equation E)
with the second solution У2 = у2(х) taken in the form
2/2 = 2/1 / -^-2- dx, where F= f ^-dx, W = e~F, G)
J y( J h
In Subsections 2.1.2—2.1.8, mainly homogeneous equations are given; the corresponding
nonhomogeneous equations may be solved by means of the formulae F) and G).
2.1.2. Equations Containing Power Functions
у" +ау = О.
Solution:
{С\ sm\i(x\f\a\) + С2 co$\i(x\f\a\) if a < О,
d + С2х if a = О,
Ci sin^v^) + С2 008@:-^) if a > 0.
2- V'L ~ (ax + Ь)у = О, аф О.
The substitution ? = a~2/3 (ax + b) leads to the Airy equation
2/« - Су = o, (i)
which is often met with in various applications. The solution of equation A) can be
written as
where Ai(?) and Bi(?) are the Airy functions of the first and second kind, respectively.
The Airy functions admits the following integral representation:
1 f00 /1 \
-J cos^jt3 + &) dt,
v Г M-ih3+**)+H?3+#)]dt-
© 1995 by CRC Press, Inc.

The Airy functions can be expressed in terms of the Bessel functions and the modified
Bessel functions of the order 1/3 with the formulae
h/3(
Ai(-C) = jVt [J-i/s(z) + Ji/3(z)],
where z = -|-?3/2.
For large values of ?, the leading terms of the asymptotic expansions of the Airy
functions are
The Airy equation A) is a special case of the equation 2.1.2.7 with n = 1.
3- V'L ~ (a2*2 + а)У = 0-
О
/ ax \
Particular solution: у$ = exp( —— j.
4- v'L - (ax2 + b)y = o-
The transformation z = x2^/a, и = ez^2y leads to the degenerate hypergeometric
equation 2.1.2.65:
5- У'хХ + a3xB - ax)y = 0.
О
/ (XX
Particular solution: yo = expf \- ax
The substitution ? = ж Н leads to an equation of the form 2.1.2.4:
2a
© 1995 by CRC Press, Inc.

v'L -
= о-
1°. For n = —2, this is the Euler equation 2.1.2.118, while for n = —4, this is the
equation 2.1.2.198 (in both cases the solution is expressed in terms of elementary
function).
2°. Assume 2/(n + 2) = 2m + 1, where m is an integer. Then the solution is
жр -*—ж« +С2ехр—*—ж« if то > О,
-У^ХЛ] ifm<0,
d n + 2 1
where D = ——, q = = .
dx 2 2m +1
3°. For any n, the solution is expressed in terms of Bessel functions and modified
Bessel functions of the first or second kind (see 2.1.2.122):
'—a
CXJ
where q = -y (n + 2).
y'L ~ a(ax2n + nxn~
Particular solution: yo = expl
V n+ 1
y'L ~ ахп~2(ахп + n + l)y = 0.
Particular solution: yo = x exp I
C2Y
'—a
if a < 0,
if a > 0,
= 0.
„n+l
v'L + (ax2n + ьхп-г)у = о.
The substitution ? = rrn+1 leads to an equation of the form 2.1.2.103:
(n + lJ?2/« + n(n + Щ + К + b)y = 0.
V^ + o»i + bl/= 0.
The equation of damping oscillation.
1°. Solution with A2 = a2 - 46 > 0:
A — a
у =
2°. Solution with A2 = 46 - a2 > 0:
xj +
— A — а
j.
у =
3°. Solution with a2 = 46:
/ ax \ (_, . Xx _, Аж \
= exp(—— J (Ci sin — + C2 cos — J.
J.
© 1995 by CRC Press, Inc.

v'L + avL + (bx + c)v = °-
This is a special case of equation 2.1.2.103.
v'L + avL - (bx2 + c)v = o-
The substitution у = и ещ>(-^-х2 Vb) leads to an equation of the form 2.1.2.103:
+ a)u'x + (aVbx - c+ Vb)u = 0.
v'L + avL + H-bx2 + ax + l)y = 0.
/ bx2 \
Particular solution: yo = exp I — 1.
15* Ухх Н~ аУх Н" Ьж(—Ьж3 + аж + 2)у = 0.
/ bx \
Particular solution: yo = expl — 1.
\ о /
16. у'^х + ау'х + Ь(-Ьх2п + ахп + пхп~х)у = 0.
/ Ь ,1 \
Particular solution: у0 = expl х ).
V п + 1 /
17* Ужж + °Уж + b(—bx2n — ахп + пхп~1)у = 0.
Particular solution: yn = expl — xn+1 — ах).
V п + 1 /
18- »»» + ХУ'Х + (п + 1)у = 0, п = 1, 2, 3, ...
dn Г / х2 \ г /" / ж2 ^
Solution: у = < exp I ) Ci + Сг / exp I
ах11 у V 2 / L J V 2 /
19. у^ — 1ху'х -\- 1пу ^0, п ^ 1, 2, 3, ...
Solution: у = ехр(ж2) < ехр(-ж2) \Ci + С2 / ехр(ж2) dx\>.
dx11 у L J J J
This is a special case of equation 2.1.2.46 with n = 1, m = 2.
Particular solution: yo = expl ——x — bx j.
22- V'L + (ax + b)y'x -ay = 0.
Particular solution: yo = ax + b.
23. y^, + (ож + b)ya, + с(ож + b — c)y = 0.
Particular solution: yo = e~cx.
© 1995 by CRC Press, Inc.

24- y'L + (ax + 2b)y'x + (abx - °
Particular solution: y0 = xe~bx.
25- v'L + (ax + b)y'x + (cx + d)y = o-
This is a special case of equation 2.1.2.103.
26- V'L + (ax + Ь)У'Х + c[(o - c)x2 + bx + l]y = 0.
О
Particular solution: у$ = expf
О
CX
27- y'L + 2(ax + b)y'x + (°2a;2 + 2abx + c)y = °-
The substitution и = уехр(-^-аж2 + Ьх) leads to an equation of the form 2.1.2.1:
<х + (с-а-Ь2)« = 0.
28- y'L + (ax + b)y'x + (ax2 + $x + i)v = °-
Assuming у = uexp(sa;2), where s is a root of the quadratic equation 4s2 + 2as+a = 0,
yields an equation of the form 2.1.2.103:
u'xx + l(a + As)X + b\U'x + KP + 2bs)x + 7 + 2s\u = 0-
29- l/i» + («ж + Ъ)у'х + c(-cx2n + ax^1 + bxn + nxn~^)y = 0.
Particular solution: y0 = exp I
V n + 1
-xn+1
30. y'L + a(x2 - Ь2)у'х - a(x + b)y = 0.
Particular solution: yo = x — b.
Particular solution: yo = e~cx.
32- Ухх + (ax2 + 2Ь)У'Х + (abx2 ~ax + Ь2)у = О.
Particular solution: yo = xe~bx.
33. yxx + Bx2 + a)y'x + (ж4 + ax2 + 2x + b)y = 0.
The substitution и = у exp(-g-rr3) leads to a constant coefficient equation: u'xa
bu = 0.
34. yxx + (ax2 + Ъх)у'х + (Зож + 2b)y = 0.
Particular solution: yo = жехр(—^ах — \bx ).
35. у^ + (оЬж2 + Ьж + 2а)у'х + а2(Ьх2 + 1)у = 0.
Particular solution: yo = (ах + 1)е~ах.
© 1995 by CRC Press, Inc.

36- V'L + (°ж2 + Ьх + с)у'х + х(аЬх2 + Ьс+ 1а)у = 0.
Particular solution: уо = expl х — сх ).
37- V'L + (°ж2 + Ьх + с)у'х + (аЬх3 + асх2 + Ъ)у = 0.
Particular solution: уо = expl ——х — сх).
38- y'L + (ах3 + 2Ь)у'х
Particular solution: yo = хе~Ьх.
39- V'L + (°ж3 + Ьх)у'х + 2Bож2 + Ь)у = 0.
а 4 Ь
Particular solut
. / а 4 Ь 2\
ion: уо = хещ)[ х х 1.
40- y'L
Particular solution: yo = (ах + 1)е~ах.
This equation is encountered in the theory of diffusion boundary layer.
) dx.
- y'L + ахпу'х = 0.
is equation is en
// axn+1
exp (
V n+ 1
42- y'L + axny'x + Ъхп~^у = 0.
For n = —1, we obtain the Euler equation 2.1.2.118. For n ф —1, the substitution
z = xn+1 leads to an equation of the form 2.1.2.103:
(n + \fzy"zz + (n + l)(az + n)y'z + by = O.
43- y'L + 2axny'x + a(ax2n + nx^y = 0.
Particular solution: y0 = жехр( -xn+1).
V n + 1 /
44. y'L + axny'x + (bx2n + cxn~x)y = 0.
The substitution ? = xn+1 leads to an equation of the form 2.1.2.103:
(n + lJ?y« + (n + 1)K + n)y't + (&? + c)y = 0.
45- У^ + oa;"y^, - Ь(ахп+т + Ьх2т + тхт~1у = 0).
Particular solution: yn = expf xm+1).
V m + 1 /
46- У^ + 1axny'x + (a2x2n + bx2m + anxn~^ + cxm~1)y = 0.
The substitution w = У expf xn+1) leads to an equation of the form 2.1.2.10:
V n + 1 /
W'L + (bx2m + «"-> = 0.
© 1995 by CRC Press, Inc.

47- v'L + (ож™ + Ь)У'Х + с(ож™ + Ь-с)у = О.
Particular solution: yo = е~сх.
48- v'L + (°ж™ + 2Ъ)у'х + (аЬхП - ахП~г + ь2)у = о-
Particular solution: yo = хе~Ьх.
49. y'L + (abxn + Ъх™-1 + 2а)у'х + а2(Ьхп + 1)у = 0.
Particular solution: yo = (ах + 1)е~оа:.
50- V'L + (аЬхп + 2Ьхп~1 - а2х)у'х + а(аЬхп + Ъхп~г - а2х)у = 0.
Particular solution: yo = {ах + 2)е~ах.
51- V'L + хп[ах2 + {ас + Ъ)х + Ъс]у'х - хп{ах + Ъ)у = 0.
Particular solution: yo = х + с.
52- V'L + (°ж™ + ЬхТП)у'х ~ (ож"-1 + Ъхт~г)у = 0.
Particular solution: yo = х.
53- V'L + (°ж™ + ЬхГП)у'х + {апх™-1 + Ътхт-^)у = 0.
Integrating, we obtain a first order linear equation: y'x + {axn + bxm)y = C.
54- V'L + (°ж™ + ЬхТП)у'х + [°(n + I)»" + b{m + 1)ж"г-1]у = 0.
Particular solution: yo = жехр( xn+1 xm+1).
Vn+1 m +1 /
55- v'L + (°ж™ + ЬхТП)у'х + с(°ж" + ЬхГП - c)v = °-
Particular solution: уд = e~cx.
56. y^ + {axn + bxm)y'x + [abxm+n + Ь{т + 1)ж"г-1 - axn~x\y = 0.
Particular solution: yn = жехр( xm+1
V m + 1
57- У^ + (ож" + Ьжт + c)y4 + {abxm+n + bcxm + anxn~^)y = 0.
Particular solution: yn = exp (
V n + 1
rrn+1 - ex
58- жу1 +
Solution:
{C\ cos л/4аж + Сг sin л/4аж if аж > О,
Ci cosh \/4|аж| + C<i sinh \/4|аж| if аж < О.
59- ЖУ^ + aVx + {bx + С)У = 0-
This is a special case of equation 2.1.2.103.
© 1995 by CRC Press, Inc.

- xv'L + nvL + ьх1-2^ = o.
For n = 1, this is the Euler equation 2.1.2.118. For n ф 1, the solution is
У =
n — 1
n — 1
Ciexp
n-1
n-1
if b > 0,
if 6 < 0.
61- «yL + (i - 3n)y4 - а2п2х2п-гу = o.
Particular solution: y0 = {axn + 1) exp(—axn).
62-
«yL + °y^ + ЬхПу = о-
If n = —1 and 6 = 0, we have the Euler equation 2.1.2.118. If n
solution is expressed in terms of Bessel functions:
= x 2
n+1
n+1
¦ж 2
—1 and b ф 0, the
1 — al
63.
^ + bxn(-bxn+1 +a + n)y =
Particular solution:
= exp (
V n + 1
„n+1
64- XV'LX + ахУ'х + ay = 0.
Particular solution: yo = xe~ax.
65-
The degenerate hypergeometric equation.
If Ь ф 0, —1, —2, —3, ..., Kummer's series is a particular solution:
Ф(а, b;x) =
0)fc ж*
where (a)^ = a(a + 1)... (a + к — 1), (a)o = 1- If b > a > 0, this solution may be
written in terms of the definite integral
Ф(а,Ь;х) =
T(b)
Г(а)Г(Ь-а) Л
where F(z) = /0°° e *i0 -1 di is the gamma-function.
If b is not an integer, then the general solution has the form
у = Ci<?(a, b; x) + С2ж1~ьФ(а -6+1, 2-6; x).
© 1995 by CRC Press, Inc.

TABLE 2.1
Special cases of Kummer's function Ф(о, Ь;ж).
a
a
1
a
1
2
—n
—n
—n

n+1
a
2
a+1
3
2
1
2
3
2
2.+1
2n+2
z
X
2x
—X
2
— Ж
Ж2
2
ж2
2
ж
2ж
2ж
Ф
—ех вшЬж
ж
аж-«7(а,ж)
2 ег?ж
W D) ""*>"<*>
п! / 1уп
Bп+1)! V 2 J ^+Да;)
п! ?(ь i)( }
гA+,),-(|)-(х)
г("+тУ(т)^1/-н(«>
Conventional notation
Incomplete gamma-function
Г t l
7(а,ж)= let dt
Jo
Error function
2 /"^
erf ж——— / exp(—t2)dt
V 7Г Jo
Hermite polynomials
n = 0, 1, 2, 3, ...
Laguerre polynomials
exx~a dn
K ' n! dx11 y '"
Modified Bessel functions
In Table 2.1 are given some special cases where Ф is expressed in terms of simpler
functions.
Function Ф possesses the properties
dn (a)
Ф(а,Ь;х) = ехФ(Ь — а, b; —ж); п Ф(а,Ь;х) = У1 Ф(а + п, b + n; ж).
(XX \®)n
The following asymptotic relations hold:
Ф(а, 6; ж)
Г(Ь)
Г(Ь - а)
—- , as ж —>• +оо,
-ж)"а 1 + 0( ) , авж^-оо.
The following function is a solution of the degenerate hypergeometric equation:
T(a — 6 + 1) ' Г(а)
© 1995 by CRC Press, Inc.

Calculate the limit as 6 —>• n (n is an integer) to obtain
(l) f
Ф(а, 6; ж) = -i—r^ <^ Ф(а, n+1; ж) In ж
п!Г(а — п) [
OO i n
^ (n + 1
(n-1)! ^ (a-n)r
Г(а) ?- A - n)r r\ '
r=0
where n = 0, 1, 2, ... (the last sum is omitted for n = 0), VK2) = [inlX-z)]^, is the
logarithmic derivative of the gamma-function:
n-l
fe=i
-y = 0.5572 ... is the Euler constant.
If b is a negative number, then function Ф may be presented with the formula
Ф(а, 6; ж) = ж1Ф(а -6 + 1, 2-6; ж)
which holds for any value of ж.
For b ф 0, —1, —2, —3, ..., the general solution of the degenerate hypergeometric
equation may be written in the form
y = СгФ{а,Ъ;х) + С2Ф(а,6;ж),
while for 6 = 0, —1, —2, —3, ..., it may be written as
1-ь
[С1Ф(а-Ъ+1, 2-6; ж) + С2Ф(а - 6 + 1, 2-6;
у = х1ь[С1Ф(а-Ъ+1, 2-6; ж) + С2Ф(а - 6 + 1, 2-6; ж)].
The functions Ф and Ф are described in the books by Abramowitz & Stegun A964)
and Bateman & Erdelyi A953, vol. 1) in more detail (see also Suplement 2).
66- xv'L + (ax + b)vL + CK° - c)x + % = °-
Particular solution: yo = e~cx.
67- xy'^x + Bax + b)y'x + a(ax + b)y = 0.
Solution: у = e-ax(d + C2x1~b).
68. xyxx + [(a + b)x + n + m]y'x + (abx + an + bm)y = 0,
where n and m are positive integers; а ф b or n ф то.
Solution:
y = C1e-ax—-—x "e- ~'~+u2e
~ne{a-b)x + C2e~bx
69- xv'L + (ax + b)v'x + (cx + d)v = °-
This is a special case of equation 2.1.2.103.
© 1995 by CRC Press, Inc.

70- xv'L ~ (ax + l)t/i - Ьх2(Ьх + a)y = 0.
/ bx2 \
Particular solution: yo = exp ( — 1.
a?x + a)y = 0.
- xv'L ~ Bax
Solution: у = eax\ d sin I —- Vb ) + C2 cos —
72- ЖУ^ + (ax + b)vL + cx(-cx2 + ax + b + 1) = 0.
Particular solution: yo = exP (""
ex
xv'L -
= о.
Solution: у = C\ exp \_\ (a + л/в2 — b)x2] + Ci exp [ у (a — Va2 — b
x2].
74- «yL + (abx2 + b~5)v'x + 2°2(b-2)x3y
Particular solution: yo = (еж2 + 1) exp(—ax2).
75-
c2)x + b + 2c]y = 0.
Particular solution: yg = xec
76- XV'L
2)y'x + by = O.
Particular solution: yn = a -\ .
x
77- XV
'L
c)y'x + Bax + b)y = 0.
This equation can be integrated to obtain the first order linear equation: xy'x
(ax2
c-l)y =
78- XV'L + (ax2 +bx + c)y'x + (c - 1)(ож + b)y = 0.
Particular solution: yo = x ~c.
79- xv'L + (ax2 + bx + c)vL + (Ax2 + Bx + c)v = °-
1°. Let A = ak, В = k(b — к), С = ck, where к is an arbitrary number.
Particular solution: yg = e~ x¦
2°. Let A = a(b + k), B = a(c + 1)- k(b + k),C = -ck.
О
Particular solution: yo = exp ( \- kx ).
V 2 /
3°. Let A = a(b + к), В = 2a-bk - к2, С = b(c - 1) + k(c - 2).
О
Particular solution: yo = xl~c exp ( \- kx J.
4°. Let A = -ak, B = a(c-1)- k(b + k),C = b(c- 1) + k(c- 2).
Particular solution: yo = x1~cekx.
© 1995 by CRC Press, Inc.

80- xv'L + (ax2 + bx + 2)v'x + (cx2 + dx + b)y = 0.
The substitution u = xy leads to an equation of the form 2.1.2.103: uxx
(cx + d — a)u = 0.
xv'L
Particular solution: yo = x1~b.
82- xv'L + x(ax2 + b)vL + Cax2 + b)y = °-
ax
/ ax
Particular solution: yo = xexpl bx
83- xv'L + (ax3 + bx2 + 2)vL + bxv = о-
Particular solution: yn = a-\ .
x
84. xyxx + (abx3 + bx2 + ax — l)y'x + a2bx3y = 0.
Particular solution: yo = (ax + l)e~ax.
85. xy" + (ax3 + bx2 + cx + d)yf + (d — l)(ax2 + bx + c)y = 0.
Particular solution: yo = x .
86- ЖУ^ + ож"у4 + (оЬж" - ож"-1 - Ь2ж + 2Ъ)у = 0.
Particular solution: yo = хе~Ьх.
Particular solution: yo = ж.
88- ЖУ^ + (ж" + 1 - n)y'x + bx2n~xy = 0.
For b ф -j-, the general solution has the form
у = Ci expf ^ж") + С2 expf ^"),
where /?i and /32 are the roots of the quadratic equation /32 + /3 + b = 0.
For b = \, the solution is
89- XV'L + (°ж" + b)vL + anxn~1y = 0.
Particular solution: yo = хг~ь ехр ( ).
V n J
90. xyxx + (axn + b)y'x + o(b — "
Particular solution: yo = x1~b.
© 1995 by CRC Press, Inc.

xv'L + (ож™ + Ь)У'Х + а(Ь + п- 1)хп-1у = 0.
/ С1Хп \
Particular solution: уп = exp ( ).
V n /
92- xv'L + (ож™ + b)vL + c(axn -cx + b)y = 0.
Particular solution: yo = e~cx.
93- xv'L + (abxn + b - 3n + l)y'x + a2n(b - nfx^^y = 0.
Particular solution: yo = (axn + 1) exp(—axn).
This is a special case of equation 2.1.2.141 with 7 = 0.
95. xy" -\- {axn -\- bxn~1 + 2)y' + bxn~2y = 0.
Particular solution: yo = a -\ .
x
96. xyxx -\- {axn -\- bx)y'x -\- {abxn -\- anxn~1 — b)y = 0.
/ aXT
Particular solution: yo = rrexpl
97. xyxx + {abxn + bxn г + ax — l)y'x + a2bxny = 0.
Particular solution: yo = {ax + l)e~ax.
98- XV'L + (°ж™ + ЪхГП + c)v'x + (c - 1)(ожгг-1 + bxm~1)y = 0.
Particular solution: yo = x ~c.
"• XV'L + {abxn+m + anxn + bxm + 1 - 2n)y'x + a2bnx2n+m-1y = 0.
Particular solution: y0 = {axn + 1) exp(—axn).
100. (ж + a)yxx + {bx + c)y'x + by = 0.
/fbx + c—1,\
Particular solution: yo = exp — / dx .
V J x + a I
101. @1Ж + do)yxx + {bs_x + bo)y'x — mb-j_y = 0.
If m = 1, 2, 3, ..., a polynomial of order m in x is a particular solution of the equation,
which may be presented as
m 1 \ fe
fe=0 X
d xv+1
where D = , Ixv = with v ^ —1.
dx v + l r
© 1995 by CRC Press, Inc.

TABLE 2.2
Solutions of equation 2.1.2.103 for different
values of the determining parameters
Constraints
«2 = 0,
«2 7^0,
a2 = 4ao<i2
U2 = «1 = 0,
h
D-d!
2a2
a0
d\
d\
2a2
h
2b2
Notation: D2
Solution: у -
A
«2
A(h)
1
1
L
= ax — 4aoct2?
-- ehxz{?)
b2
«2
2b2h + b
«i
h
«2
ib0b2-b:
4ao62
A(h) =
, where ? -
г
L
J(
(a
2a2/i + ai,
x — /л
A
z
a,b,0
1 2\
to
a(/?V? )
l/3(k?3/2)
B(h) = b2l
Parameters
a = B(h)/A(h),
/c = -ai/B62)
1 2b2h + h
2 2a2 '
/3 = 2^
2/аоЧ1/2
3 U2 J
г2+ 61Л. + 60
102. (ож + Ь)у^, + 5(сж + d)y^ - s2[(o + c)x + Ь + d]y = 0.
Particular solution: yo = esx.
103. (a2x
b1)y'x + (aox + bo)y = 0.
Let J(a, b; x) be an arbitrary solution of the degenerate hyperheometric equation
xy'^,x + (b—x)y'x — ay = 0 (see 2.1.2.65), and Zv(x) be an arbitrary solution of the Bessel
equation (see 2.1.2.121). The results of solving the original equation are presented in
Table 2.2.
104. (ж-
Particular solution:
(axn + bxm + c)y'x + (anx™-1 + bmxm-1)y = 0.
axn + bxm + с - 1
x + A
¦ cte
)¦
This is a special case of equation 2.1.2.118. The substitution x = e* leads to a constant
coefficient equation: y^ — y't+ ay = 0.
106. ж2?/^ + (ож + b)y = 0.
This is a special case of equation 2.1.2.127.
107. x^y'^ + [о2ж2 - n(n + l)]y = 0, n = 0, 1, 2, ...
Cxeax + C2e-aa:
Solution: yxn+1 = {xzD)n(K—
or"
, where D = .
ax
© 1995 by CRC Press, Inc.

108. x2y'^x - [a2x2 + n(n + l)]y = 0, n = 0, 1, 2, ...
Solution: yxn+1 = (x3D)n( — ^—^ J, where D = —.
\ X s (XX
109. x2y'^x - (a2x2 + 2abx + b2 - b)y = 0.
Particular solution: yo = x eax.
110. x2y'lx + (ax2 + bx + c)y = 0.
The substitution у = x^u, where A is a root of the quadratic equation A2 — A + с = 0,
leads to an equation of the form 2.1.2.103: xu'^x + 2\u'x + (ax + b)u = 0.
For a = — -j-, 6 = fc, c= -j- — m2, the original equation is reffered to as Whittaker's
equation.
111. x2y'lx - (ax3 + ¦?)!/ = 0.
Particular solution: y0 = ж~1^4ехр( — л
112. x2y'lx - [о2ж4 + aAb - \)x2 + b(b + l)]y = 0.
2
Particular solution: yo = x~b expf o~j-
113. x2y'lx + (axn + b)y = 0.
This is a special case of equation 2.1.2.127.
114. x2y'ix - [a2x2n + aBb + n - l)xn + 6F - l)]y = 0.
Particular solution: y0 = xb exp ( —xn).
V n J
115. x2y'lx + (ax2n + bxn + c)y = 0.
This is a special case of equation 2.1.2.141.
)v =
та-1
The transformation ^ = axn + b, w = yx 2 leads to an equation of the form 2.1.2.7:
w'^ + (an)-2?w = 0.
2n(bxn + c)m 1~П
117. x2y'lx + \ax2n(bxn + c)m + 1П ]y = 0.
n-l
The transformation ? = bxn + c, w = yx 2 leads to an equation of the form 2.1.2.7:
© 1995 by CRC Press, Inc.

118. x2y'x[x + axy'x + by = O.
The Euler equation.
Solution:
У= S
l-a
+ С2\х\->*
if A - af > 46,
if A - aJ = 46,
. |rr| 2 [Cisin(/iln|rr|)+C2cos(/iln|rr|)] if A - aJ < 46,
where ц = -§-1A - aJ - 46I1/2.
119- x2vL + xy'x + [x2 - (n + \J]y = 0, n = 0, 1, 2, ...
This is a special case of equation 2.1.2.121.
sin ж
SCution: » =
X
+C2
cos ж
120- ж2У^ + xy'x - [x2 + (n+ \J]y = 0, n = 0, 1, 2, ...
This is a special case of equation 2.1.2.122.
Solution: У = Хп+1
121. x2y'xlx + xy'x + (x2 - v2)y = 0.
The Bessel equation.
1°. Let v be an arbitrary noninteger. Then the general solution is
where Jv and Yv are Bessel functions of the first and second kind:
/ ч _ y> (-l)k(x/2)v+2k Jv{i
fc—0
A)
B)
Solution A) is denoted by у = Zv(x) which is reffered to as the cylindric function.
The cylindric functions possess the following properties:
2vZv{x) = x\Zv_x(x) + Zv+1{x)],
Functions Jv and Yv may be presented in terms of definite integrals (with x > 0):
nJv(x) =
Г
I
Jo
— vV) (Ш — sin ttv
/•CO
/ exp(—ж sinh t — vt) dt,
Jo
(eut + e~vt cos wv)e~x sinh* dt.
© 1995 by CRC Press, Inc.

2°. In the case v = n+ -j, where n = 0, 1, 2, ..., the Bessel functions are expressed
in terms of elementary functions:
. . /~2~ „ , i / 1 d \nsina;
Jn+i(x) = \ —xn+i( —) ,
2 V 7Г V x dx J x
1 d \ n cos x
x dx J x
3°. Let v = n be an arbitrary integer. The the following relations hold:
The solution is given by formula A), wherein function Jn(x) is obtained by substi-
substituting v = n into formula B), while function Yn(x) is found by taking the limit as
v —>• n and for positive n becomes
7Г
fe=0
ir ^y J \2J k\(n
К—(J
where tp(l) = —7, ф{п) = —j + Y^kZi k~x, 7 = 0.5572... is the Euler constant,
ф(х) = [1пГ(ж)]^, is the logarithmic derivative of the gamma-function.
For nonnegative integer n and large x, we may write
x) = (—l)n(cosa? + sina;) + O(x~ ),
тле J2n+i(a;) = (-l)n+1(cosa; -sina;) + O(a;).
Function Jn may be presented in terms of the definite integral
1 Г
Jn(x) = — / cos(x sin t — nt)dt; n = 0,1,2,...
к Jo
The Bessel functions are described in the books by Abramowitz & Stegun A964)
and Bateman & Erdelyi A953, vol. 2) in more detail (see also Suplement 2).
122. x2y^x + xy'x - (ж2 + u2)y = 0.
The modified Bessel equation.
It can be reduced to the equation 2.1.2.121 by means of the substitution x = ix.
Solution:
у = dlv(x) + C2Kv{x),
where Iv and Kv are modified Bessel functions:
© 1995 by CRC Press, Inc.
( 1 V*
0
jfcin
[v + k + l)'
( ) Ж
Ax) 2
1-й - IV
sinTr;/

Iv (ж) may be expressed in terms of Bessel function:
Iv[x) = е-™'2Мхё*«2), i2 = -1.
The case v = n + -j, where n = 0, 1, 2, ..., is given in 2.1.2.120.
If v = n is a nonnegative integer, we have
\ (Jm"(
m=0
m=0
m\(n + m)\
where VKZ) is the logarithmic derivative of the gamma-function (see 2.1.2.121); for
n = 0, the first sum is omitted.
As x —>• +oo, the leading terms of the asymptotic expansion are
v{x), v{x)^e
л/2тгж у2ж
The modified Bessel functions are described in the books by Abramowitz & Stegun
A964) and Bateman & Erdelyi A953, vol. 2) in more detail (see also Suplement 2).
123. x2y^x + 2xy'x - (a2x2 + 2)y = 0.
Solution: x2y = d(ax - l)eax + C2(ax + l)e-oa:.
124. x2y'^x - 2axy'x + [b2x2 + o(o + l)]y = 0.
Solution: у = xa(Ci sin bx + C2 cos Ьж).
125. x2y'lx - 2axy'x + [-b2x2 + 0@ + l)]y = 0.
Solution: у = ха{СгеЪх + C2e-6a:).
126. x2y'lx + Лжу^ + (ож2 + Ьх + с)» = 0.
The substitution у = xku, where к is a root of the quadratic equation /c2 + (A — l)/c +
с = 0, leads to an equation of the form 2.1.2.103:
xv!'xx + (A + 2k)u'x + (ax + b)u = 0.
127. x2y'lx + ожу^ + (Ьхп + c)y = 0, n ^ 0.
The case 6 = 0 corresponds to the Euler equation 2.1.2.118.
For b ф 0, the solution is
±=± [„ . /2 г i\ „.,/2 ^ i
y = x 2
where rv = -i- -^/A — aJ — 4c, Jv and 1^ are Bessel functions of the first and second
kind.
© 1995 by CRC Press, Inc.

128. x2y'x[x + axy'x + xn(bxn + c)y = 0.
The substitution ? = xn leads to an equation of the form 2.1.2.103:
n2C^? + n(n - 1 + аЦ + (bf + c)y = 0.
129. x2y'lx + (ax + b)y'x + cy = 0.
The transformation x = ?~1, у = t^e^w, where к is a root of the quadratic equation
k2 + A — a)fc + с = 0, leads to the equation of the form 2.1.2.103:
?«4'? + [B - &)? + 2/c + 2 - a]«4 + [A - &)? + 2/c + 2 - a - bk]w = 0.
130. ж2»^ + ax2y'x + (bx2 +cx + d)y = 0.
The substitution у = uexp(— \ax) leads to an equation of the form 2.1.2.110:
х<2у'хх + Кт + ь)х<2 + cx + d}u = о.
131. x2yZa + (ax2 + b)y'x + c[(a - c)x2 + b]y = 0.
132. x2y'lx + (ax2 + bx)y'x - by = 0.
Particular solution: yo = x~be~ax.
133. x2yxx + (ax2 + bx)y'x + [k(a — k)x2 + (an-\-bk — 2kn)x + n(b — n— l)]y = 0.
Particular solution: yo = x~ne~kx.
134. a2x2yxx + @1Ж2 + b\x)y'x + (a0x2 + box + co)y = 0.
The substitution у = xkw, where к is a root of the quadratic equation a2k2+(b\— й2)к+
Cq = 0, leads to the equation of the form 2.1.2.103:
- 2a2k + b\)wx + (uqx + a\k + 6q)w = 0.
135. x2y'x'x + [ax2 + (ab - l)x + b]y'x + a2bxy = 0.
Particular solution: y0 = (ax + l)e~ax.
136- x^v'L
For n = 0, 1, 2, ..., particular solutions are polynomials: yo = Pn(x), where Pq = 1,
Рг = x, P2 = 2x2 — 1 — 2a, P3 = 2x3 — C + 2a)x, ... The polynomials contain only
even powers of x for even n and only odd powers of x for odd n.
137. x2y'x'x + x(ax2 + bx + c)y'x + (Ax3 + Bx2 + Cx + D)y = 0.
1°. The substitution у = xkw, where к is a root of the quadratic equation k2 +
(c-l)k + D = 0, leads to an equation of the form 2.1.2.79 (see also 2.1.2.75-2.1.2.78):
xw"xx + (ax2 + bx + с + 2k)w'x + [Ax2 + (B + ak)x + C + bk]y = 0.
2°. Let s and r be arbitrary parameters.
For A = ar, В = as + br — r2, С = bs + cr — 2rs, D = s(c — s — 1), a particular
solution is yo = x~se~rx.
For A = a(b - г), В = a(c - s + 1) + rib - г), С = bs + cr - 2rs, D = s(c - s - 1),
a particular solution is yo = x~s ещ>(—\ах2 — rx).
© 1995 by CRC Press, Inc.

138. x2y'x'x + axny'x - (abxn + acxn~x + b2x2 + Ibex + c2 - c)y = 0.
Particular solution: yo = xcebx.
139. x2y'x'x + axny'x + (abxn+2m - b2x4m+2 + amxn~x - m2 - m)y = 0
Particular solution: y0 = х~тещ>(-- -x2m+1
V 2m + 1
140. х2у'х[х + x(axn + b)y'x + b(axn - l)y = 0.
Particular solution: yo = x~ .
141. x2yZa + x(axn + b)y'x + (ax2n + f3xn + 7)y = 0.
The transformation ? = xn, w = у?~к, where к is a root of the quadratic equation
n2k2 + n(b — l)k + 7 = 0, leads to an equation of the form 2.1.2.103:
п2Ыхх + In«C + 2/cn2 + n(n - 1 + b)]w'x + (a? + kna + fi)w = 0.
142. x2y'lx + xBaxn + b)y'x + [a2x2n + a(b + n - l)xn + otx2m + Cxm + i\y = 0.
The substitution w = у expf —xn) leads to the equation of the form 2.1.2.141:
x2w'^x + bxw'x + (ax2m + Cxm + j)w = 0.
143. x2y'lx + (axn+2 + bx2 + c)y'x + (anxn+1 + acxn + bc)y = 0.
Particular solution: yo = expf — xn+1 — bx).
\ It ~\ i- s
144. A - x2)y'lx + n(n - l)y = 0, n = 0, 1, 2, ...
This equation is encountered in Hydrodynamics when decribing axially symmetric
Stokes Hows.
For n > 2, the solution is
y = C1Jn(x)+C2nn(x),
where Jn and 7in are the Gegenbauer functions which may be presented in terms of
the Legendre functions of the first and second kind (see 2.1.2.147) as follows:
„ , , Pn-2(x) - Pn(x) ^ , л Qn-2{x) - Qn{x)
For n = 0 and n = 1, the solution is у = C\ + C2x.
145. (ж2 - a2)y'lx + by'x - 6y = 0.
Particular solution: yo = Dж — b)\x + a
© 1995 by CRC Press, Inc.
2a+b 2a-b
X — a
•la

146. (ж2 - \)y'lx + ху'х + ау = О.
1°. For a = к2 > О, the solution is
С\ cos(/cArcosh |ж|) + С2 sin(/c Arcosh |ж|) if \x
>1,
У =
I C\ exp(/c arccos x) + C2 exp(—к arccos x) if \x < 1,
where Arcosh x = In (x + \Jx2 — 1).
2°. For a = —k2 < 0, the solution is
C\ exp(/cArcosh|a;|) + C2 exp(—к Arcosh |ж|) if \x > 1,
C\ cos(/c arccos x) + C2 sin(/c arccos x) if \x < 1.
3°. For a = —n2, where n is a nonnegative integer, partial solutions are the Tcheby-
cheff polynomials Tn(x) = 21~ncos(n arccosx).
147. A - x2)yxx - 2xy'x + n(n + l)y = 0, n = 0, 1, 2, ...
The Legendre equation.
The solution is
where the Legendre polynomials Pn (x) and the Legendre functions of the second kind
Qn(x) are given by the formulae
i
(x) = -Pn(
J2
-L X Tib
m—1
Functions Pn = Pn{x) can be conveniently calculated by the recurrence relations
1 2 P 2n+1p np
Three leading functions Qn = Qn(x) are
_ 1 1 + ж _ x 1 + x г)_3ж2-1,1 + ж 3
All n zeros of the polynomial Pn{x) are real and lie on the interval —1 < x < +1;
functions Pn{x) form an orthogonal system on the closed interval —1 < x < +1, with
the relations taking place
0 if n ф m,
Pn{x)Pm{x)dx= I 2
2n
if n = m.
148. A - x2)y'^x - 1xy'x + v(y + l)y = 0.
The Legendre equation; v is an arbitrary number.
The case v = n where n is a nonnegative integer is considered in 2.1.2.147.
The substitution z = x2 leads to the hypergeometric equation. Therefore, with
< 1 the solution can be written as
where F is the hypergeometric series (see 2.1.2.158).
The Legendre equation is discussed in the books by Abramowitz & Stegun A964),
Bateman & Erdelyi A953, vol. 1), and Kamke A976) in more detail (see also Suple-
ment 2).
© 1995 by CRC Press, Inc.

149. (ж2 - \)y'lx + 2(n + 1)ху'я - (у + п + l)(i/ - п)у = 0, п = 1, 2, 3, ...
Solution: у = ——yv(x), where yv is the general solution of the Legendre equation
dxn
2.1.2.148.
150. (ж2 - \)y'lx - 2(n - 1)ху'я - (у - n + l)(i/ + n)y = 0, n = 1, 2, 3, ...
Solution: у = \x2 — l\n yv(x), where yv is the general solution of the Legendre
equation 2.1.2.148.
151. (ож2
/dx
—-^^^^^ leads to a constant coefficient equation: y"z+cy = 0.
Wax2 + b
152. (ж2 + a)y'lx + 2bxy'x + 2(b - l)y = 0.
Particular solution: yo = (ж + а) ~ .
153. (ож2 + b)y'^x + {In + l)axy'x + cy = 0, n = 1, 2, 3, ...
This equation can be obtained by n-fold differentiation of the equation of the form
2.1.2.151:
(ax2 + b)u'lx + axu'x + (c — an2)u = 0.
dnu
Solution: у =
dxn
154. A — x2)yxx — xy'x + Bож2 + b)y = 0.
This is an algebraic form of the Mathieu equation.
The substitution x = cosz leads to the Mathieu equation 2.1.6.4: у"г + (a + b +
acos2z)y = 0.
155. A — x2)yxx + (ож + b)y'x + cy = 0.
The substitution 2z = 1 + x leads to the hypergeometric equation 2.1.2.158:
z(l — z)yzz + [az + -jF — a)]y'z + cy = 0.
156. (ож2 + h)y'lx + (ex2 + d)y'x + A[(c - оА)ж2 + d - b\]y = 0.
Particular solution: yo = c~Xx ¦
157. (ож2 + b)y'lx + [X(c + a)x2 + (c - a)x + 2ЬХ]у'х + А2(еж2 + b)y = 0.
Particular solution: yo = (Xx + l)e~Xx.
© 1995 by CRC Press, Inc.

158. x(x - l)yxx + [(cx+ /3 + l)x - i]y'x + cx/3y = 0.
The Gauss hypergeometric equation.
For 7^0, —1, —2, —3, ..., a solution can be expressed in terms of the hyper-
hypergeometric series:
00 ( \ (R\ к
F(a, /3,r,x) = l + ^2 ) ^p (a)fe = <*(a + l)...(a + k-i),
which is a fortiori convergent for |ж| < 1.
For 7 > /3 > 0, this solution can be expressed in terms of the definite integral
where Г(/3) is the gamma-function.
If 7 is not an integer, the general solution of the hypergeometric equation has the
form
y = C1F(a,l3,T,x)+C2x1-'<F(a-j+l, /З-7+l, 2 - 7; x).
In degenerate cases 7 = 0, —1, —2, —3, ..., a particular solution of the hyper-
hypergeometric equation corresponds to C\ = 0, C2 = 1. If 7 is a positive integer, another
particular solution corresponds to C\ = 1, C2 = 0. In both these cases, the general
solution can be constructed by means of the formula given in 2.1.1.
Table 2.3 represents some special cases where F is expressed in terms of elementary
functions.
In Table 2.4 are given the general solutions of the hypergeometric equation for
some values of the determining parameters.
Function F possesses the following properties:
F(a,p,r,x)=F(p,a,r,x),
F(a,C,r,x) = (l-x)-aF(a, 7 - /3, 7; -^-),
X -L
——F(a, [3,7; x) = ,) F(a + n, /3 + n, 7 + n; x).
ftiyTi У J/VJ
The hypergeometric functions are discussed in the books by Abramowitz & Stegun
A964) and Bateman & Erdelyi A953, vol. 1) in more detail.
159. x(x + a)yxx + (bx + c)y'x + dy = 0.
The substitution x = —az leads to the hypergeometric equation 2.1.2.158:
2A - z)y'lz + [(c/a) - bz}y'z -dy = 0.
160. 2x(x — l)yxx + Bx — l)y'x + (ax + b)y = 0.
The substitution x = cos2? leads to the Mathieu equation 2.1.6.4:
y'cc — (a + 26 + a cos 2?)y = 0.
© 1995 by CRC Press, Inc.

TABLE 2.3
Some special cases in which the hypergeometric function
-F(a,/3,7; 2) is expressed in terms of elementary functions
a
—n
—n
a
a
a
a
a
a
a
a
a
a
a
a
l
2
1
2
h^
1
2
n+1
/3
/3
/3
/3
« + T
a + |
—a
1-a
a-i
1 a
2 a
1 a
a + 1
a + i
a + i
l
2
1
1
1
n+m+1
7
7
—n — m
/3
l
2
3
2
1
2
1
2
2a-l
3
2
3
2
1
2
}«
2a
3
2
3
2
2
3
2
n+m+Z+2
z
X
X
X
X2
X2
-X2
X2
X
sin2 x
. 2
sin x
sin2 ж
ж
X
X
X2
-X2
— X
X2
X
F
к—О
(l-x)~a
2rr(l-2a)
[VTTx^+x] 2a~x + [VT+x^-x] 2a~x
2VT+X2
22«-2[l + Vl^x-}2-2a
sin[Ba-l)rr]
(a-l)sinBrr)
sin[Ba-2)rr]
(a-l)sinBrr)
cos[Ba-l)rr]
cos ж
A + ж)A-ж)-«-1
f1+^r^Y2»
\ 2 )
1 /i + yr3^y-2«
VT=x~ V 2 у
1
— arcsm ж
ж
1
— arctan ж
ж
— 1п(ж + 1)
ж
1 1 + ж
2ж 1-ж
(, i; ^n + m + t + i;. a j +ia^ [
n!/!(n + m)!(m + /)! сгжп+т \v" "' dxlj'
F 1ПA"Ж) am' 0 1-
ж
© 1995 by CRC Press, Inc.

TABLE 2.4
General solutions of the hypergeometric equation
for some values of the determining parameters
a
0
a
a
a
1
a
a
/3
/3
a + |
a + |
/3
/3
/3
7
7
2a + l
l
2
3
2
7
a
a + 1
Solution: у = у(ж)
C1 + C2 / |ж-7|ж-1Г"/3сгж
c1(i+^Ia+c2(i-^Ia
l_2al
¦\J X L J
ж 1~7 ж —1 7~/3~1f C1 + C2 / а
х-1\~Р(Сл +Ci \x~a
«'-l.-il'-*.)
,-i|-'*)
161. (ж2 + 2ож + b)?/^ + (x + a)v'x — т2У = 0-
Solution: у = C\ (x + a + л/ж2 -
C2(x + a + Vx2 + 2ах + Ь) ~
162. (ож2 + Ьх + с)у^х + (dx + к)у'х + (d - 2а)у = 0.
Integrating yields a first order linear equation:
(ax2 + bx + c)y'x + [(d - 2a)x + k-b]y = C.
163. (ож2 + Ьх + c)y'lx + (kx + d)y'x -ky = 0.
Particular solution: yo = kx + d.
164. (ож2 + 2Ьж + c)y'lx + (ax + b)y'x + dy = 0.
The substitution ? =
1
dx
Vax2 + 2bx + с
leads to a constant coefficient equation:
165. (ож2 + 2Ьж + c)y'^ + 3(ож + Ь)у'х + dy = 0.
The substitution и = y\/\ax2 + 2bx + c\ leads to an equation of the form 2.1.2.164:
(ax2 + 2bx + c)u" + (ax + b)u' +(d- a)u = 0.
© 1995 by CRC Press, Inc.

166. (о2ж2 + Ь2х + c2)iC + (bix + сг)ух + coy = 0.
Let Ai and A2 be the roots of the quadratic equation агА2 + 62A + C2 = 0.
x — Ai
1°. For Ai ф А2, the substitution z = — — leads to the hypergeometric equation
A2 — Ai
2.1.2.158:
z(l - z)y'lz - (Ax + B)y'z -Cy = 0,
where A =—, B=— —, C=—.
Cl2 «2(A2 — AiJ U2
2°. For Ai = A2 = A, the transformation x = A + ^~1, у = ?ku, where к is a root of
the quadratic equation аг/с2 + (аг — Ь\)к + Co = 0, leads to an equation of the form
2.1.2.103:
«гС4'? - [(ci + Abi)? + 61 - 2a2(k + 1)]^ - k(d + Xb^u = 0.
3°. Let Co = — ci2n(n — 1) — bin, where n is a positive integer. Then, amongst solutions
there exists a polynomial of the degree < n.
167. (ax2 + bx + c)y^x - (x2 - k2)y'x + (x + k)y = 0.
Particular solution: у$ = x — k.
168. (ax2 + bx + c)y'lx + (x3 + k3)y'x - (x2 - kx + k2)y = 0.
Particular solution: yo = x + k.
169. x3iC + (ax + b)y = 0.
This is a special case of equation 2.1.2.127 with n = —1.
170. x3y'lx + (ax2 + b)y'x + cxy = 0.
The substitution x = 1/z leads to an equation of the form 2.1.2.141:
z2y'^ + zB-a-bz)y'z+cy = 0.
171. x3y'lx + (ax2 + bx)y'x + by = 0.
b
Particular solution: yo = a — 2 -\ .
X
172. x3y'lx + (ax2 + bx)y'x + cy = 0.
The substitution x = 1/z leads to an equation of the form 2.1.2.103:
zy"zz + B - a - bz)y'z + cy = 0.
173. x3y'lx + (ax2 + bx)y'x + (ex + d)y = 0.
The substitution у = xku, where к = —d/6,leads to the equation of the form 2.1.2.129:
x2v!'xx + [(a + 2k)x + b}u'x + [k(a + к - 1) + c]u = 0.
If с = 0 and d = b(a — 2), a particular solution is yo = eb^x-
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174. x3y'x'x + (ax3 - x2 + abx + b)y'x + a2bxy = 0.
Particular solution: yo = (ax + l)e~ax.
175. x3y'x'x + x(axn + b)y'x - (axn - abxn~x + b)y = 0.
/ b \
Particular solution: yo = x exp ( — 1.
\ X У
176. x(ax2 + b)y'^x + 2(ax2 + b)y'x - 2axy = 0.
b
Particular solution: yo = ax -\ .
x
177. x(x2 + a)iC + (bx2 + c)y'x + sxy = 0.
The substitution az = —x2 leads to the hypergeometric equation 2.1.2.158:
(l - z)y"zz + I [l + ? - A + b)z]y'z - jsy = 0.
178. x2(ax + Ь)у?а + [ex2 + Bb + aX)x + b\]y'x + X(c - 2a)y = 0.
Particular solution: yo = exP ( —
179. x2(ax + b)y'lx - 2x(ax + 2b)y'x + 2(ax + 3b)y = 0.
ClX2 + C2x3
solution: у = .
ax + b
180. x2(ax + b)y'lx + [oB - n - m)x2 - b(n + m)x]yx
+ [am(n — l)x + bn(m + l)]y = 0.
dxn + C2xm+1
ax + b
Solution: у =
181. x2(x + a2)yxx + х(Ьгх + oi)y^ + (&ож + ao)y = 0.
The substitution у = xku, where к is a root of the quadratic equation а2к2+к(а\— а2)
ao = 0 leads to an equation of the form 2.1.2.159:
x(x + a<2)v!'xx + [Bk + bx)x + 2ka2 + a^ + [k2 + k(h - 1) + bo]u = 0.
182. (ож3 + bx2 + cx)y'lx + (ax2 + Cx + 2c)y'x + (C - 2b)y = 0.
Particular solution: yg = 2a — о. Л •
X
183. (ож3 + bx2 + cx)y'lx + (ax2 + Cx + 2c)y'x - (ax + 2b- C)y = 0.
Particular solution:
, та м , A v, л са +F-/3)B6-/3)
yo = ax + 2@ -b)-\ , where A =
x a — a
© 1995 by CRC Press, Inc.

184. (ож3 + bx2 + сх)у'х'х + [-2ax2 - (b + l)x + k]y'x + 2(ax + l)y = 0.
Particular solution: yo = (dk + b — l)x2 + (c + k)Bx — k).
185. (ож3 + bx2 + cx)yxx + (nx2 + mx + k)y'x + (k — 1) [(ra — ak)x + m — bk]y = 0.
Particular solution: yo = x .
186. (ож3 + bx2 + cx)y'lx + [(m - о)ж2 + Bcm - l)x - c]y'x + (-2mx + l)y = 0.
Particular solution: y0 = (a + m)x2 + B6 + 4cm — l)(x + c).
187. (ож3 + bx2 + cx)y'lx + (nx2 + mx + k)y'x + [-2(o + ri)x + l]y = 0.
With the constraint
2Ba + n)(c+ k) + B6 + 2m + l)[m + 1 + 2k(a + n)} =0,
a particular solution has the form yo = Ba + n)x2 + Bb + 2m + l)(x — k).
188. (ож3 + x2 + b)y'^ + о2ж(ж2 - b)y'x - a3bxy = 0.
Particular solution: yo = {ax + 2)e~ax.
189. 2ж(ож2 +bx + c)y'^ + (ax2 - c)y'x + Xx2y = 0.
Лх \1/2
—5 ; ) dx leads to a constant coefficient equation:
ax2 + bx + с )
2y'^ + Xy = 0.
190. ж(ож2 + bx + \)y'lx + (ax2 + /3x + -y)y'x + (nx + m)y = 0.
The substitution у = x1~~iw leads to an equation of the similar form:
x(ax2 + bx + l)wlx + [(a + 2a- 2aj)x2 + (C + 2b- 2bj)x + 2- j]w'x+
{[n + (l-j)(a- aj)}x + m + A - j)(/3 - bj)}w = 0.
{(a + C + 1)ж2 - [a + C + 1 + 0G + 6) - 6}x + aj}y'x + (aCx - q)y = 0.
Heun's equation.
For \a\ > 1 and 7 ф 0, —1, —2, —3, ..., a solution can be represented as the
power series
00
F(a, q; a, C,7,6, x) = ^ cnxn,
n=0
wherein the coefficients are determined by the recurrence formulae
c0 = 1, G7C1 = q,
a(n + 1)G + n)cn+i = aG + 6 + n — I) + a + [3 — 6 + n-\ \ncn
L nJ
- [(n - l)(n - 2) + (n - l)(a + /3 + 1) + a/3]c_i.
The series is a fortiori convergent for |ж| < 1.
© 1995 by CRC Press, Inc.

TABLE 2.5
Some transformations retaining the form of Heun's equation
No
1*
2
3
4
5
6
7
8
9
10
11
12
New variables
? = x, w = y
? = l-x, w = y
? = x, w =
t x
ж
хГгу
X
ay
t x
? = —, w = y
a
X
C = l , w = y
a
a
, x-1
s — > ^
X
? а(х-1)
s x(a-l) '
t x
x — 1
, x(a-l)
s a(x-l) '
X
ay
— \x
w |
ay
x\ay
x-l\ay
-\x-lay
Parameters of transformed equation for w = w(^)
a
1-a
a
1
a
1
a
1
a
1-1
a
a
I-1
a
a
a-1
a
a-1
I-1
a
q
aC-q
<?i
<n_
a
q
q
q
q
q
q
q
a
a
a —7+1
a —7+1
a
a
a
a —7+1
a
a
a
a
a
/3-7+1
/3-7+1
a-7+1
P
P
a+7-1
a—7+1
a—7+1
a-6+1
a-6+1
7
6
2-7
2-7
a-/3+l
7
a+/3-7
-6+1
a-/3+l
6
6
7
7
6
7
6
a+p—j
-6+1
6
a+p—j
-6+1
7
a+f3—7
-6+1
a-p+1
ot+ii—7
-6+1
a-/3+l
a+/3—7
-6+1
Notation: q1 = q+(a-j+l)(p-j+l)-af3+6(j-l), q2 = qi+a6(l-j),
<fe =qa~1+a(a — j+l)+aa~1F—/3) — a6.
* This row corresponds to the original equation, while the others refer to the transformed
equation for w = «;(?)
If 7 is not an integer, the general solution of Heun's equation can be presented as
follows:
y = C1F(a,q;a,C,j,6,x)+C2\x\1-'1F(a, 9l; a - 7 + 1, /3- 7 + 1, 2 - 7, 6, x),
where qi = q + (a - j + l)(/3 - 7 + 1) - a/3 + 6G - 1).
In Table 2.5 are listed some transformations retaining the form of Heun's equation.
(Whenever at least one of the indicated equations is integrable by quadrature with
some values of parameters, all the other equations are also integrable for those values
of the parameters.)
For Heun's equation, see also the book by Bateman & Erdelyi A955, vol. 3).
192. (ож3 + bx2 +cx + d)y'lx - (x2 - X2)y'x + (x + X)y = 0.
Particular solution: yo = x — A.
© 1995 by CRC Press, Inc.

193. 2(ож3 + bx2 + cx
2y'^ + Xy = 0.
/
—
V ax3
(Зож2 + 2Ьх + с)у'х + Ху = 0.
dx
V ax3 + bx2 + ex + d
leads to a constant coefficient equation:
194. 2(ож3 + bx2 + cx + d)y^x + 3Cож2 + 2bx + c)y'x + (бож + 2b + X)y = 0.
tion is obtained by differentiating the equation 2.1.2.193.
This equation is
195. (ож3 + bx2 +cx + d)y'lx + [ax2 + (a-y + C)x + Cf]y'x - (ax + C)y = 0.
Particular solution: yo = x + j.
196. (ож3 + bx2 +cx + d)y'lx + (ж3 + X3)y'x - (ж2 - Лж + Х2)у = 0.
Particular solution: yo = x + A.
[oB - k)x2 + b(l - k)x - ck]y'x + Xxk+1y = 0.
197. 2ж(ож2 +bx
/x dx
—-^^^=^^^= leads to a constant coefficient equation: 2yl',
Vax2 + bx + с 44
198. x4yxx + ay = 0.
The transformation z = 1/x, и = y/x leads to a constant coefficient equation: u"zz +
аи = О.
199. x4y'x'x + (ax2 + bx + c)y = 0.
The transformation z = 1/x, и = y/x leads to an equation of the form 2.1.2.110:
z2uzz + (cz2 + bz + a)u = 0.
200. x4y'lx - (a + b)x2y'x + [(a + b)x + ab]y = 0.
Solution:
У =
The substitution z = 1/x leads to a constant coefficient equation: yzz — 2ay'z + by = O.
202. x4y'lx + +axny'x - (ож" + оЬж" + Ъ2)у = 0.
Particular solution: yn = xe~b'x.
203.
= 0.
Solution: C1\x\m\x-a
1-m
1—m
x — a
l, where m is a root of the quadratic
equation m(m — 1)а = —b.
© 1995 by CRC Press, Inc.

204. ж2(ж — аJухх + by = cx2(x — аJ.
Solution:
У =
ж — а
У1+ аBт-1)
- а\
х — а
where m is a root of the quadratic equation m(m — l)a2 = —b.
205. ax2(x - lJyxx + (bx2 + ex + d)y = 0.
Find roots p and q of the quadratic equations
ap(p - 1) + d = 0, aq(q -l) + b + c + d = 0.
Then, the substitution y = xp(x — l)qw leads to the hypergeometric equation 2.1.2.158:
ax(x - l)w'lx + 2a[(p + q)x - p\w'x + Bapq - с - 2d)w = 0.
206. ж2(ж2 + a)y'lx + (bx2 + c)xy'x + dy = 0.
The substitution ? = x2 leads to an equation of the form 2.1.2.181:
4?2(? + «)y« + 2ф + 1)C + a + c}y'^ + dy = 0.
207. (ж2 + \Jy'lx + ay = 0.
Solution:
У
С\ cos(/3 arctan ж) + С2 sin(/3 arctan ж) if а + 1 = (З2 > О,
= \ С\ cosh(/3 arctan ж) + С2 sinh(/3 arctan ж) if а + 1 = —/З2 < О,
С\ + С2 arctan ж if а = — 1.
208. (ж2 - \Jy'ix + ay = 0.
Solution:
У=
x-1
C2sinf/31n
x-1
if а - 1 = 4/32 > О,
if а - 1 = -4/32 < О,
if a = 1.
209. (ж2 ± o2Jyl + b2y = 0.
This is the bending equation of a double-walled compressed bar with a parabolic
cross-section.
For the upper sign (constricted bar), the solution is as follows:
у = ух2 + a2 [C\ cos и + C2 sin и), и =
¦ arctan ( —
а
For the lower sign (bar with salients), the solution is
у = у a2 — x2 (C\ cos и + C2 sin и), и = -^ ^— In
zct
a — x
< a.
© 1995 by CRC Press, Inc.

210. (ax2 + bJyxx + 2ax(ax2 + b)y'x + cy = 0.
/dx
—ц — leads to a constant coefficient equation: y'L + cy = 0.
axz + b ??
211. (x2 - \Jy'lx + 2x(x2 - l)y'x - [1/A/ + l)(x2 - 1) + n2]y = 0,
where v is an arbitray number, n is a nonnegative integer.
1°. With n = 0, this equation coinsides with the Legendre equation 2.1.2.148. Denote
its general solution by yv(x).
2°. With n = 1, 2, 3, ..., the general solution of the original equation is given by
the formula
_ 2 _ 1|n/2_^_
212. A - X2Jy'ix - 2жA - x2)y'x + [1/A/ + 1)A - x2) - fi2]y = 0,
where v and ц, are arbitrary numbers.
The Legendre equation.
The transformation x = 1 — 2?, у = (x2 — l)M/2«; leads to the hypergeometric
equation 2.1.2.158:
m ~ 1)<? + (A» + 1)A " 2CL + (^ " ^(^ + M + 1)^ = 0
with parameters a = ц, — v, C = ц, + v + 1, 7 = // + 1.
In particular, the original equation is integrable by quadrature with v = /л or
rv = —/x — 1.
See Supplement 2 for more detail on the Legendre equation.
213. a(x2 - \Jy'ix + bx(x2 - l)y'x + (ex2 + dx + e)y = 0.
The transformation
where p and q are parameters which are determined by solving the second order
algebraic system
Aaq(q - 1) + 2bq + с + d + e = 0, (p-q) [2a(p + q-l) + b]=d
leads to the hypergeometric equation 2.1.2.158 inw = u;(?).
214. (ax2 + bJy'lx + Bax + c)(ax2 + b)y'x + ky = 0.
/dx
— — leads to a constant coefficient equation: y'L + cy'c +
ax2 + b 44 4
ky = 0.
215. (ax2 + bJy'ix + (ax2 + b)(cx2 + d)y'x + 2(bc - ad)xy = 0.
/O j
OH* -\— /7 \
—о—г dx ,
axz + b
© 1995 by CRC Press, Inc.

216. (ж2 + оJу^ + Ьж"(ж2 + а)у'х -
Particular solution:
217. (ж2 + aJy'lx + Ьхп(х2 + а)у'х - т[Ьж7
Particular solution: yo = (х2 + а)т'2.
218. (ж - оJ(ж - ЪJу'^х - су = О, афЪ.
х —
а)у = 0.
+ (т - 1)х2 + а]у = 0.
The transformation ? = In
x — о
, у = (х — Ъ)г] leads to a constant coefficient equation:
(а — Ь) ML — rfe) — сц = 0. Therefore, the solution is as follows:
„, _ Л I. _ |(l+A)/21 _ ii(l-A)/2 , r< \~ _ ~|(l-A)/2 _
where A2 = 4c(a - b)~2 + 1 ф 0.
219. (ж - оJ(ж - ЬJу^'а, + (ж - о)(ж - Ь)Bж + Х)у'х + \ху = 0.
Let /ci and k2 be the roots of the quadratic equation
(a - bJk2 + (a - b)(a + b + X)k + /л = 0.
1°. With /ci ^ /C2, the solution is
fei
x — a
x —
C2
x — a
x — b
2°. With /ci = fo = fc> the solution is
ж — a
x-t
220. (ож2 + Ьж + сJ?/^ + Ay = 0.
The transformation
dx
C2 In
x — a
x —
ax2 + bx + с '
w =
+ bx + с
leads to a constant coefficient equation: w'L + (A + ac — -j-b2)w = 0.
221. (ож2 + bx + cJy'lx + {lax + k)(ax2 + bx + c)y'x + my = 0.
/dx
— leads to a constant coefficient equation: y'
ax1 + bx + с «
(к —
my = 0.
222.
'^x - xby'x +ay = 0.
The transformation ? = x~2, w = yx~2 leads to a constant coefficient equation:
4«^ + aw = 0.
© 1995 by CRC Press, Inc.

223. x6y'^x + (Зж2 + a)x3y'x + by = 0.
The substitution ? = x~2 leads to a constant coefficient equation: Ay'L — 2ayi + by = 0.
_ . bn
224.
тг=1
— ar
У
where \ (an + /?n) = 1,
n=l
and 6n+3 = bn.
Denote this equation by
— о2)(Ь3ж — o3) r^L bnx — an
> 0, An = an6n+i — an+ibn ф 0, ап+з = an,
«1 «2 «3
b\ 62 63
У
A)
With ai = 62 = 0, аз = 63 = 1, ai = «3 = 0, ct2 = a, /?i = 1 — 7, /З2 = /3, and
/?з = 7 — a — C, equation A) transforms into the hypergeometric equation 2.1.2.158.
The transformation
Ax + В \b\x — ai\r\b3X — аз
с. = ————, w =
-у,
B)
Cx + D' \b2x-a2\r+s
where AD — ВС ф 0, brings the original equation into an equation of the similar form:
Аг А2 A3
в1 в2 в3
04 + r a2 — r — s as + s
0i+r 02-r-s 03 + s
= 0,
C)
where An = Aan + Bbn, Bn = Can + Dbn.
In B), assume r = —«i, s = —аз, А = 61/А3, В = —ai/A3, С = —Ь2/А2, and
-D = a2/A2 to obtain the hyperheometric equation C).
225. xny^x + c(ax + b)n~4y = 0.
X XI
The transformation ? = 1—j-, w = ;—j- leads to an equation of the form 2.1.2.7:
ax + b
ax + b
= 0.
226. xniC + axy'x - (b2xn + 2Ьхп~1 + abx + a)y = 0.
Particular solution: y0 = xebx.
227. xny'^x + (ax + b)y'x - ay = 0.
Particular solution: yo = ax + b.
228. xny'^x + (ax™'1 + bx)y'x + (a - l)by = 0.
Particular solution:
= x
1~a
© 1995 by CRC Press, Inc.

229. xniC + Bxn~1 + ax2 + bx)y'x + by = 0.
Particular solution: yn = a-\ .
x
230. xny'x\x + (axn + b)y'x + c[(a - c)xn + b]y = 0.
Particular solution: yo = e~cx.
231. xny'^x + (axn - ж" + abx + b)y'x + a2bxy = 0.
Particular solution: yo = (ax + l)e~ax.
232. xniC + (axn+m + l)y'x + axm(l + тшв")» = О.
Particular solution: уп = expf xm+1).
V m + 1 /
233. (axn + b)y'^ + (cxn + d)y'x + Л[(с - aX)xn + d- ЬХ]у = 0.
Particular solution: yo = e~Xx.
234. (axn + bx + c)yxx = an(n — l)xn~2y.
Particular solution: yo = axn + bx + с
235. x(xn + \)y'lx + [(a - b)xn + a - n]y'x + b(l - a)xn-1y = 0.
Particular solution: y0 = (xn + l)b/n.
236. x(x2n + а)у'1х + (x2n + a- an)y'x - b2x2n~^y = 0.
Solution: у = Ci (xn + Vx2n + a)b/n + C2 (xn + \J x2n + a )~b/n.
237. x2(a2x2n - \)y'ix + x[a2(n + l)x2n + n - l]y'x - u(u + l)a2n2x2ny = 0.
Solution: y = yv (ax11), where yv (x) is the general solution of the Legendre equation
2.1.2.148.
238. x2(axn - \)y'lx + x(apxn + q)y'x + (arxn + s)y = 0.
Find the roots Ai, A2 and B\, B2 of the quadratic equations
A2 - (q + l)A - s = 0, B2 - (p - 1)B + r = 0
and define parameters c, a, f3, and 7 by the relations
A1+B1 Аг + В2 Ах-А2
c = A1: a= , 13= , 7= hi.
n n n
Then the solution of the original equation has the form
у = xcw(axn),
where «;(?) is the general solution of the hypergeometric equation 2.1.2.158:
© 1995 by CRC Press, Inc.

n~1y = 0.
239. (xn + aJy'lx - bxn~2[(b - l)xn + a(n - l)]y = 0
Particular solution: yo = (%n + a) •
240. (axn + bJy'lx + (axn + b)(cxn + d)y'x + n(bc - ad)xn~1y
Particular solution: yo = exp ( — / — dx).
V J axn + b J
241. (xn + aJy'^ + bxm(xn + a)y'x - xn~2{bxm+1 + an - а)у = 0.
Particular solution: y0 = (xn + aI/n.
242. (ox" + bJy'lx + cxm(axn + b)y'x + (cxm - anx^1 - l)y = 0.
/ f dx \
Particular solution: yo = exp I — / 1.
V J ax11 + b I
243. x2{axn + bJy'lx + (n + l)x(a2x2n - Ъ2)у'х + су = 0.
1 / axn \
The substitution ? = In I 1 leads to a constant coefficient equation:
nb \axn + bJ
b(n + 2) yl + cy = 0.
244. (ажгг+1 + bxn + c)yxx + (axn + (Зхп г т ,)Ух
+ [n(a - a - on)»" + (n - l)(/3 - bn)xn~2\y = 0.
Particular solution: yo = exp / — ; dx .
y [J axn+1+bxn+c J
245. (axn + bxm + c)y'x'x + (Л - x)y'x + у = 0.
Particular solution: yo = x — A.
246. (ox" + bxm + c)y'x-x + (A2 - x2)y'x + (x + X)y = 0.
Particular solution: yo = x — A.
247. 2(ож" + bxm + c)y'x[x + (ото" + bmxm-1)y'x + dy = 0.
/dx
—-^^^^=^^^^ leads to a constant coefficient equation:
Vaxn + Ъхт + с
2yl', + dy = 0.
248. (ox" + b)m+1yxx + (axn + b)y'x - anmxn~1y = 0.
г, Г f dx 1
Particular solution: yo = exp — / — — .
У0 P[ J (axn + b)m\
249. xPny'^x + [2Pn + (ax2 + bx)Qn^]y'x + bQn_2y = 0,
where Pn = Pn{x) and Qn-i = Qn-i{x) are arbitrary polynomials of the degrees n
and n — 2, respectively.
Particular solution: yo = a -\ .
or.
x
© 1995 by CRC Press, Inc.

2.1.3. Equations Containing Exponential Functions
!• V'L + aeXxV = О, А ф 0.
Solution:
where Jo and lo are Bessel functions.
2- V'L + (™x ~ b)y = 0.
Solution:
у = С, J2V-bB^e*/2) + C2Y2V-bB^e*/2)
where Jv and Yv are Bessel functions.
3- V'L + a(XeXx - ae2Xx)y = 0.
Particular solution: y0 = exp ( — — eXx ).
Particular solution: yo = ехр(аеж + Ьх).
5- V'L ~ (ae2Xx + beXx + c)y = 0.
The transformation z = eXx, w = z~ky, where к = \fc/\, leads to an equation of the
form 2.1.2.103:
A2z«4 + A2B/c + l)w'z - (az + b)w = 0.
6- V'L + (ae4Xx + be3Xx + ce2Xx - \\2)y = 0.
The transformation ? = eXx, w = yeXx^2 leads to an equation of the form 2.1.2.6:
7- y'L + [ae2Xx(beXx + c)n - \\2]y = 0.
The transformation ? = beXx + c, w = yeXx^2 leads to an equation of the form 2.1.2.7:
8- V'L + аУ'х + Ье2аху = 0.
The transformation ? = eax, и = уеах leads to a constant coefficient linear equation:
u'^ + ba~2u = 0.
9- y'L - аУ'х + Ье2аху = 0.
The substitution ? = eax leads to a constant coefficient equation: y'L + ba~2y = 0.
© 1995 by CRC Press, Inc.

y'L + пу'х + (ЬеХх + с)у = о-
Solution:
\ А / \ Л
where Jv and Yv are Bessel functions.
The substitution z = ex leads to an equation of the form 2.1.2.116:
12- y'L -У'х+ Ue2Xx(beXx + c)n + * * 1 у = 0.
The substitution z = ex leads to an equation of the form 2.1.2.117:
2\bz* + с)" + -Ц^-] у = 0.
y'x + aeXx(aeXx + X)y = 0.
Solution: у = ещ>(-^-еХх) (С\ + C2x).
V Л
Particular solution: yo = expl —— e ж 1.
V Л
y4 + oeAa;(beAa; + Л)у = О.
—
Л
; - Ье»х(еХх + Ье»х + ц)у = О.
Particular solution: yn = expl —ёхх I.
y^ + (ae2Xx + beXx + k2e2>lx + кце^ + c)y = 0.
The substitution w = у exp( —e^x) leads to an equation of the form 2.1.3.5:
(ae2Xx + beXx + c)w = 0.
y'L + аеХхУ'х + b(axneXx - bx2n + nxn~x)y = 0.
Particular solution: wq = exp( — xn+1
V n + 1
V'L + 2aeXxy'x + (a2e2Xx + a\eXx + bx2n + cxn~x)y = 0.
The substitution w = у expf -r?Xx) leads to an equation of the form 2.1.2.10:
21
© 1995 by CRC Press, Inc.

y'L ~ (° + 2beax)y'x + b2e2axy = 0.
Particular solution: y0 = exp( —eax ).
V a I
27
30
20. y'L + (ae2Xx + X)y'x - aXe2Xxy = 0.
Particular solution: y0 = aeXx + Xe~Xx.
21- y'L + (aeXx — X)y'x + be2Xxy = 0.
The substitution ? = eXx leads to a constant coefficient linear equation: A2yl', + aAyl +
by = O.
22- y'L + (aeXx + b)y'x + c(aeXx + b — c)y = 0.
Particular solution: yo = e~cx.
23. y'ix + (о + Ье2Аа;)у; + Л(о - A - be2Aa;)y = 0.
Particular solution: y0 = beXx + ae~Xx.
24. y^ + (abeXx + b- 3X)y'x + a2X(b - X)e2Xxy = 0.
Particular solution: y0 = (aeXx + 1) ещ>(-аеХх).
25. y'L + BaeXx — X)y'x + (a2e2Xx + be'xx)y = 0.
This is a special case of equation 2.1.3.30.
26. yxx + BaeXx + b)y'x + [a2e2Xx + a(b + X)eXx + c]y = 0.
The substitution w = yexpl — e x) leads to a constant coefficient linear equation:
V Л /
wxx + bwL + cw = 0.
'• V'L + (aeXx + 2b - A)y^ + (ce2Xx + abeXx + b2 - ЪХ)у = 0.
The transformtion ? = \eXx, w = ebxy leads to a constant coefficient linear equation:
w'Je + aw'? + cw = 0.
28. y'L + (aeX + b)y'x + [c(a — c)e2x + (ofe + bc + с — 2ck)ex + k(b — k)]y = 0.
Particular solution: yo = exp(—cex — kx).
29. y'L + (aeXx + b)y'x + (ae2Xx + /3eXx + 7)y = 0.
The substitution ? = ex leads to an equation of the form 2.1.2.141:
?2y^e + («?A + b + l)?y? + (а^2Л + /3?л + 7)y = 0.
i. y'lx + BaeXx - X)y'x + (a2e2Xx + be2>xx + ce^ + k)y = 0.
The substitution w = у expf — eXx J leads to an equation of the form 2.1.3.5:
V A A J
© 1995 by CRC Press, Inc.

31- V'L + BoeAa: + b - X)y'x + (a2e2Xx + abeXx + ce2»x + de»x + k)y = 0.
The substitution w = yexp( — eXx -\ —x) leads to an equation of the form 2.1.3.5:
\ A Z /
W'L + [ce^X + de}1X + k-\{b- AJ]to = 0.
32- V'L + (aeXx + Ье>хх)Ух + aeXx(be»x + X)y = 0.
Particular solution: yo = expl —— e x).
V Л *
33. y'ix + eXx(ae2»x + b)y'x + »[eXx(b - ae2»x) - (л]у = О.
Particular solution: y0 = ae^x + be~^x.
34. y'ix + (aeXx + be»* + c)y'x + (a\eXx + Ьце^у = О.
Particular solution: wq = expf eXx ёхх —ex).
V А /л I
35. yxx + (aeXx + be»x + c)y'x + [abe^x+^x + aceXx + Ьце^у = 0.
Particular solution: yo = expl e^x —ex).
V /л I
36. y'L + (aeXx + 2be»x - X)y'x +
[abe(x+^x + ce2Xx + Ь2е2»х + Ь(ц - X)e»x]y = 0.
1°. With A = 0, the equation transforms into 2.1.3.26, and with /л = 0 it transfroms
into 2.1.3.27.
2°. With Х/л ф 0, the transformation ? = \eXx, w = yexp(—емж) leads to a constant
coefficient equation: w'L + awi + cw = 0.
37. yxx + [abe^x+^x + a\eXx + be»x - 2X]y'x + a2bXe^2X+^xy = 0.
Particular solution: yo = {чеХх + 1) exp(—aeXx).
38. yxx + (ax + b)eXxy'x - aeXxy = 0.
Particular solution: yo = ax + b.
39. yxx + (axeXx + 2b)y'x + (abxeXx - aeXx + Ь2)у = 0.
Particular solution: yo = xe~bx.
40. y'L + x(aeXx + be»x)y'x - (aeXx + be»x)y = 0.
Particular solution: yo = x.
41- V'L + (ax11 + beXx)y'x + (abxneXx + anxn~x)y = 0.
Particular solution: yn = exp( — xn+1
\ n + 1
© 1995 by CRC Press, Inc.

42- V'L + ° exp(ba;")y^ + c[a exp(bxn) - c]y = 0.
Particular solution: yo = e~cx.
43- V'ix + (ax + b) e*P(*xn)yx - a exp(\xn)y = 0.
Particular solution: yo = ax + b.
44. y'lx + axn exp(bxm)y'x - ax11-1 exp(bxm)y = 0.
Particular solution: yo = x.
45- XV'L ~ Bож2 + l)y'x + 4bx3 ехрBЛж2)у = О.
Solution:
where Jv and Yv are Bessel functions.
46- XV'L + axeXxy'x + aeXx(l + Xx)y = 0.
Particular solution: yo = x exp( —— eXx ).
47- ny'L + axeXxy'x - [a(bx + l)eXx + b(bx + 2)]y = 0.
Particular solution: yo = xebx.
48- XV'L + (axeXx + b)y'x + a(b - l)eXxy = 0.
Particular solution: yo = x ~ ¦
49- XV'L + ia(bx + l)eAa: + bx- l]y'x + ab2xeXxy = 0.
Particular solution: y0 = (bx + l)e~bx.
50. жу^ + [(ax2 + bx)eXx + 1\y'x + beXxy = 0.
Particular solution: y$ = a -\ .
x
51- xy'lx + (axn + beXx)y'x + axn~^(beXx + n - l)y = 0.
Particular solution: yo = exp( xn ).
V n /
52- xy'^x + (axeXx + bxn)y'x + [a(bxn - l)eXx + bnx^^y = 0.
b
Particular solution: yo = x exp( xn).
V n /
53- xy'lx + [(axn + l)eXx + anxn + 1 - 2n]y'x + a2nx2n~xeXxy = 0.
Particular solution: yo = (axn + 1) exp(—axn).
© 1995 by CRC Press, Inc.

54- xv'L + (aeXx + Ье>лх)у'х + (a\eXx + bne»x)y = 0.
Integrate to obtain a first order linear equation: xy'x + (aeXx + be^x — l)y = C.
55- xv'L + [°ж" ехр(Ьж"г) + c]y'x + a(c - l)»" ехр(Ъхт)у = О.
Particular solution: yo = x ~c.
56. (ж + a)yZa + (beXx + c)y'x + b\eXxy = 0.
al- с- ЬеХх \
dx .
x + a J
57. 4x2iga + iax2n ехр(Ьж") + 1 - n2]y = 0.
та-1
The transformation ^ = bxn, w = yx 2 leads to an equation of the form 2.1.3.1:
2e^w = 0.
58. х2Ух-х + 2axy'x + [(b2e2- - ^)с2ж2 + o(o - l)]y = 0.
Solution: у = x~a[C\Jv{becx) + C2Y^(becx)], where Jv and Yv are Bessel functions.
59- x2vL + axeXxy'x + b(aeXx - b - l)y = 0.
Particular solution: yo = %~b'•
60- x2vL + x(aeXx + 2b)y'x + [a(cx + b)eXx - c2x2 + 6F - l)]y = 0.
Particular solution: yo = x~be~cx.
61- ж4у^ + (e2/- - L2)y = 0.
Solution: у = x[C\Jl/(e1^x) + СгУ^Де1/21)], where Jv and Yv are Bessel functions.
62- «4yL + foexpf ) + bexpf — ) + c]y = 0.
L \J'/ \J'/ J
The transformation ? = l/ж, w = у/ж leads to an equation of the form 2.1.3.5: w'Je +
(ае2Л? + 6ел? + c)w = 0.
л о ,„4 // | ,„2 \х / | г Aъ ™\ Ааз ^2l г»
Do. X XI -\- О>Х в XI Ч- О (О — X)S — О Ш ^ U.
Particular solution: yo = жехрF/ж).
64. (ж2 + оJу4'а; + ЬеХх(х2 + а)у'х - (ЬхеХх + а)у = 0.
Particular solution: yo = ух2 + а.
65. (ж" + afy'^ + Ъ(хп + а)еХху'х - хп~2(ЬхеХх + an - а)у = 0.
Particular solution: yo = (хп + аI'11.
© 1995 by CRC Press, Inc.

66. (axn + bJy'^ + c(axn + b)eXxy'x + (ceXx - anxn~x - \)y = 0.
т. • , f f dx
Particular solution: yn = expl — /
\ J axn +
67. (a2e2Xx + Ь)Ух-х - b\y'x - a?X2v2e2Xxy = 0.
Solution: у = Cx{aeXx + Va2e2Xx + b)v + C2(aeXx + Va2e2Xx + b) v.
68. 2(aeXx + b)y'^x + aXeXxy'x + cy = 0.
/dx
— leads to a constant coefficient linear equation:
VaeXx + b
69. (aeXx + b)y'lx + (ceXx + d)y'x + k[(c - ak)eXx + d- bk]y = 0.
Particular solution: yo = e~kx¦
70. (aeXx + b)y'^ + (ceXx + d)y'x + (neXx + m)y = 0.
For a = 0, this is an equation of the form 2.1.3.29. For a/0, the transformation
? = aeXx, w = yt;~k, where к is a root of the quadratic equation bX2k2 + dXk + m = 0,
leads to an equation of the form 2.1.2.159:
</L + A[Ba?:A + aX + c)? + aB6A;A + bX + d)}w* + (ak2X2 + ckX + n)w = 0.
71. (ex + k)y'^x + (aeXx + be»x + c)y'x + (a\eXx + Ьце»х - ex)y = 0.
Integrating yields a first order linear equation:
(ex + k)y'x + {aeXx + be»x - ex + c)y = С
72. (aeXx + ЪJуЧх + сеХх(ХЬ - ceXx)y = 0.
Particular solution: у = (aeXx + 6)fc, where к = —.
73. (aeXx + ЬJу?а + cr(aeXx + b)y'x + ceXx(cr + Л - ceXx)y = 0.
Particular solution: у = (aeXx + b)k, where к = —.
ал
74. (aeXx + ЬJ^ + (аХеХх + c)(aeXx + b)y'x + my = 0.
/dx
—г — leads to the constant coefficient linear equation:
aeXx + b
y'ss + q/j + my = 0.
Xx + ЪJуЧ + ke»x(aeXx + b)y' + ceXx(ke»x - ceXx
75. (aeXx + ЪJуЧх + ke»x(aeXx + b)y'x + ceXx(ke»x - ceXx + Xb)y = 0.
Particular solution: у = (aeXx + b)k, where к = —.
ctX
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76. 4(aeXx + Ъ)пу'?х + [ke2Xx(ceXx + d)n~4 - X2(aeXx + b)n]y = 0.
The transformation
ae\x + b \x/2
leads to an equation of the form 2.1.2.7: 4:w'/e + k(AX)~2^~nw = 0, where A = ad—be.
ceXx + d ' ceXx + d
77. (aeXx + bx + c)y'^x - a\2eXxy = 0.
Particular solution: yo = ae>~x + bx + c.
78. [(ax + b)eXx + c]y'^x - c\2y = 0.
Particular solution: yo = ce~Xx + ax + b.
2.1.4. Equations Containing Hyperbolic Functions
!• V'L - (a-2q cosh 2x)y = 0.
The modified Mathieu equation.
The substitution x = i? leads to the Mathieu equation 2.1.6.4:
For eigenvalues a = an(q) and a = bn(q), the corresponding solutions of the modified
Mathieu equation are
Ge2n+p(x,q) = ce2n+P(ix,q) = ^2 A22^ cosh[Bk +p)x],
k=0
oo
^2n+p(x,q) = -ise2n+P(ix,q) = У^ Blktp smn[Bfc + p)x],
k=o
where p may be equal to 0 and 1, and coefficients ^42fe+P an<^ ^2k+P are indicated in
2.1.6.4. P P
The modified Mathieu equation is discussed in the books by Abramowitz & Stegun
A964) and Bateman & Erdelyi A955, vol. 3) in more detail.
Utilize the formula cosh2rr = 2 cosh2 ж — 1 to obtain an equation of the form 2.1.4.1:
// (a i a \
Уxx + (~^" + о + ~^" cosh 2x 1 у = 0.
**• Уxx ~ °[° cosh2(bx) + bsinh(ba;)]y = 0.
Particular solution: yo = exp — sinhFa;) .
L b J
^' Уxx ~ o-[a sinh2(bx) + bcosh(ba;)]y = 0.
Particular solution: yo = exp — coshFa;) .
© 1995 by CRC Press, Inc.

5. ухх + (о cosh2 ж + Ъ sinh2 ж + с)у = 0.
Utilize the formulae 2 sinh x = coshBrr) — 1 and 2 cosh x = coshBrr) + 1 to yield
an equation of the form 2.1.4.1:
a — b a + b , .„
6- V'L + [° tanh(Aa:) + b]y = 0.
The transformation
- tanh(AaQ _ k/x
K ~ 1 + tanh(AaO '
where к is a root of the quadratic equation 4/c2 + b — a = 0, leads to an equation of
the form 2.1.2.159:
4A2C(C + l)n& + 4AB/c + A)(C + 1)«4 + D/c2 + a + b)w = 0.
7- V'L - 4°2 tаnh2Coж)y = 0.
Particular solution: yo = sinhCaa;)[coshCaa;)]~1'3.
8- у4'а. +[оЛ-o(o + A)tanh2^)]y = 0.
Particular solution: yo = [cosh(Aa;)]~0'A.
9- V'ix + 13оЛ - A2 - o(o + A) tanh2^)]y = 0.
Particular solution: yo = sinh(Aa;)[cosh(Aa;)]~0'A.
10- V'L + [° coth(Aж) + b]y = 0.
The transformation
_ 1 - tanh(AaQ _ k/x
K ~ 1 + tanh(AaO '
where к is a root of the quadratic equation 4/c2 + b — a = 0, leads to an equation of
the form 2.1.2.159:
4A2C(C - 1)«4'? + 4AB/c + A)(C - 1)«4 + D/c2 + a + b)w = 0.
l/i» - 4o2 соШ2(Заж)у = 0.
Particular solution: yo = coshCaa;)[sinhCaa;)]~1'3.
12- l/i» + [«A - o(o + A) coth2(Aж)]y = 0.
Particular solution: yo = [sinh(Aa;)]~o'A.
13- V'L + i3aX - A2 - o(o + A) coth2(Aж)]y = 0.
Particular solution: yo = cosh(Aa;)[sinh(Aa;)]~0'A.
© 1995 by CRC Press, Inc.

14- У'хХ + аУ'х - -МЛ + ° tanh(Aa;)]y = 0.
Particular solution: yo = совЬ(Аж).
15. yxx + osinh(Aa;)y^, + b[osinh(Aa;) — b]y = 0.
Particular solution: yo = e~bx.
16. yxx + osinh(Aa;)y^, — A[A + cosh(Aa;)]y = 0.
Particular solution: yo = вшЬ(Аж).
17. yxx + ocosh(Aa;)y^, — A[A + sinh(Aa;)]y = 0.
Particular solution: yo = совЬ(Аж).
18. y'xx + 2 tanh x y'x -\- ay = 0.
Solution:
f Ci cos(frr) + C2 sinCbx) if a - 1 = 62 > 0,
у cosh ж = <
I Ci со8Ь(ож) + Co вшщож) if a — 1 = —о < 0.
y4 + biatanh(Aa;) - b]y = 0.
Particular solution: yo = e~ .
20- l/i» - Atanh(Aa;)y4 - o2 cosh2(Aa;)y = 0.
Solution: у = C\ exp — sinh(Aa;) + C2 exp —— sinh(Aa;) .
LA -I LA -I
21- V'lv ~ tanh ж y'x + a2 coth2 x (sinhxJm-2y = 0.
Solution:
y = Vsinhrr \dJ i f — sinhm ж) + С2У i (— sirihmx)\,
where Jv and Yv are Bessel functions.
22- v'ix + 2 tanh ж v'x + (ax2 + bx + c)v = °-
The substitution и = у cosh ж leads to an equation of the form 2.1.2.6:
v!'xx + (ax2 + bx + c-l)u = 0.
23- v'L + 2 tanh x v'x + (°ж" + i)y = о.
The substitution u = ycosha; leads to an equation of the form 2.1.2.7: uxx + axnu = 0.
24- v'L + 2 tanh x y'x + (ax2n + ЪхП~г + i)y = о.
The substitution и = у cosh ж leads to an equation of the form 2.1.2.10:
© 1995 by CRC Press, Inc.

25- V'L + 2ncothxy'x + (n2 - o2)y = 0, n = 1, 2, 3, ...
Solution: у = (^^^-Y(Cieax + C2e~ax).
V smh x dx J
26. y'lx + B tanh ж + a)y'x + (a tanh ж + b)y = 0.
The substitution и = у cosh ж leads to a constant coefficient linear equation: u'xx +
au'x + (b- l)u = 0.
27- V'L + atanh"(Aa;)t? - Л[Л + atanh"+1^)]y = 0.
Particular solution: yo = cosh(Aa;).
28- V'L + (ax + ь) 81пЬгг(Лж)у4 - a sinhn(\x)y = 0.
Particular solution: yo = ax + b.
29- V'ix + (ax + b) taпhrг(Лж)y^ - о taпhrг(Лж)y = 0.
Particular solution: yo = ax + b.
30. y'lx + axn coshm(\x)y'x - ax™-1 coshm(Xx)y = 0.
Particular solution: yo = x.
31. yxx + axn tanhm(Xx)yx — ахп г tanh"г(Лж)y = 0.
Particular solution: yo = x.
32. жу^, + ax coshn(Xx)y'x — [a(bx + 1) соэЬгг(Лж) + b(bx + 2)]y = 0.
Particular solution: yo = xebx.
33. жу^, + ax tanhn(Xx)y'x — [a(bx + 1) taпhrг(Лж) + b(bx + 2)]y = 0.
Particular solution: yo = xebx.
34. жу^, + [ax соэЬгг(Лж) + b]y'x + a(b — 1) coshn(Xx)y = 0.
Particular solution: yo = x .
35. xyxx + [ax tanh™(^) + b]y'x + a(b — 1) tanh™(^)y = 0.
Particular solution: yo = x .
36. xyxx + [(ax2 + bx) соэЬгг(Лж) + 2]y^, + ЬсоэЬгг(Лж)у = 0.
Particular solution: yn = a-\ .
x
37. жу^'з, + [(ож2 + bx) tanh™(^) + 2]y^ + btanh™(^)y = 0.
Particular solution: yo = a -\ .
x
© 1995 by CRC Press, Inc.

38. xyxx + (о sinh™ x + bxrn~^~1)yx + bxm(a sinh™ x + m)y = 0.
b
( °
Particular solution: Vq = exp I
V m + 1
39- xv'L + (° tanh™ x + bxm+1)y'x + bxm(a tanh™ x + m)y = 0.
Particular solution: yo = expf xm+1).
V m + 1 /
40. x2y^x + ax coshn(Xx)y'x + b[a cosh"(Aa;) - b - l]y = 0.
Particular solution: yo = %~b'•
(Aa;)y4 + b[atanh"(Aa;) - b - l]y = 0.
Particular solution: yo = ж~ь.
42. (osinha; + b)yxx + (csinha; + d)y'x + A[(c — oA) sinha; + d — bX]y = 0.
Particular solution: yo = e~Xx¦
43. (o tanh x + b)yxx + (c tanh x + d)y'x + A[(c - oA) tanh x + d — b\]y = 0.
Particular solution: yo = e~Xx¦
44. [otanh(Aa;) + b]yxx + [ctanh(Aa;) + d\y'x + [ntanh(Aa;) + m]y = 0.
The transformation
_ 1 + tanh(AaQ _ k/x
K ~ 1 - tanh(Aa;) '
where /c is a root of the quadratic equation 4(a — b)k2 + 2(c — d)k + n — m = 0, leads
to an equation of the form 2.1.2.159:
+ [4(a + 6)/c2 + 2(c + d)fc + n + m]w = 0.
45. [a + b coth(Xx)]yxx + [c + dcoth(Aa;)]y^, + [n + m coth(Xx)]y = 0.
Multiply this equation by tanh(Aa;) to obtain the equation 2.1.4.44.
46. cosh2(oa;)y4/a; - by = 0.
The substitution аж = hiW — @ < ^ < 1) leads to the hypergeometric equation
2.1.2.158:
?(? - l)a? + BC " Щ + a~2by = 0.
47. sinh2(oa;)y4/a, - by = 0.
The substitution ax = ±ln —, (? > 0) leads to the equation 2.1.2.177:
V?Ti
© 1995 by CRC Press, Inc.

48. y^ sinh2 ж - [о2 sinh2 ж + п(п — 1)]у = 0, о ^ 0; п = 1, 2, 3, ...
/ 1 /I \П
Solution: y = sinhna;( — ) (Cieaa; + С2е~ах).
V sinh х ах J
49. sinh"(Aa:)tC + [осо3Ьгг-4(Аж) - A2 sinhn(Xx)]y = 0.
у
The transformation ? = tanh(Arr), u; = r-тт—r- leads to an equation of the form
cosh(Aa?)
2.1.2.7: w'^ + a\~2?-nw = 0.
50. coshn(Xx)y'x'x + [asinhn-4(As) - A2 coshn(Xx)]y = 0.
V
The transformation f = coth(Aa;), w = ;—— leads to an equation of the form
sinh(Aa;)
2.1.2.7: w'^ + a\-2?~nw = 0.
51. [оэтЬ(Аж) + bx + c]yxx — oA2 этЬ(Аж)у = 0.
Particular solution: yo = asinh(Aa;) + bx + с
52. [осоэЬ(Аж) + bx + c]yxx — oA2 соэЬ(Аж)у = 0.
Particular solution: y0 = acosh(Arr) + bx + с
2.1.5. Equations Containing Logarithmic Functions
!• v'xx ~ (°2a;2 In2 ж + о In ж + а)у = 0.
Particular solution: y0 = е~ах2/4хах2/2.
2- V'L - (a2x2n In2 ж + оггж" In ж + ахп~г)у = 0.
Particular solution: yo = e~Fx^-n+1^F, where F =
(n + IJ
3- V'L + ° 1п"(Ьж)у4 + с[° 1п"(Ьж) - c]y = 0.
Particular solution: yo = e~cx.
4- y'L + [o b"(baj) + c]y'x + ac lnn(bx)y = 0.
Particular solution: yo = e~cx.
**' v'xx "I" (ax "I" ^) 1пГг(сж)Уж "" ° 1пГг(сж)у = 0.
Particular solution: yo = ax + b.
6- y'L + <™n 1пт(Ьх)у'х - a^ lnm(^)y = 0.
Particular solution: yo = x.
x + a)y = 0.
Particular solution: yn = e~axxax.
© 1995 by CRC Press, Inc.

8- XV'L ~ [°2ж 1п2гг(Ьж) + an In™ (Ьж)]у = 0.
Г f 1
Particular solution: yo = exp a / In™ (bx) dx .
L J J
9- XVxx "I" ОЖ ln X Ух "I" O.(ln X + l)j/ = 0.
Particular solution: y0 = е^ж121.
10. жу^, + (ax In ж + b)y4 + (ob In ж + o)y = 0.
Particular solution: y0 = eaxx~ax.
11. жу^ + Bож In ж + 1)у4 + (о2ж In2 ж + о In ж + о)у = 0.
Solution: у = еахх~ах{Сг + С2 In ж).
12- «yL + ln «(«ж + Ь)у4 + o(b In2 ж + 1)у = 0.
Particular solution: y0 = еахх~ах.
xv'L + ax 1пП(Ьх)у'х + ап ьгг-1(Ьж)у = о.
Г f 1
Particular solution: уд = exp —a / 1ппFж) dx .
14- XV'L + ах ln" ХУ'Х + (°ln" x + an In™ ж)у = 0.
( f \
Particular solution: yo = х exp I —a / In™ xdx \.
15- XV'L + (°ж" In ж + 1)у'х - ах™-1 у = 0.
Particular solution: yo = In ж.
16- XV'L + (ах 1п" ж + !)у4 - ° In" ж у = 0.
Particular solution: yo = In ж.
!7- жу^ + (ож In™ ж + Ь)у4 + а(Ь - 1) In™ ж у = 0.
Particular solution: yo = ж ~ .
18- ЖУ1 + [(«ж2 + Ьж) 1п™(сж) + 2]у'х + Ъ 1п™(сж)у = 0.
Particular solution: y0 = а-\ .
ж
19- ЖУ^ + (ож™ + ь ln™ х)у'х + ахп-г(Ь In™ ж + п - 1)у = 0.
Particular solution: yo = ехр( ж™ ).
V п /
20- ЖУ^ + (°ж" + Ьж ln™ Ж)У^ + [Ь(ож™ - 1) \пт х + апхп~1]у = 0.
Particular solution: yo = ж ехр( ж™ ).
V п /
© 1995 by CRC Press, Inc.

21. x2y'x[x + (о In ж + b)y = 0.
The transformation ? = а1пж + 6— —, ад = уж'2 leads to an equation of the form
2.1.2.7: ад^+а?ад = 0.
22. x2yxx + (o In2 ж + b In ж + c)y = 0.
The transformation ? = In ж, ад = уж/2 leads to an equation of the form 2.1.2.6:
23. x2^^ + [a(b In ж + с)" + \\у = 0.
The transformation ? = 61пж + с, ад = уж-1/2 leads to an equation of the form 2.1.2.7:
w'^ + ab~2inw = 0.
24. x2yxx + xy'x + a \nn(bx)y = 0.
Solution:
у = л/1пFж) [d J i
v L~2
where m = -o"(n + 2); Jv and ^ are Bessel functions.
25. ж2»^ + жу^ + (о In2" x + b In" x)y = 0.
The substitution ? = In ж leads to an equation of the form 2.1.2.10: y'J? + (a?2n +
&f*-i)y = 0.
26. x2yxx + жBо In ж + 1)у4 + (ж2 + о2 In2 ж + b)y = 0.
The substitution у = адехр(— -j«hi ж) leads to the Bessel equation 2.1.2.121: ж2ад^',+
xw'^ + (ж2 + b — a)w = 0.
27. ж2у4'а, + жB In ж + о + l)y'x + (In2 ж + о In ж + b)y = 0.
The transformation ? = In ж, ад = у exp(-i- In ж) leads to a constant coefficient equa-
equation: w'^ + aw'^ + (b - 1)ад = О.
28. x2yxx + жB1пж + a)y'x + [In2 ж + (о — 1) In ж + Ьж" + c]y = 0.
The substitution ад = у exp(-j In ж) leads to an equation of the form 2.1.2.127:
u'x + (bxn + с - 1)ад = 0.
29. x2y'lx + ax lnn(bx)y'x + c[a 1п"(Ьж) - с - l]y = 0.
Particular solution: yo = ж~с.
30. x2y'lx + x(axn + b In x)y'x + b(axn In ж - In ж + l)y = 0.
Particular solution: yo = exp ( In2 ж J.
© 1995 by CRC Press, Inc.

31. х(х + а)у^х + х(Ыпх + с)у'х + Ьу = 0.
т. • , ( Г Ыпх + с-1 , \
Particular solution: vq = exp I — / dx .
V J x+a )
32- xAv'L + ax2 ^n(Px)y'x + [a(c - x) \nn(bx) - c2]y = 0.
/ с \
Particular solution: y0 = x exp — ).
V x J
33. (o In x + b)y'^x + (c In x + d)y'x + Л[(с - оЛ) In x + d - ЬХ]у = 0.
Particular solution: yo = e~Xx.
34. ж In ж y'lx - ny'x - о2жAп хJп+1у = 0.
Solution: у = deaF + C2e-aF, where F = j\nnxdx.
= j
35. ж \п(ах)у'х'х - [п 1п(ож) + т]у'х - b2x2n+1 ln2rn+1 (ож)у = 0.
Solution: у = Cie6F + C2e~bF, where F = j' xn\nm(ax)dx.
36. ж2 \n(ax)y'Jx + у = 0.
Solution: у = С\ 1п(аж) + С21п(аж) / [1п(аж)]~2 dx.
37. ж In2 ж ухх + (ax + 1) In ж y'x + bxy = 0.
/dx
leads to a constant coefficient linear equation: y'L
mx K
ay'e + by = 0.
38. x(ax In ж + bx + c)yxx — ay = 0.
Particular solution: yo = ax\nx -\-bx -\- c.
39. ж2 (o In ж + Ьж + ^y'ln + ay = 0.
Particular solution: yo = a In x + bx + с
40. \nn(ax)y'x-x + (b2 - x2)y'x + (x + b)y = 0.
Particular solution: yo = x — b.
2.1.6. Equations Containing Trigonometric Functions
1. yxx + o2y = Ьэт(Аж).
Solution:
C\ sin(ax) + C2 cos(ax) -\ r -y sin(Aa;) if а ф А,
a — A
C\ sin(ax) + C2 cos(ax) —— xcos(ax) if a = A.
© 1995 by CRC Press, Inc.

TABLE 2.6
The general solution of the Mathieu equation expressed in
terms of subsidiary periodical functions (fi(x) and (f2(x).
Constraint
2/iO)>l
2/i(tt)<-1
|У1GГ)|<1
У1(тг) = ±1
General solution у = y(x)
C1e2^(fi1(x) + C2e-2l"x(fi2(x)
С1е2">1(Ж) + С2е-2">2(Ж)
(Сi cos vx + C2 sin vx)(fii(x)
+ (C\ cos vx — C2 sin vx)ifi2 (x)
Ciipi(x) + C2Xlf2(x)
Period of
ipi and (fi2
IT
2тг
7Г
7Г
Index
/i is a real number
H = p+\i, i2 = -l
p is imaginary part of ц,
/л = iv is a pure
imaginary number,
cosB?rrv) = уДтг)
ц = 0
2. у'?х + а2у = bcos(Xx).
Solution:
sin(aa;) + C2 cos{ax)
У =
a2-A2
cos(Arr) if а ф А,
C\ sin(aa:) + C2 сов(аж) H ж sin(aa:) if a = A.
3.
4.
The substitution Xx = 2? + -|- leads to the Mathieu equation 2.1.6.4:
DaA cos
= 0.
V'L + (° - 2qcos2x)y = 0.
Т/ге Mathieu equation.
1°. Given numbers a and q, there exists the general solution y{x) and a characteristic
index ц, such that
For small values of q, the approximate value of \i can be found from the equation
„„2
coshGr/i) = l + 2sin2(-|-v/«)
A — a)y/a
sinGrv/a)+O(94).
If y\ (x) is the solution of the Mathieu equation, which satisfies the initial conditions
y1 @) = 1 and y[ @) = 0, the characteristic index can be determined from the relation
coshB7r/i) = 2/1G1").
The solution y\(x) and hence /л can be determined with any degree of accuracy by
means of numerical and approximate methods.
The general solution differs depending on the value of y\ (тг) and can be expressed
in terms of two subsidiary periodical functions <fii(x) and <fi2(x) (see Table 2.6).
© 1995 by CRC Press, Inc.

TABLE 2.6a
Periodical solutions of the Mathieu equation cen = cen(x, q) and
se^, = sen(x,q) (for odd n, functions ce.,,, and se^, are 27r-periodical,
and for even n, they are тг-periodical); definite eigenvalues
о = an(q) and b = bn(q) correspond to each value of parameter q.
Mathieu functions
CO
Ce2n = E A2m cos 2mx
m=0
oo
ce2n+i = E ^2m"+i cosBm+l)rr
m=0
oo
se2n = E B2m sin 2тож>
m=0
seo = O
CO
se2n+i = E B2l++\ sinBm+l)a;
m=0
Recurrence relations
for coefficients
qA22n=a2nA20n;
qA2n = (a2n-4)A2n-2qA2n;
qA22?n+2 = (a2n-4m2)A2?l
- qA2^_2, m > 2
qA23n+1 = (a2n+1-l-q)A2n+1;
^t+з = [а2п+1-Bт+1J]^++\
дВ42« = F2п-4)В22";
9B22™+2 = (&2n-4m2)B2t
-qB2^_2, m>2
qB3 =(b2n+i-i--q)B1 ;
^22Г+з = [b2n+i-{2m+lJ]B22^+\
-qB2Z+_\, m>l
Normalization
conditions
oo
№2+Е(^J
m=0
Г2 ifn = 0
~ \ 1 if n > 1
CO
m=0
oo
E(^J=1
m=0
oo
Z^yB2m+l) =1
m=0
2°. In applications, of major interest are periodical solutions of the Mathieu equation
which exist for definite values of parameters a and q (those values of a are referred to
as eigenvalues). The most important solutions are listed in Table 2.6a.
The Mathieu functions possess the following properties:
y-rr, qj,
^-x, qj, ce2n+i(rr, -q) = (-l)nse2n
2n(x, -q) = (-l)n~1se2n[^Y~x' «), se2n+i(rr, -q) = (-l)nce2n+i (у-ж, q
ce2n(x, -q) =
Selecting sufficiently large number m and omitting the term with the maximum num-
number in the recurrence relations (indicated in Table 2.6a), we can obtain approximate
relations for eigenvalues an (or bn) with respect to parameter q. Then, equating
the determinant of the corresponding homogeneous linear system of equations for
coefficients A^ (or B^) to zero, we obtain an algebraic equation for finding an(q)
(or bn(q)).
For fixed real q ф 0, eigenvalues an and bn are all real and different, while
if q > 0 then ao < b\ < a,\ < b2 < a2 < ¦ ¦ ¦
if q < 0 then ao < ai < bi < b2 < a2 < аз < Ьз < b<± < ¦ ¦ ¦
The eigenvalues possess the properties
a2n(-q) = a2n(q), b2n(-q) =b2n(q), a2n+1(-q) = b2n+1(q).
© 1995 by CRC Press, Inc.

The solution of the Mathieu equation corresponding to eigenvalue an (or bn) has
n zeros on the interval 0 < ж < тг (q is a real number).
Listed below are two leading terms of asymptotic expansions of the Mathieu func-
functions cen(x,q) and sen(x,q), as well as of the corresponding eigenvalues an(q) and
bn(q), as <? ^ 0:
ce0 =
2 ' 128'
cei = cos ж —— cos Зж, a\ = 1 + q;
8
q / cos 4ж \ d 5g2
се2 = со82ж+тA-^), «2 = 4+—;
q Г cos(n + 2)ж cos(n — 2)ж 1 2 q2
сеп = со8пж + -^ —^ — j, an = n + 2^2 _ ^ (n > 3);
sei = sin ж —— sin Зж, b\ = 1 — q;
8
sin 4ж q
se2 = sm 2ж - q , 62 = 4 --—;
зш(п + 2)ж sin(n — 2)a
The Mathieu functions are discussed in the books by Abramowitz & Stegun A964)
and Bateman & Erdelyi A955, vol. 3) in more detail.
5- y^ + (osin2a; + b)y = 0.
Utilizing the formula 2 sin2 ж = 1 — cos2rr, we obtain the Mathieu equation 2.1.6.4:
У + {\а + ь
6- V'L + (° cos2 x + b)y = 0.
Utilizing the formula 2 cos2 ж = 1 + сов2ж, we obtain the Mathieu equation 2.1.6.4:
?• v'xx ~ o[osin2(ba;) + bcos(ba;)]y = 0.
Particular solution: yo = exp\—-r cos(bx) .
8- V'ix ~ o[ocos2(ba;) + bsin(ba;)]y = 0.
Particular solution: yo = exp — — sintbx) .
L b J
9- y^ + o[A+(A-o)tan2(Aa;)]y = 0.
Particular solution: yo = [сов(Аж)]0'л.
Ю- V'L + (°tan2 x + b)y = 0.
The transformation ? = sin2 ж, w = ycosmx, where m is a root of the quadratic
equation m2 + m + a = 0 leads to the hypergeometric equation 2.1.2.158:
?(? - l)w»t + [A - m)C - ±Щ -i(m + b)w = 0.
© 1995 by CRC Press, Inc.

y'x'x + a[\+(\-a)cot2(\x)}y = 0.
Particular solution: yo = [sin(Aa;)]o'A.
= 0.
The substitution ж = ? + -|- leads to the equation 2.1.6.10: y'J? + (a tan2 ж + b)y = 0.
V'L + ° sin(bx)y'x + c[axn sin(bx) - cx2n + nxn~x\y = 0.
Particular solution: «n = exp I
V n + 1
xn+1
14. y^ + (o sin ж + b)y'x + o(b sin ж + cos x)y = 0.
Particular solution: yo = exp(acosx).
15. yxx + o,sinn(bx)y'x + c[a sin"(foc) — c]y = 0.
Particular solution: yo = e~cx.
16. yxx + [asinn(bx) + c]y'x + ac sin" (Ьж)у = 0.
Particular solution: yo = e~cx.
17. yxx + (o sin" x -\- b sin ж)у^, + b(a sin""* ж + cos ж)у = О.
Particular solution: yo = ехр(бсовж).
18. yxx + (ax + b) sin"(cx)y'x — asinn(cx)y = 0.
Particular solution: yo = ax + b.
19. y'lx + ож" sinm(bx)y'x - ож"-1 sinr"(te)y = 0.
Particular solution: yo = ж.
20. yxx + ож" sin (bx)y'x + с[ож"+'в э1п"г(Ьж) — еж2*1 + /гж'в—1]у = 0.
х у
21. Ухх ~^~ (° COS Ж "I" ^)Уа; "I" °(^ COS Х — S^n Ж)У ^ 0-
Particular solution: yo = ехр(—a sin ж).
( с
Particular solution: yo = exp f — ———
\ /C ~t~ -1-
22- У^ + °cos"(fa;)y4 + c[acosn(bx) - c]y = 0.
Particular solution: yo = e~cx.
23. y^ + [acosn(bx) + c]y'x + accosn(bx)y = 0.
Particular solution: yo = e~cx.
24. yxx -\- (a cos" x -\- b cos ж)у^, + b(a cosTl+1 ж — sin ж)у = 0.
Particular solution: yo = exp(—6 sin ж).
© 1995 by CRC Press, Inc.

25. yxx + (ax + b) cosn(cx)y'x — acosn(cx)y = 0.
Particular solution: yo = ax + b.
26- V'L + ахП cosm(bx)y'x - ax'1 cosm(bx)y = 0.
Particular solution: yo = %¦
27. y'lx + axn cosm(bx)y'x + c[axn+k cosm(bx) - cx2k + кхк-г]у = О.
Particular solution: yo = exp( — xk+1).
V к + 1 /
28- V'L + (o - A) ta.n{Xx)y'x + aXy = 0.
Particular solution: yo = [cos(Aa?)]0'A.
29- V'L + atanxy'x + by = 0.
1°. The substitution ? = sinrr leads to an equation of the form 2.1.2.155:
(C2 - 1)J& + A - аЩ -by = O.
2°. Solution with а = -2:
sin(/crr) + C2 cos(/crr) if b + 1 = /c2 > 0,
V cos ж = .
'. Cisinh(/crr)+C2cosh(/crr) if b + 1 = -k2 < 0.
3°. Solution with a = 2, 6 = 3:
у = C\ cos3 x + Ci sin x(l + 2 cos2 ж).
30- Ухх + ntanxy'x + a2(cosxJny = 0.
Solution: у = C\ sin(cra) + C2 cos(au), where и = f cos™
31- t/i» + tan xy'x + a2 cos2 «(sin хJп~2у = 0.
Solution:
у = vsinrr C\ J_^_ ( — sin™ x J + C2Y_^_ ( — sin™:
where Jv and Yv are Bessel functions.
32. yxx -\- a tan xy'x -\- (b tan2 ж + c)y = 0.
This is a special case of equation 2.1.6.55.
33. yxx -\- tan xy'x — a(a — 1) cot2 ж у = 0.
Solution:
© 1995 by CRC Press, Inc.
sina;| (Ci + C2ln | sin ж|) iia=\.

34- v'L - 2Л tan(\x)y'x + (ax2 + bx + c)y = 0.
The substitution и = ycos(Xx) leads to an equation of the form 2.1.2.6:
v!'xx + (ax2 + bx + с + X2)u = 0.
35- v'L ~ 2Atan(Aa;)y4 + (ax2n + bxn~x - A2)y = 0.
The substitution и = ycos(Xx) leads to an equation of the form 2.1.2.10:
36- V'L + аЬапп(Ъх)у'х + c[otan"(ba;) - c]y = 0.
Particular solution: yg = e~cx.
37- V'L + atann(\x)yx + blotan^+^Aa;) + (A - b) tan2(Aa;) + А]у = 0.
Particular solution: yo = [cos(Aa;)]b'A.
38- У'1Х + °tan" x У'х + (° tan"+1 x - a tan" x + 4)y = 0.
Particular solution: yo = sin ж cos ж.
39- y'L + [atan"(fec) + c]y'x + actann(bx)y = 0.
Particular solution: yo = e~cx.
40- У'1Х + tan ж(°tan" x + b- l)y'x + (ab tan"+2 x - a tan" x + 1b + 2)y = 0.
Particular solution: yo = sin x cos x.
41- y'L + (ax + b) tann(cx)y'x - atann(cx)y = 0.
Particular solution: yo = ax + b.
42. y'L + axn taiim(bx)yx - ax™'1 tanm(bx)y = 0.
Particular solution: yo = %¦
43. y'lx + axn tanm(bx)y'x + c[axn+k tanm(bx) - cx2k + кхк-г]у = 0.
Particular solution: yo = exp( —
V к + 1
xk+1).
44- V'L + cot x У'х + "{У
The substitution ? = cos ж leads to the Legendre equation 2.1.2.148:
; + (b2 - a2)y = 0.
„ ,. , ... cos(bx)
Particular solution: yo = —-
smiax)
© 1995 by CRC Press, Inc.

46- V'L + (А - о) cot(Aa;)y; + аХу = 0.
Particular solution: yo = [sin(Aa;)]o^A.
47- V'L + ° cot(Xx)y'x + by = 0.
The substitution ? = Аж + -у leads to an equation of the form 2.1.6.29:
y^t — aA tan ? y^ + b\~ у = 0.
48. y^ - 2o cotBoa;)y4 - b2 sin2Bax)y = 0.
Solution: у = C\ exp — вш2(аж) + Ci exp sin2(aa:) .
L Qj J L Qj J
49- V'L -ncotxy'x+ a2(sin жJггу = О.
Solution: у = C\ sin(cra) + C<i cos(au), where и = /sin™ xdx.
50- v'L - 2 со*Bж)у^ + °tan2 ж у = o.
The substitution ? = cos ж leads to the Euler equation 2.1.2.118: ?22/^t — ?2/^ + ay = 0.
51- У'1Х + acot(\x)y'x + b[X+ (X-a-b) cot2(Xx)]y = 0.
Particular solution: yo = [sin(Aa;)]b^A.
52- У^ + ocota;y4 + (b cot2 x + c)y = 0.
This is a special case of equation 2.1.6.55.
53- У'1Х + 2Л cot(Xx)y'x + (ax2 + bx + c)y = 0.
The substitution и = ysin(Xx) leads to 2.1.2.6: u'^x + (ax2 + bx + с + X2)u = 0.
54- y'L + 2Acot(Xx)y'x + (ax2n + bx71-1 - X2)y = 0.
The substitution и = ysin(Arr) leads to 2.1.2.10: u"xx + (ax2n + бж™)^ = 0.
55- Уха + (°tan x + b cot x)y'x + (a tan2 x + f3 cot2 x + j)y = 0.
The transformation ? = sin2 x, у = w sin™ ж cosm ж, where n and m are roots of the
quadratic equations
n2 + (b-l)n + 0 = O, m2 - (a+l)m + a = 0,
leads to the hypergeometric equation 2.1.2.158:
56- y'L + acotn(bx)y'x + c[a cotn(bx) - c]y = 0.
Particular solution: yo = e~cx.
© 1995 by CRC Press, Inc.

57. у'?х + [a cotn(bx) + c]y'x + ас cot"(bx)y = 0.
Particular solution: yo = e~cx.
58- V'ix + (ax + b) cotn(cx)y'x — acotn(cx)y = 0.
Particular solution: yo = ax + b.
59- V'L + ахП cotm(bx)y'x - ож"-1 cotm(bx)y = 0.
Particular solution: yo = x.
60. y'lx + ож" cotm{bx)y'x + c[axn+k cotm(bx) - cx2k + кхк~г]у = О.
Particular solution: yo = exp( xk+1).
V к + 1 /
61- XV'L + Kax2 + bx) sinn(cx) + 2]y'x + b sinn(cx)y = 0.
Particular solution: yn = a-\ .
x
62- XV'L + (axn+1 + b sin x)y'x + axn(b sin x + n)y = 0.
Particular solution: yo = exp( xn+1).
V n + 1 /
63. xyxx + (ож" + bx sin ж)у^, + [Ь(ож" — 1) sin ж + апхп~1]у = 0.
Particular solution: y0 = x exp ( xn ).
V n /
64- XV'LX + ож" siiim(bx)yx — [a(cx + 1)ж"~1 sinm(bx) + c2x + 2c]y = 0
Particular solution: yo = xecx.
fi^ I4i" 4- \nrrn iinm(hv\ 4- fAii1 4- n( f 1 \vn~l iinm(hv\ii — fl
Particular solution: yo = x1~c.
66. жу^, + ож соэ"(Ьж)у^, — [о(сж + 1) соэ"(Ьж) + с2х + 2с]у = 0.
Particular solution: yo = хесх.
67. жу^, + [(ож2 + bx) соэ"(сж) + 2]у'х + bcosn(cx)y = 0.
Particular solution: yo = а -\ .
х
68- XV'L + (ож"+1 +bcosm х)у'х + axn(bcosm ж + п)у = 0.
Particular solution: yo = ехр( хп+1).
V п + 1 /
69- XV'LX + [°ж" cosm(bx) + с]у'х + а(с — 1)ж"~1 cosm(bx)y = 0.
Particular solution: yo = х1~с.
© 1995 by CRC Press, Inc.

7ft 'Vtl —I— InT^ —I— Г%1(* €*C\^^' 1" ill —I— \\~%\ Cl It*^1 1 1 €*C\^}^' It* —I— ClTi'K*'^' 111 ft
I \J • cf *w \^ 11**1/ ^^ UtJLr LUB / У-т> ^^ 11/11**1/ II LUB *l/ ^^ i* I fc*l/ Jy VJ.
Particular solution: yo = x exp( xn).
V n /
71- XV'L ~ 2Xx tan(Xx)y'x + (ax + b)y = 0.
The substitution и = ycos(Xx) leads to an equation of the form 2.1.2.59:
xv!'xx + [{a + X2)x + b}u = 0.
72- XV'L + ax tann(bx)y'x - [a(cx + 1) tan"(ba;) + c2x + 2c]y = 0.
Particular solution: yo = xecx.
XV'L + Kax2 + bx) tann(cx) + 2]y'x + b tann(cx)y = 0.
Particular solution: yn = a-\ .
x
XV'L + (axn+1 +btanm x)y'x + axn(b tanm x + n)y = 0.
Particular solution: yo = exp( xn+1).
V n + 1 /
75- XV'L + (°ж" + bx tanm x)v'x + [b(axn - 1) tan™ x + anxn~x\y = 0.
Particular solution: yo = x exp( xn).
V n /
76- XV'L + \ахП tan™(ba;) + c]y'x + a(c - I)*™ tanm(bx)y = 0.
Particular solution: yo = x1~c.
77'• XV'L + 2Xx cot(Xx)y'x + (ax + b)y = 0.
The substitution и = ysin(Xx) leads to an equation of the form 2.1.2.59:
xv!'xx + [{a + X2)x + b}u = 0.
78- XV'L + ax cotn(bx)y'x - [a(cx + 1) cot"(ba;) + c2x + 2c]y = 0.
79- XV'L + (axn+1 + b cotm x)y'x + axn(b cot™ x + n)y = 0.
Particular solution: yo = exp (
V n + 1
xn+1
80- XV'L + (°ж™ + bx cotm x)v'x + [b(axn - 1) cot™ x + anxn~^\y = 0.
Particular solution: yo = x exp( xn ).
V n /
XV'L + \ахП cotm(bx) + c]y'x + a(c - I)*™ cotm(ba;)y = 0.
Particular solution: yo = x1~c.
© 1995 by CRC Press, Inc.

82. x2y'x[x + x(a sin" ж + l)y'x + b(a sin" x - b)y = 0.
Particular solution: yo = x~ .
83. x2y'Jx + x(a sin" x + b)y'x + b(a sin" x - l)y = 0.
Particular solution: yo = Х~Ь¦
84- x2y'L + ахП sinm(bx)y'x + фж"-1 sin™^) - с - l]y = 0.
Particular solution: yo = ж~с.
85- x2v'L + x(a cos" ж + !)У4 + b(° cos" x-b)y = 0.
Particular solution: yg = x~ .
86. ж2?/^ + x(a cos" ж + b)y'x + b(a cos" ж - l)y = 0.
Particular solution: yo = ж~ .
87- x2v'L + ахП cosm(bx)y'x + c\axn~x соът(Ъх) - с - \\y = 0.
Particular solution: yo = ж~с.
88- ж2У^ - 2Xx2 tan(Xx)yx + (ax2 + bx + c)y = 0.
The substitution и = ycos(Xx) leads to an equation of the form 2.1.2.110:
x2ulx + [(a + X2)x2 + bx + c]u = 0.
89. x2yxx + жA - 2ж tan x)y'x — (ж tan ж + v2)y = 0.
Solution: у cos ж = Ci ./„(ж) + С21^(ж), where Jv and 1^ are Bessel functions.
90. x2y'xx — xBx tan ж + fe)y^, + (ax2 -\- bx -\- c-\- kx tan ж)у = 0.
The substitution и = у cos ж leads to an equation of the form 2.1.2.126:
x uxx ~ kxu'x + [(a + l)x + bx + c]u = 0.
91- x2v'L + x(a tan" x + !)< + b(°tan" х~ь)у = o.
Particular solution: yo = x~b.
92- x2v'L + x(a tan" x + b)v'x + Hatan" x-l)y = 0.
Particular solution: yo = ж~ .
93. x2y'x'x + ож" tanr"(^)y^, + фж" tanr"(te) - с - l]y = 0.
Particular solution: yo = ж~с.
94- ж2У^ + 2Xx2 cot(\x)y'x + (ax2 +bx + c)y = 0.
The substitution и = ysin(Xx) leads to an equation of the form 2.1.2.110:
x2uxx + [(a + X2)x2 + bx + c]u = 0.
© 1995 by CRC Press, Inc.

95. x2yxx + жBж cot ж + k)y'x + (ax2 + bx + с + кх cot ж)у = О.
The substitution и = у sin ж leads to an equation of the form 2.1.2.126:
x2uxx + kxu'x + [(a + l)x2 + bx + c]u = 0.
96. x2yxx + ж(о cot" ж + l)y'x + b(a cot" ж — b)y = 0.
Particular solution: yo = x~b.
97. x2yxx + ж(о cot" ж + b)y^, + b(a cot" ж — l)y = 0.
Particular solution: yo = x~b.
98. x2yxx + ож" cot (bx)y'x -\- c[axn~1 cotm(bx) — с — l]y = 0.
Particular solution: yo = x~°.
99. ж4у" + osinf — ] +b \y = 0.
L \x J J
The transformation ? = l/ж, w = у/ж leads to an equation of the form 2.1.6.3:
w'^ + [asin(A?) + b]w = 0.
100. xAyxx + ax2 sinn(bx)yx + [a(c — x) эт"(Ьж) - c2]y = 0.
/ с \
Particular solution: y0 = xexpf — j.
\ X У
101. x4y'x'x + ax2 cosn(bx)y'x + [a(c - ж) cosn(bx) - c2]y = 0.
/ С \
Particular solution: yo = ж exp I — ).
V ж /
102. ж4у4'а, + ож2 tann(bx)y'x + [а(с - ж) tan"(fa;) - с2]у = 0.
/ с \
Particular solution: yo = ж exp I — 1.
V ж /
103. хАухх + ах2 cotn(bx)y'x + [а(с — ж) cot"(foc) - с2]у = 0.
/ С \
Particular solution: у0 = хexpf — j.
\ X У
104. sinBx)yxx -y'x + 2o2 sin ж у = 0.
Solution: у = C\ sin(cra) + C<i cos(cra), where и = f \ftaiaxdx.
105. sinBx)y'x'x - 2ny'x + 2o2 sin2 ж^апжJ™"^ = 0.
Solution: у = C\ sin(au) + C<i cos(au), where и = /tan™ xdx.
106. sin ж yxx + cos ж y'x + u(u + 1) sin ж у = 0.
The substitution ? = cos ж leads to the Legendre equation 2.1.2.148:
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107. sin ж yxx + Bn + 1) cos ж y'x + (y — n) (y + n + 1) sin ж у = 0,
where rv is an arbitrary number, n is a positive integer.
The substitution ? = cos ж leads to an equation of the form 2.1.2.148:
2 + (n-v)(v + n + l)y = 0.
108. sin2 x y'lx +ay = 0.
This is a special case of equation 2.1.6.110.
109. sin2 x yxx — [a sin2 x + n(n — l)]y = 0, n = 1, 2, 3, ...
1 d \n г- г-
) (C^ + C"^)
Solution: у = 8шп
sin x ax
110. sin2 x yxx + (a sin2 x + b)y = 0.
Set x = 2?. Utilizing the trigonometric formulae sin 2? = 2 sin ? cos ? and 6 = 6(sin ? +
cos2 ?J and dividing both sides of the equation by sin x, we arrive at an equation of
the form 2.1.6.55:
y'tf + (b tan2 С + b cot2 С + 4a + 2b)y = 0.
111. sin2 ж j/^ - {[(o2b2 - (a+lJ]sin2# + a(a + l)bsin2# + a(a- l)}y = 0.
Particular solution: yo = e°ba: sm° ж (cos % + b sin ж).
112. sin2 x yxx + sin x cos ж y^, + [v(v + 1) sin2 ж — n2]y = 0,
where v is an arbitrary number, n is a nonnegative integer.
The transformation ? = cos x, у = w sin™ ж leads to an equation of the form 2.1.2.149:
(C2 - 1)«4'? + 2(n + 1)?«4 + (n - i/)(n + v + l)w = 0.
113. sin2 ж y^, + sin ж(о cos ж + b)y'x + (a cos2 ж + /3 cos ж + 7)y = 0.
Set x = 2?. Utilizing the trigonometric formulae
sinB?) = 2 sin ? cos ?, cosB^) = cos2 ? - sin2 ?, 6 = 6(sin2 ^ + cos2 ^),
C = /3(sin2 С + cos2 ?), 7 = 7(sin2 ^ + cos2 ^J,
and dividing all the terms by sin x, we arrive at an equation of the form 2.1.6.55:
114. cos2 ж yxx — [a cos2 ж + n(n — l)]y = 0, n = 1, 2, 3, ...
Solution: у = cosn x( —^—-^Y(Ciex^ + C2e-X^).
V cos x ax / '
115. cos2 ж у^'з, + (a cos2 ж + Ъ)у = 0.
The substitution ж = ? + -§¦ leads to 2.1.6.110: sin2 ?y^ + (a sin2 ? + b)y = 0.
© 1995 by CRC Press, Inc.

116. cos2 ж yxx + asinBx)y'x + [ЬсоэBж) + c]y = 0.
Dividing the equation by cos ж and utilizing the formulae
sinBa;) = 2sina;cosa;, cosBrr) = cos2 ж — sin2 x, c = c(sin2 x + cos2 x),
we obtain an equation of the form 2.1.6.55:
yxx + 2atanxy'x + [(c - b) tan2 x + b + c]y = 0.
117. cos2(ax)yxx + (n — l)osinBo«)y^, + na2[(n — 1) эт2(аж) + соэ2(аж)]у = О.
Particular solution: yo = совп(аж).
118. cos2 xyxx + cos#(asin# + b)y'x + (a sin2 x + Csinx + j)y = 0.
The substitution x = ? + -|- leads to an equation of the form 2.1.6.113: sin2?yl', —
sin ? (a cos ? + b)y^ + (a cos2 ? + /3 cos ? + j)y = 0.
119. sin ж cos2 x yxx + cos#(asin2 x + b)y^, + с sin ж у = 0.
1°. Dividing the equation by sin ж cos2 ж and assuming b = b(sin2 x + cos2 ж), с =
c(sin2 ж + cos2 ж), we obtain the equation 2.1.6.55:
) ]y^ + с(^ап2ж + 1)у = 0.
2°. Particular solutions:
Уо = cos° ж for с = aF + 1),
yo = tan1"^ for c= (a + 2)F-l),
y0 = sin1 жсов^6 ж for с = 2(a + b - 1).
120. sin ж cos2 ж y^ + созж(азт2 ж — l)y^, + bsin3 ж у = 0.
Solution: у = Ci(cosx)kl + ^(совж)'02, where /ci and fe are the roots of the
quadratic equation /c2 — ak + b = 0.
121. sin2 ж cos2 ж y^, + (o sin2 x -\-b cos2 ж + с sin2 ж cos2 ж)у = 0.
Dividing the equation by sin ж cos2 ж and assuming a = a(sin ж+сов2ж), b = b(sin x+
cos2 ж), we arrive at the equation 2.1.6.55:
v'xx + (a *ап2 ж + b cot2 ж + a + b + c)y = 0.
122. [оэт(Лж) + bx + c]yxx + a\2 эт(Лж)у = 0.
Particular solution: yo = авш(Аж) + Ьж + с.
123. [осоэ(Лж) + bx + с]ухх + оЛ2 соэ(Лж)у = 0.
Particular solution: yo = йсов(Аж) + Ьж + с.
124. в-тп(ах)Ух-х + (ж2 - Ь2)у^ - (ж + Ъ)у = 0.
Particular solution: yo = х — b.
© 1995 by CRC Press, Inc.

125. sinn(Xx)y'^x + [A2sin"(Aa;) + acos"-4(Aa;)]y = 0.
У
The transformation ? = tan(Aa;), w = 7-—г leads to an equation of the form 2.1.2.7:
cos(Aa;)
w'^ + a\-2?-nw = 0.
126. (o sin" x + b)y'J,x + (c sin" x + d)y'x + A[(c - oA) sin" x + d - b\]y = 0.
Particular solution: yg = e~ .
127. cosn(Xx)y'^x + [A2 cos"(Aa;) + osin"-4(Aa;)]y = 0.
The substitution Xx = -|- — A? leads to an equation of the form 2.1.6.125.
128. cosn(ax)y'^x + (x2 - Ь2)у'х - (x + b)y = 0.
Particular solution: yg = x — b.
129. (о cos" x + b)yxx + (с cos" x + d)y'x + A[(c - oA) cos" x + d- bX]y = 0.
Particular solution: yg = e~Xx.
130. (o tan" x + b)y'lx + (ex + d)y'x - cy = 0.
Particular solution: yg = ex + d.
131. (o tan" x + b)y'lx + (ctan" x + d)y'x + X[(c - oA) tan" x + d- bX]y = 0.
Particular solution: yg = e~Xx.
132. (o cot" x + b)y'lx + (c cot" x + d)y'x + X[(c - oA) cot" x + d- bX]y = 0.
Particular solution: yg = e~Xx.
2.1.7. Equations Containing Inverse Trigonometric Functions
1* Уxx "I" (ax + ^ + carcsina;)y^, + [c(ax + b) arcsina; + a]y = 0.
Particular solution: yg = exp(—\ax2 — bx).
2. yxx + b(arcsinx)nyx + c[b(arcsina;)" — c]y = 0.
Particular solution: yg = e~cx.
3. yxx + b(arcsinx)ny'x + a [bxm (arcsina;)" — ax2m + mxm~1]y = 0.
Particular solution: yg = expl x ).
V m + 1 /
4- y'xx "I" (ax "I" b)(arcsina;)"y^, — o(arcsina;)"y = 0.
Particular solution: yg = ax + b.
© 1995 by CRC Press, Inc.

5. y'^x + axn(arcsinx)my'x — axn 1(arcsina;)r"y = 0.
Particular solution: yo = x.
6* Ухх + (ax + & + с arccos x)y'x + [c(ax + b) arccos x + a]y = 0.
Particular solution: yo = exp(—-i-аж2 — bx).
7- Ухх + b(arccos x)ny'x + c[b(arccos x)n — c]y = 0.
Particular solution: yo = e~cx.
8- V'xx + b(arccos x)ny'x + a[ba;m(arccos x)n - ax2m + тхт~1]у = О.
Particular solution: yn = exp( xm+1).
\ m + 1 /
9- Vxx + (ax + b) (arccos x)ny'x — o(arccos x)ny = 0.
Particular solution: yo = ax + b.
10. yxx + axn (arccos x) my'x — axn~1 (arccos x)my = 0.
Particular solution: yo = %¦
11. yxx + (ax + b + carctana;)y^, + [c(ax + b) arctana; + a]y = 0.
Particular solution: yo = exp(—уаж2 — Ьж).
12- Ухх + b(arctan a;)"y4 + c[b(arctan x)n - c]y = 0.
Particular solution: yo = e~ca:.
13. y^ + b(arctana;)"y4 + a[bxm(arctana;)" - ax™ + тхт~1]у = 0.
Particular solution: yo = exp( xm+1).
V m + 1 /
14. y^, + (oa; + b)(arctana;)Tly^, — o(arctan x)ny = 0.
Particular solution: yo = ax + b.
15. y^ + aa;™(arctana;)"^ - oa;"-1(arctana;)r"y = 0.
Particular solution: yo = x.
16. yxx + (ax + b + с arccot ж)у^, + [с(ож + b) arccot x + o]y = 0.
Particular solution: yo = exp(—-i-аж2 — bx).
17. у'1х + b(arccot ж)"у^, + c[b(arccot x)n - c]y = 0.
Particular solution: yo = e~cx.
18. Ухх "Ь b(arccot х)пу'х -\- а [Ьж™1 (arccot ж)" — ах2т -\- тхт г]у = 0.
Particular solution: yo = ехр( хт~
V т + 1
© 1995 by CRC Press, Inc.

У'хХ + (ax + b)(arccot x)ny'x - o(arccot x)ny = 0.
Particular solution: yo = ax + b.
20- yxx + axn(arccotx)my'x — axn 1(arccot x)my = 0.
Particular solution: yo = x.
21. xyxx + аж arcsin ж y^, — [a(bx + 1) arcsin ж + b(bx + 2)]y = 0.
Particular solution: yo = xebx.
22. жу^, + [a(bx + 1) arcsin ж + bx — l]y^ + ab2x arcsin ж у = 0.
Particular solution: yo = (bx + 1)е~Ьх.
23. xyxx + [(ax2 + bx) arcsin ж + 2]y^, + b arcsin ж у = 0.
Particular solution: yn = a-\ .
x
24. xyxx + [ax (arcsin x)n + b]y'x + a(b — 1) (arcsin x)ny = 0.
Particular solution: yo = x1~b.
25. xyxx + (ож""* + b arcsin ж)у^, + axn(b arcsin ж + n)y = 0.
Particular solution: yo = exp( xn+1).
V n +1 /
26. жу^ + (axn + Ьж arcsin ж)у^, + [b(axn — 1) arcsin ж + anxn~1]y = 0.
Particular solution: y0 = x exp ( xn ).
V n /
27. x2yxx + Ьж arcsin ж y^, + o(b arcsin ж — a — l)y = 0.
Particular solution: yo = x~a.
28. x2yxx + ж(Ь arcsin ж + 2)y^, + [b(ax + 1) arcsin ж — о2ж2]у = 0.
Particular solution: yo = —e~ax.
x
29. жу^, + ax arccos xy'x — [a(bx + 1) arccos ж + b(b + l)]y = 0.
Particular solution: y0 = xebx.
30. жу^, + [a(bx + 1) arccos ж + bx — l]y'x + ab2x arccos ж у = 0.
Particular solution: y0 = (bx + l)e~bx.
31. жу^ + [(ax2 + bx) arccos ж + 2]y^, + b arccos ж у = 0.
Particular solution: yo = a -\ .
x
© 1995 by CRC Press, Inc.

32- xv'Lx + [аж(агссоэ x)n + b]y'x + a(b - l)(arccos x)ny = 0.
Particular solution: yn = x .
33. xyxx + (ож""* + b arccos x)y'x + axn(b arccos x + n)y = 0.
Particular solution: yn = exp (
V n + 1
xn+1
34. жу^, + (аж™ + bx arccos ж)у^, + [b(axn — 1) arccos ж + anxn x]y = 0.
Particular solution: yn = x exp I ж™ ).
35. x2yxx + bx arccos ж y'x + a(b arccos ж — о — l)y = 0.
Particular solution: yn = x~a.
36. x2yxx + ж(Ь arccos ж + 2)y^, + [b(ax + 1) arccos ж — о2ж2]у = 0.
Particular solution: yo = —e~ax.
x
37. жу^, + ax arctan ж y'x — [a(bx + 1) arctan ж + b(b + l)]y = 0.
Particular solution: yo = жеЬж.
38. жу^, + [a(bx + 1) arctan x -\- bx — l]y'x -\- ab2x arctan ж у = 0.
Particular solution: yo = (bx + l)e~bx.
39. жу^ + [(ax2 + bx) arctan ж + 2]y'x -\- b arctan ж у = 0.
b
Particular solution: yn = a-\ .
x
40- xv'ix + [аж (arctan ж)" + b]y'x + a(b — 1) (arctan ж)"у = 0.
Particular solution: yn = x .
41. жу^ + (ож""* + b arctan ж)у^, + axn(b arctan ж + n)y = 0.
Particular solution: yn = exp( xn+1).
V n + 1 /
42. жу^ + (axn + Ьж arctan ж)у^, + [^(ож" — 1) arctan ж + anxn~1]y = 0.
Particular solution: yn = x exp I xn\.
43. жу^, + (^(arctan" ж + b)y^, — o(arctan" ж + b)y = 0.
Particular solution: yn = x.
44. жу^, + b arctan" xy'x + a(b arctan" ж — ож)у = 0.
Particular solution: yn = e~ax.
© 1995 by CRC Press, Inc.

45. xyxx + o(arctanTl ж + bx)y'x + obarctan™ жу = О.
Particular solution: yo = e~bx.
46. xyxx + barctan™ жу^, + oж(barctanrг ж — ож + l)y = 0.
Particular solution: yo = exp(—-^-аж2).
47. x2yxx + Ьж аг^апжу^, + о(Ьаг^апж — a — l)y = 0.
Particular solution: yo = x~a.
48. ж2у4'а, + ж(Ьаг^апж + 2)y'x + [b(ax + 1) аг^апжа2ж2]у = О.
Particular solution: yo = тге~ах¦
49. x2yxx + ож^г^ап™ ж + b)y^, — o(arctan" ж + b)y = 0.
Particular solution: yo = x.
50. x2yxx + barctan" жу^, + o(barctan" ж — ож2)у = 0.
Particular solution: yo = e~ax.
51. x2yxx + o(arctan" ж + Ьх2)у'х + obarctan" ж у = 0.
Particular solution: yo = e~ .
52. x2yxx + ж[(ож + b) arctan" ж + 2]y^, + barctan" ж у = 0.
Particular solution: yo = a -\ .
x
53. xyxx + ож arccot ж y'x — [о(Ьж + 1) arccot ж + Ь(Ьж + 2)]у = 0.
Particular solution: yo = xebx.
54. жу^, + [о(Ьж + 1) arccot ж + Ьж — l]y^, + оЬ2ж arccot ж у = 0.
Particular solution: yo = (bx + l)e~bx.
55. xyxx + [(ож2 + Ьж) arccot ж + 2]у^, + b arccot ж у = 0.
Particular solution: yo = a -\
x
56- XV'LX + [аж(arccot ж)" + b]y^, + a(b — 1) (arccot ж)"у = 0.
Particular solution: yo = x .
57. xyxx + (ож""* + b arccot ж)у^, + axn(b arccot ж + n)y = 0.
/ft ~>
Particular solution: yo = exp( — xn+1
\ n + 1
© 1995 by CRC Press, Inc.

58. xyxx + (axn + bx arccot x)y'x + [^(аж™ — 1) arccot x + anxn x]y = О.
Particular solution: yn = жехр( xn).
V n J
59. x2yxx + ^ж arccot xy'x-\- a(b arccot ж — о — l)y = 0.
Particular solution: yo = x~a.
60. x2yxx + x(b arccot ж + 2)y^, + [b(ax + 1) arccot ж — о2ж2]у = 0.
Particular solution: «n = —e
x
61. (ож2 + 6)^ + с(ож2 + b)(arcsina;)n^ - 2a[ca;(arcsina;)n + l]y = 0.
Particular solution: yo = ax + b.
62. (ож2 + b)yxx + c(ax2 + b)(arccosx)nyx — го^жСагссоэж)" + l]y = 0.
Particular solution: yg = ax + b.
63. (ж2 + l)y^ - [о2 (ж2 + 1)(аг^апжJ + о]у = 0.
Particular solution: yo = (ж2 + 1)~°'2 ехр(аж arctana;).
64. (ож2 + Ь)ухх + с(ах2 + b)(arctar^)"y4 - 2o[cж(arctanж)rг + 1]у = 0.
Particular solution: yo = ах + Ъ.
65. (ож2 + b)y'^ + с(ах2 + Ъ)(arccot ж)"у^, - 2o^(arccot ж)" + 1]у = 0.
Particular solution: yg = ах + Ъ.
66. х4ухх + ож2 агсэтжу^, + [а(Ь — ж) агсэ1пж — Ь2]у = 0.
/ Ь
Particular solution: yo = х ехр —
V х
67. х4ухх + ож2 arccos ж у'х + [о(Ь — ж) arccos ж — Ь2]у = 0.
Ь
х.
Particular solution: yo = х ехр ( — ).
V х /
68. х4ухх + ож2 аг^апжу^, + [а(Ь — ж) arctaпж — Ь2]у = 0.
Ь
х.
Particular solution: yo = х ехр ( — J.
69. х4ухх + ож2 arccot ху'х-\- [а(Ь — ж) arccot ж — Ь2]у = 0.
/ Ь
Particular solution: yo = х ехр —
V х
2 + 1Jухх + [о(аг^апжJ
у
The transformation ? = arctana:, w = —1^=^= leads to an equation of the form
V2 + 1
70. (ж2 + 1Jухх + [о(аг^апжJ + Ьаг^апж + с]у = 0.
2.1.2.6: w'^ + (a?2 + b? + с + l)w = 0.
© 1995 by CRC Press, Inc.

71. (ж2 + 1Jу'пХ + [b(arctana;)" - 1]у = 0.
У
The transformation ? = arctanrr, w = —-f^=^= leads to an equation of the form
Vx2 + 1
2.1.2.7: w'^ + b?nw = 0.
72. (ж2 + lJyxx + [o(arccot жJ + barccot ж + с]у = 0.
The transformation ? = arccotrr, w = —-^=^= leads to an equation of the form
Vx2 + 1
2.1.2.6: w'^ + (a?2 + b? + c+l)w = 0.
73. (ж2 + lJyxx + [b(arccot ж)" - l]y = 0.
У
The transformation ? = arccotx, w = —, leads to an equation of the form
2.1.2.7: w'^ + b?nw = 0.
74. (ож2 + bJy'^x + (ex + rf)(arcsini)nj/^, — с(агсэтж)ггу = О.
Particular solution: yo = ex + d.
75. (ж2 + aJy'lx + b(x2 + a)(arcsinx)ny'x - [Ьж(агс3тж)гг + а]у = 0.
Particular solution: yo = V x2 + a.
76. (ож2 + ЪJухх + c(ax2 + ЬХагсэтж)™^ + [с(агсэ1пж)гг - 2ож - l]y = 0.
Particular solution: yn = exp ( — / —» r-).
V J ax2 + b)
11. (ax2 + bJyxx + (ex + с?)(агссозж)ггу^, — с(агссоэж)ггу = 0.
Particular solution: yo = ex + d.
78. (ж2 + aJyxx + b(x2 + o)(arccos ж)"у^, - [Ьж(агссоэ ж)" + a]y = 0.
Particular solution: yo = V x2 + a.
79. (ож2 + bJy'lx + c(ax2 + b)(arccosx)ny'x + ^(агесовж)" - 2ож - l]y = 0.
dx
Particular solution:
= exp I — /
ax2 + b
80. (ож2 + bJy'lx + (ex + d)(arctanx)nyx - c(arctanx)"y = 0.
Particular solution: yo = ex + d.
81. (ж2 + oJy4'a, + Ь(ж2 + о)(аг^апж)ггу4 - [Ьж(аг^апж)" + o]y = 0.
Particular solution: yo = V x2 + a.
82. (ож2 + ЪУу'^ + c(ax2 + b)(arctanx)ny'x + [c(arctan ж)" - 2ож - l]y = 0.
Particular solution: yo = exp I — / ¦
ax2 + b J'
© 1995 by CRC Press, Inc.

83. (ож2 + bJyxx + (ex + d)(arccot x)ny'x — c(arccot ж)"у = О.
Particular solution: yo = ex + d.
84. (ж2 + aJy'lx + b(x2 + o)(arccot ж)"у^, - [fcc(arccot ж)" + a]y = 0.
Particular solution: yo = \J x1 + a.
85. (ож2 + bJyxx + c(ax2 + b)(arccot ж)"у^, + [c(arccot ж)" - 2ож - l]y = 0.
/ f dx \
Particular solution: yo = exp I — / — 1.
V J axz + b I
2.1.8. Equations Containing Combinations of Exponential, Logarithmic,
Trigonometric, and Other Functions
!• V'L + aeXxy'x + b[b + aeXx tan(bx)]y = 0.
Particular solution: yo = cos(bx).
2- У'1Х + аеХхУх + b[b - aeXx cot(bx)]y = 0.
Particular solution: yo = sinFrr).
3- VZ* + ° coshn(\x)y'x + b[b + а со5Ь"(Лж) tan(^)]y = 0.
Particular solution: yo = cosFrr).
4- V'L + acoshn(\x)y'x + b[b - осо3Ьгг(Лж) cot(bx)]y = 0.
Particular solution: yg = sinFa;).
5- V'L + °coshn(kx)y'x + beXx[acoshn(kx) - beXx + X]y = 0.
b
Particular solution: y0 = exp ( — — eXx ).
6- V'L + ° sinhn(Ai)^ + b[b + a sinh™^) tan(^)]y = 0.
Particular solution: yo = cos(bx).
7- V'L + ° sinhn(Xx)yx + b[b - а 31пЬгг(Лж) cot(bx)]y = 0.
Particular solution: yg = sinFrr).
8- V'L + asinhn(kx)y'x + beXx[asinhn(kx) - beXx + X]y = 0.
Particular solution: yo = expf—— e x\.
9- VZ* + ° tanhrг(Лж)y4 + b[b + a taпhrг(Лж) tan(^)]y = 0.
Particular solution: yo = cosFrr).
© 1995 by CRC Press, Inc.

- y'L + atanhn(\x)y'x + b[b - otanh"(Aa;) cot(ba;)]y = 0.
Particular solution: yo = sinFa;).
- V'L + atanhn(kx)y'x + beXx [a tanhn (kx) - beXx + X]y = 0.
Particular solution: y0 = exp ( — — eXx j.
Aa;)y; + b[b + a coth"(Aa;) tan(ba;)]y = 0.
Particular solution: yo = cos(bx).
[Xx)y'x + b[b — ocoth"(A«) cot(ba;)]y = 0.
Particular solution: yo = sinFa;).
14- y'L + acotbn{kx)y'x + beXx[acothn(kx) - beXx + A]y = 0
Particular solution: yo = exp( eXx
V Л
15- V'L + ° 1п"(Лж)у4 + Ь[Ь + а 1гГ(Аж) tan(ba;)]y = 0.
Particular solution: yo = cos(bx).
16- y'L + ° 1п"(Лж)у4 + Ь[Ь- a ln"(Aa;) cot(ba;)]y = 0.
Particular solution: yo = sinFa;).
17- y'L + °^n(kx)y'x + beXx[aln"(fea;) - beXx + X]y = 0.
Particular solution: y0 = exp ( — — eXx ).
18- y'L + acosn(kx)y'x + beXx[acosn(kx) - beXx + X]y = 0.
Particular solution: yo = exp(—— eXx
V Л
19- y'L + as\nn(kx)y'x + beAa:[osin"(fea;) — beXx + X]y = 0.
Particular solution: yo = exp( eXx
V Л
20- y'L + atann(kx)y'x + beAa:[otan"(fea;) - beXx + X]y = 0.
Particular solution: y0 = exp ( — — eXx ).
21- y'L + acotn(kx)y'x + beXx[acotn(kx) - beXx + X]y = 0.
Particular solution: yo = exp( eXx
V Л
© 1995 by CRC Press, Inc.

22- ухх + (аеХх + Ь In" ж)у^ + aeXx(b In" ж + Л)у = 0.
Particular solution: yo = expl—— е х 1.
23. у^, + (оеАа: + b cos ж)у^, + b(aeXx cos ж — sin ж)у = 0.
Particular solution: yo = ехр(—6 sin ж).
24. ухх + (аеХх + Ь cos" ж)у^ + aeXx(b cos" ж + Л)у = 0.
Particular solution: yo = expl—— е ж 1.
25. у^, + (аеХх + Ьсоэ"ж)у^ + bcos" ж (оеАа:созж — ггэ1пж)у = 0.
Particular solution: yo = expl—6 / cosn xdx).
26. ухх + (аеХх + Ь sin ж)у^, + Ь(аеХх sin ж + cos ж)у = 0.
Particular solution: yo = ехр(бсовж).
27. ухх + (аеХх + Ь sin" х)у'х + аеХх (Ь sin" ж + Л)у = 0.
Particular solution: yo = ехр ( —— еХх ).
V А /
28- У'1Х + (аеХх + Ъ sin" ж)у^, + b sin" ж (аеХх sin ж + п cos ж)у = 0.
Particular solution: уо = ехр(— Ь I sinnxdx).
V J '
29. Ухх + (аеХх + btanx)y'x + (b + 1)(аеХх tan ж + 1)у = 0.
Particular solution: yo = cosb+1 ж.
30- У^ + (°eAa: + btan" ж)у^, + aeXx(btann x + Л)у = 0.
Particular solution: yo = exp I —— eXx
V Л
31- t/i» + (aeXx + b cot ЖХ + (Ь - !) (°eAa: cot ж - l)y = 0.
Particular solution: yo = sin ~ x.
32- У^ + (oeAa: + b cot" ж)у^, + aeXx(b cot" ж + Л)у = 0.
Particular solution: yo = expl —— e x 1.
V Л /
33. y^, + (o cosh" ж + b cos ж)у^, + b(a cosh" ж cos ж — sin ж)у = 0.
Particular solution: yo = exp(—6 sin ж).
© 1995 by CRC Press, Inc.

34. yxx + (о cosh" ж + bcosm x)y'x + bcosm г x (a cosh" ж cos ж — msinx)y = 0.
Particular solution: yo = expf —6 / cosm xdx).
35. yxx + (o cosh" x + b sin x)y'x + b(a cosh" ж sin ж + cos x)y = 0.
Particular solution: yo = ехр(бсовж).
36. yxx + (a coshn x + b sinm x)y'x + bsinm~1 x (a cosh" ж sin ж + га cos ж)у = 0.
Particular solution: yo = exp( —6 / sinmxdx).
37 • V'lv + (° cosh" x + btanx)y'x + (b+l)(a cosh" ж tan ж + l)y = 0.
Particular solution: yo = cosb+1 x.
38. y^ + (° cosh" ж + b cot ж)у4 + (b — !) (° cosh" ж cot ж - l)y = 0.
Particular solution: yo = sin ~ x.
39. y^ + (o sinh" x -\- b cos ж)у^, + b(a sinh" ж cos ж — sin ж)у = 0.
Particular solution: yo = exp(—6 sin ж).
40. yxx + (o sinh" x -\-b cos ж)у^, + b cos ж (о sinh" ж cos ж — га sin ж)у = 0.
Particular solution: yo = exp( —6 / cosmrr<i2:J.
41. y^, + (o sinh" x + b sin ж)у^, + b(o sinh" ж sin ж + cos ж)у = 0.
Particular solution: yo = expFcosrr).
42. yxx + (o sinh" ж + b sin ж)у^, + b sin ж (о sinh" ж sin ж + га cos ж)у = 0.
Particular solution: yo = expf — b / sinm x dx).
43. y'lx + (osinh"ж + btanж)y4 + (b + 1) (a sinh" ж tan ж + l)y = 0.
Particular solution: yo = cosb+1 x.
44. y'lx + (a sinh" ж + b cot ж)у^, + (b — l)(o sinh" ж cot ж - l)y = 0.
Particular solution: yo = sin1~ba;.
45. y^ + (o tanh" x + b cos ж)у^, + b(o tanh" ж cos ж — sin ж)у = 0.
Particular solution: yo = exp(—6 sin ж).
46. y^, + (o tanh" x + b cos ж)у^, + b cos ж (о tanh" ж cos ж — га sin ж)у = 0.
Particular solution: yo = expf—6 / cosm xdx).
© 1995 by CRC Press, Inc.

47. yxx + (о tanh" x + b sin x)y'x + b(a tanh" x sin x + cos ж)у = О.
Particular solution: yo = ехр(бсовж).
48- V'ix + (° tanh" x + b sin ж)у?, + b sin x (a tanh" x sin ж + га cos x)y = 0.
Particular solution: yo = expf —6 / sinmжdx).
49. y'lx + (a tanh" ж + b tan ж)у^ + (b + 1) (o tanh" ж tan ж + l)y = 0.
Particular solution: yo = cosb+1 x.
50. y4/a, + (otanh"ж + bcotж)y4 + (b - 1)(a tanh" ж cot ж - l)y = 0.
Particular solution: yo = sin ж.
51. y^ + (o coth" x + b cos x)y'x + b(a coth" ж cos ж — sin ж)у = 0.
Particular solution: yo = exP(~bsinx).
52. y^ + (a coth" ж + bcos ж)у^, + bcosm~1 x (a coth" ж cos ж — тэтж)у = 0.
Particular solution: yo = exp(— b / cosm xdx).
53. yxx + (o coth" x + b sin ж)у^, + b(a coth" ж sin ж + cos ж)у = 0.
Particular solution: yo = ехр(бсовж).
54. yxx + (a coth" x + b sin ж)у^, + b sin ж (о coth" ж sin ж + rn cos ж)у = 0.
Particular solution: уо = ехр(— b / smmxdx).
55. y4/a, + (ocoth"ж + btaпж)y4 + (b + l)(ocoth" ж tan ж + l)y = 0.
Particular solution: yo = cosb+1 ж.
56. y^ + (o coth" ж + b cot ж)у^, + (b - 1) (o coth" ж cot ж - l)y = 0.
Particular solution: yg = sin ж.
57. y^ + (o In" x + b cos ж)у^, + b(a In" ж cos ж — sin ж)у = 0.
Particular solution: yo = exp(—6 sin ж).
58- Ухх + (alnnx + bcosmx)yx + bcos ж (о In" ж cos ж -гаэтж)у = 0.
Particular solution: yo = exp( — b / cosm xdx).
59. y^ + (o In" x + b sin ж)у^, + b(a In" ж sin ж + cos ж)у = 0.
Particular solution: yo = ехр(бсовж).
© 1995 by CRC Press, Inc.

60- Ухх + (a In" ж + b sin x)y'x + b sin ж (a In" x sin x + m cos x)y = 0.
Particular solution: yo = exp( —6 / sinmxdx).
61- Ухх + (o In" ж + b tan x)y'x + F + 1) (o In" ж tan ж + l)y = 0.
Particular solution: yo = cosb+1 x.
62- Ухх + (a In" ж + bcot x)y'x + (b- 1)(а1п"ж^ж - l)y = 0.
Particular solution: yo = sin x.
63- V'L + aeXx cos(bx)y'x + b[b + aeXx sin(fec)]t/ = 0.
Particular solution: yo = cos(bx).
64- Ухх + aeXx sin(bx)y'x + b[b — aeXx соэ(Ьж)]у = О.
Particular solution: yo = sinFa;).
65- VZ* + °совЬ(Ьж) 1п"(Лж)у^, - Ь[Ь + asinhFa;) 1п"(Лж)]у = 0.
Particular solution: yo = coshFa;).
66- Ухх + ° cosh(bx) cosn(\x)y'x — b[b + a sinh(te) соэ"(Лж)]у = 0.
Particular solution: yo = coshFa;).
67- Уxx + ° соэЬ"(Лж) cos(bx)y'x + b[b + а соэЬ"(Лж) sin(bx)]y = 0.
Particular solution: yo = cosFrr).
68- Ухх + ° соэЬ(Ьж) зт"(Лж)у^, — b[b + a sinh(foc) эт"(Лж)]у = 0.
Particular solution: yo = coshFrr).
69- Ухх + ° соэЬ"(Лж) sin(bx)y'x + b[b — а соэЬ"(Лж) соэ(Ьж)]у = 0.
Particular solution: yo = sinFrr).
70- Ухх + ° cosh(te) tan"^)y^ - b[b + a sinh(^) tan"(Лж)]y = 0.
Particular solution: yo = coshFrr).
71- Ухх + ° cosh(te) cot"(Лж)y^ - b[b + a sinh(^) со^(Лж)]у = 0.
Particular solution: yo = coshFa;).
72- Ухх + ° sinh(^) \nn(kx)y'x - b[b + а совЬ(Ьж) \пп(кх)]у = 0.
Particular solution: yo = sinhFa;).
73. yxx + a sinh(foc) cos"(kx)y'x — b[b + а соэЬ(Ьж) cos"(kx)]y = 0.
Particular solution: yo = sinhFa;).
© 1995 by CRC Press, Inc.

74- v'L + ° sinh"(Aa;) cos(bx)y'x + b[b + a sinh"(Aa;) sin(ba;)]y = 0.
Particular solution: yo = cos(bx).
75. yxx + asinh(bx) sinn(kx)y'x — b[b + acosh(bx) sinn(kx)]y = 0.
Particular solution: yo = sinhFa;).
76- v'L + ° sinh"(Aa;) sin(bx)y'x + b[b — a sinh"(Aa;) соэ(Ьж)]у = 0.
Particular solution: yo = sinFa;).
77• V'ix + ° sinh(ba;) tan"(fea;)y4 - b[b + a cosh(ba;) tan"(fea;)]y = 0.
Particular solution: yo = sinhFa;).
78- V'L + ° sinh(ba;) cot"(Aa;)y4 - b[b + a cosh(ba;) cot"(Aa;)]y = 0.
Particular solution: yo = sinhFa;).
79- Ухх + ° tanh"(Aa;) cos(bx)y'x + b[b + a tanh"(Aa;) sin(ba;)]y = 0.
Particular solution: yo = cosFrr).
80- VZ* + ° tanh"(Aa;) sin(bx)y'x + b[b - a tanh"(Aa;) cos(ba;)]y = 0.
Particular solution: yo = sinFrr).
81- V'lv + ° coth"(Aa;) cos(ba;)y^, + b[b + a coth"(Aa;) sin(ba;)]y = 0.
Particular solution: yo = cosFrr).
82- У'хх + ocoth"(Aa;) sin(bx)y'x + b[b — ocoth"(Aa;) cos(ba;)]y = 0.
Particular solution: yo = sinFa;).
83- У'1Х + ° 1п"(Аж) cos(bx)y'x + b[b + а 1п"(Аж) sin(ba;)]y = 0.
Particular solution: yo = cosFrr).
84- y'L + ° 1п"(Аж) sin(bx)y'x + b[b - а 1п"(Аж) cos(^)]y = 0.
Particular solution: yo = sinFrr).
85- VL + (o + be2Xx) lnn(kx)y'x + A[(o - be2Xx) lnn(kx) - \}y = 0.
Particular solution: y0 = beXx + ae~Xx.
86- y'L + (o + be2Xx) cosn(kx)y'x + A[(o - be2Xx) cosn(kx) - \}y = 0.
Particular solution: y0 = beXx + ae~Xx.
87- y'L + (a + be2Xx) sinn(kx)y'x + A[(o - be2Xx) sinn(kx) - \}y = 0.
Particular solution: y0 = beXx + ae~Xx.
© 1995 by CRC Press, Inc.

88- V'L + (° + be2Xx) tann(kx)y'x + Л[(о - Ье2Хх) tann(kx) - Л]у = 0.
Particular solution: y0 = beXx + ae~Xx.
89- V'L + (« + be2Aa:) cot"(fea;)y; + A[(o - be2Xx) cotn(kx) - \}y = 0.
Particular solution: y0 = beXx + ae~Xx.
90- V'L + (° sn2 x + b)y = 0.
The Lame equation in the form of Jacobi, sn x is the Jacobi elliptic function.
See the books by Bateman & Erdelyi A955, vol. 3) and Kamke A976) for more
information on this equation.
91. y'lx + [Ap(x) + B]y = 0.
The Lame equation in the form of Weierstrass, p is the Weierstrass function.
See the books by Bateman & Erdelyi A955, vol. 3) and Kamke A976) for more
information on this equation.
92- XV'L + (ax ln x + beXx)v'x + a(beXx In x + l)y = 0.
axx~ax
Particular solution: y0 = eaxx
93- XV'LX + C1 - axeXx lnx)y'x + aeXxy = 0.
Particular solution: yo = 1пж.
94. xyxx + (ax ln x + b cosh" x)y'x + a(b cosh" x ln x + l)y = 0.
95. xyxx + A — ax cosh" x ln x)y'x + a cosh" x у = 0.
Particular solution: yo = ln x.
96- XV'LX + (ax In ж + b sinh" x)y'x + a(b sinh" x ln x + l)y = 0.
Particular solution: y0 = eaxx~ax.
97. xyxx + A — ax sinh" x ln x)y'x + a sinh" x у = 0.
Particular solution: yo = In x.
98- XV'LX + (ax In ж + b tanh" x)y'x + a(b tanh" ж ln ж + l)y = 0.
Particular solution: yo = eaxx~ax.
99. жу^'з, + A — ax tanh" ж ln ж)у^ + о tanh" ж у = 0.
Particular solution: yg = lnrr.
100. xyxx + (ax ln ж + b coth" ж)у^, + a(b coth" ж ln ж + l)y = 0.
Particular solution: y0 = eaxx~ax.
© 1995 by CRC Press, Inc.

101. xy'^x + A — ax coth" ж In x)y'x + a coth" ж у = 0.
Particular solution: yo = In %¦
102. жу^ + (ax In ж + b cos" ж)у^ + °(b cos" ж In ж + l)y = 0.
Particular solution: y0 = eaxx~ax.
103. жу^, + A — ax cos" x\nx)y'x + a cos" ж у = 0.
Particular solution: yo = In ж.
104. жу^, + (ax In ж + b sin" ж)у^ + a(b sin" ж In ж + l)y = 0.
Particular solution: y0 = eaxx~ax.
105. жу^, + A — ax sin" ж In ж)у^ + a sin" ж у = 0.
Particular solution: yo = In ж.
106. жу^'з, + (ax In ж + b tan" ж)у^, + a(b tan" ж In ж + l)y = 0.
Particular solution: y0 = eaxx~ax.
107. жу^ + A - ax tan" ж In ж)у^, + a tan" ж у = 0.
Particular solution: yo = In %¦
108. жу^ + (ax In ж + b cot" ж)у^ + o(b cot" ж In ж + l)y = 0.
Particular solution: y0 = eaxx~ax.
109. жу^ + A — ax cot" ж In ж)у^ + a cot" ж у = 0.
Particular solution: yg = In ж.
110. x2yxx + ж(о1пж + beXx)y'x + a(beXx In ж - In ж + l)y = 0.
Particular solution: yo = exP(~y«ln x).
111. x2yxx + x(a In ж + b cosh" ж)у^, + o(b cosh" ж In ж — In ж + l)y = 0.
Particular solution: yo = exp(—yaln ж).
112. ж2у4'а, + ж(о In ж + b sinh" ж)у^ + o(b sinh" ж In ж - In ж + l)y = 0.
Particular solution: yo = exp(—yaln ж).
113. x2yxx + x(a In ж + b tanh" ж)у^ + o(b tanh" ж In ж - In ж + l)y = 0.
Particular solution: yo = exp(—yaln ж).
114. x2yxx + x(a In ж + b coth" ж)у^ + o(b coth" ж In ж - In ж + l)y = 0.
Particular solution: yo = exp(—yaln ж).
© 1995 by CRC Press, Inc.

115. x2yxx + ж(о In ж + b cos" ж)у^, + o(b cos" ж In ж — In ж + l)y = 0.
Particular solution: yo = exp(—-^a\n ж).
116. x2yxx + x(a In ж + b sin" ж)у^, + o(b sin" ж In ж - In ж + l)y = 0.
Particular solution: yo = exp(—yaln ж).
117. x2yxx + x(a In ж + b tan" ж)у^, + o(b tan" ж In ж - In ж + l)y = 0.
Particular solution: yo = exp(—yaln ж).
118. x2y'x'x + x(a In ж + b cot" ж)у^, + o(b cot" ж In ж - In ж + l)y = 0.
Particular solution: yo = exp(—yaln x).
119. sin2 ж y^ + sin ж (a + beXx)y'x + a(beXx — cos ж)у = О.
/ x \a
Particular solution: yo = [ cot — J .
120. sin2 ж y^, + sin ж (о + b cosh" ж)у^, + a(b cosh" ж — cos ж)у = 0.
Particular solution: yo = (cot — J .
121. sin2 ж yxx + sin ж (о + b sinh" ж)у^, + a(b sinh" ж — cos ж)у = 0.
/ ОС \ °
Particular solution: y0 = (cot — j .
Y11. sin2 ж yxx + sin ж (о + b tanh" ж)у^ + o(b tanh" ж — cos ж)у = 0.
/ ОС \ °
Particular solution: yo = (cot — j .
123. sin2 ж y^ + sin ж (о + b coth" ж)у^ + a(b coth" ж — cos ж)у = 0.
/ x \ a
Particular solution: yo = (cot — j .
124. sin2 ж yxx + sin ж (о + b In" ж)у^, + o(b In" ж — cos ж)у = 0.
/ x \a
Particular solution: Уо = [ cot — J .
125. cos2 ж yxx + cosx(a + beXx)y'x + a(beXx + sinx)y = 0.
[/ X 7Г \  a
cot ( — + — J .
126. cos2 ж yxx + cos ж (о + b cosh" ж)у^, + a(b cosh" ж + sin ж)у = 0.
[/ X 7Г \  a
cot ( — + — J .
© 1995 by CRC Press, Inc.

127. cos2 x yxx + cos x (a + b sinh" x)y'x + a(b sinh" x + sin x)y = 0.
Г / X 7Г \1°
Particular solution: yo = cot ( — + — 1 .
128. cos2 x yxx + cos x (a + b tanh" x)y'x + a(b tanh" x + sin x)y = 0.
Г ( X 7Г \1°
Particular solution: yo = cot I — + — 1 .
129. cos2 x yxx + cos x (a + b coth™ x)y'x + a(b coth" x + sin x)y = 0.
[/ X 7Г \ 1 a
cot ( — + — 1 .
130. cos2 x yxx + cos x (a + b In" x)y'x + a(b In" x + sin x)y = 0.
[/ X 7Г \  a
cot ( — + — J .
2.1.9. Equations Containing Arbitrary Functions
Notation: f = f(x) and g = g(x) are arbitrary functions; a, b, c, d, n, m, k, A, a, C,
and 7 are arbitrary parameters.
о," -L- nil — f
У XX T аУ — J ¦
Solution:
1 fx
(x — ?)] d? if a = k2 > 0,
У =
d cos(kx) + C2 sin(kx) + -3- f /(?) sin[k(
1 fx
d cosh(/crr) + C2 sinh(/ca;) + — / /(C) sinh[/c(rr - ?)] d? if a = -/c2 < 0,
k Jxo
where xq is an arbitrary number.
2- v'L + avL + by = f.
The substitution у = гиещ>(— \ах) leads to an equation of the form 2.1.9.1:
wxx + {Ъ- \o?)w = /exp(yarr).
3- v'L + fvL = 9-
Solution: у = d + / e~F(c2 + / eFgdx\ dx, where F = f dx.
4- v'L + *fv'x -fv = o.
Particular solution: yo = %¦
5- y'L + fvL + o(/ - o)i/ = o-
Particular solution: yo = e~ax ¦
© 1995 by CRC Press, Inc.

+ fy' + a(xnf - ax2n + nxn~x)y = 0.
/ a
Particular solution: yo = exp I
V n + l
xn+1
V'L + (/ + ахП + b)vL + i(axn + b)f + o,nxn~^\y = 0.
Particular solution: yo = exP( xn+1 — bx).
V n + 1 /
Particular solution: yo = xeax.
ХУ'хх + (Xf + а)Ух + (° ""
Particular solution: yo = x ~a.
rnnJf I \( nrn _|_ 1 Л f _|_ nrn 1 ~\nJ _1_ /J™ ?-. f\
Particular solution: yo = (аж + 1)е~ах.
xv'L + i(ax2 + bx)f + 2WX + bfv = °-
Particular solution: yn = a-\ .
x
•ЬУхх I \J I U-Ll /Ух ' У-l ' 1Ь)У — u'
Particular solution: yo = expf xn+1).
V n + 1 /
хУхх + (xf + axn)v'x + ИахП - !)/ + апхп-г]у = 0.
Particular solution: yo = x expl xn ).
V n /
Particular solution: yo = (axn + 1) exp(—axn).
x2yxx -\- a.xy'x -\- /3y = f.
The nonhomogeneous Euler equation.
The substitution x = e* leads to an equation of the form 2.1.9.2:
The nonhomogeneous Bessel equation.
The general solution is expressed in terms of Bessel functions:
у = dJv(x) + C2Yv{x) + уУ„(ж) fxJv{x)f{x) dx - у Jv{x) f xYv{x)f{x) dx.
© 1995 by CRC Press, Inc.

x2y" -\- xfy' -\- a(f — a — l)y = 0.
Particular solution: yo = x~a.
18- x2v'L + x(f + 2a)vL + Kbx + «)/ - b2x2 + °(°
Particular solution: yo = ж~°е~ х.
19- ж2у^ + ж/у^ + [(аж2^1 + n)/ - а2ж4™+2 - n2 - n]y = 0.
Particular solution: wq = x~n exp( — — —x2n+1).
V zn + 1 /
20. (ax2 + bx + c)y'lx + (x + k)fy'x - fy = 0.
Particular solution: yo = x + k.
21- ж4у^ + x2fy'x + [(A - ж)/ - A2]y = 0.
Particular solution: yo = x exp ( — ).
V x /
22. x2(ax2 + b)y^ + x(ax2 + b)fy'x - [(ax2 - b)f + 2% = 0.
Particular solution: yo = ax -\ .
x
23. (x2 + aJy'lx + (x2 + a)fy'x - (xf + a)y = 0.
Particular solution: yo = V ж2 + а.
24. (x2 + aJy'lx + (x2 + a)fy'x - m[xf + (m - l)x2 + a]y = 0.
Particular solution: yo = (x2 + a)m'2.
25. (axn + h)y'lx + (axn + b)fy'x - anxn~2(xf + n - l)y = 0.
Particular solution: yo = axn + b.
26. (axn + bx)y'xlx + (axn + bx)fy'x - [(ana;" + b)f + an(n - 1)ж"-2]у = 0.
Particular solution: yo = axn + bx.
27. (xn + аJу'1х + (xn + a)fy'x - xn~2(xf + an - a)y = 0.
Particular solution: yo = (xn + aI'™.
28. (axn + bJy'lx + (axn + b)fy'x + (f - anx^1 - l)y = 0.
/ f dx \
Particular solution: yo = exp — / — .
V J ax11 + b I
29. f(x)y'ix + [ax2 + (ac + b)x + bc]y'x - (ax + b)y = 0.
Particular solution: yo = x + с
© 1995 by CRC Press, Inc.

v'L - (f2 + f'x)y = o.
Particular solution: yo = exP ( I f d
3i. v'L + fvL - [«(о +1)/2 + *fL\v = o-
Particular solution: yo = expla / /dxj.
32. y^ + 2fy'x + (p + fx)y = 0.
Solution: у = (С2Ж + Ci)exp(— f dx),
33. t?, + A - o)/y^ - o(/2 + Д)у = О.
Particular solution: yo = exp la / / dx j.
34. tC + /y^ + (fg -g2+ g'x)y = 0.
Particular solution: yo = exp( — / gdx).
35. y'L + Vy'x + (f2 + fx+ о)У = 0-
The substitution и = у exp I / / dx I leads to a constant coefficient linear equation:
u'xx + аи = 0.
36. y^ + 2fy'x + (f2 + fx+ ax2n + bxn-^)y = 0.
/ dx I leads to an equation of the form 2.1.2.10: w'^x +
37. y'L + B/ + a)y'x + (f2 + af + fx + b)y = 0.
/ dx I leads to a constant coefficient linear equation:
uxx + au'x + bu = 0.
38. y'L + (/ + g)y'x + (fg + f'x)y = o.
Particular solution: yg = exp I — / / dx j.
39. жу! + ж/у; + (/ + ж/^)у = 0.
Particular solution: yo = xexp( — / f dx).
40- «y^ + (ж/ + а)у'х + (о/ + ж/^)у = 0.
Particular solution: yo = exp( — / f dx).
© 1995 by CRC Press, Inc.

41. (x + a)y'lx + (/ + b)y'x + fxy = 0.
Particular solution: yn = exp I / dx
\J x + a
42- x2y'L + xBf + l)y'x + (f2 + xfx + x2- a)y = 0.
The substitution у = wexp( — / — dx) leads to the Bessel equation 2.1.2.121:
\ J x /
x2w'lx + xw'x + (x2 - a)w = 0.
43- x^v'L + жB/ + a)vL + if2 + (° -
a/ \
— (ia: I leads to an equation of the form 2.1.2.127:
x )
x2w"xx + axw'x + (bxn + c)w = 0.
— dx I leads to an equation of the form 2.1.2.141:
x )
x2wxx + (axn + b)xw'x + (ax2n + /3xn + j)w = 0.
45- '- '
/dx
—— leads to a constant coefficient equation: 2y^ + ay = 0.
v/
46. fv'L-fLv'x-af3y = o.
r
Solution: у = deu + C?,e~u¦, where u=y/a fdx.
J
47. fy'ix - (f'x + af2)y'x + bf3y = 0.
Solution:
у = C\ exp(Ai / f dx) +C2exp(X2 / /dx),
where Ai and A2 are the roots of the quadratic equation A2 — aX + b = 0.
48. fy'L - (f'x + af)y'x - bf2(a + bf)y = 0.
Particular solution: yo = exp( —6 / /dx).
49. fy'L ~ (fL + 2af)y'x + Ы'х + о2/ - Ь2Р)У = 0.
Particular solution: yo = eax exp(b / /dx).
© 1995 by CRC Press, Inc.

50. Pv'L + f(f'x + a)y'x + by = 0.
/dx
—— leads to a constant coefficient linear equation: y'^+ay'^
J
by = O.
'L + f(fL + 29 + а)Ух + (f9'x +g2 + ag + b)y = 0.
The transformation ? = / ~~f > u = У exP I / ~F dx J leads to a constant coefficient
J J \J J /
equation: u'^ + au'^ + bu = 0.
^
52. fgy'L - (afxg + bfg'x)yx - A/2<*+Vb+1y = 0.
r
Solution: у = Cieu + C2e~u, where u=V\ fagb dx.
J
53. fy'L-fx'xy = b.
Solution: y = df + C2f I f~2 dx.
dx \
54. 4
1°. Solution with a = 0:
2°. Solution with a > 0:
У = v/ (G\(F + C2C ^)) where ip = / ——.
3°. Solution with a < 0:
У = \/f (Cicostp + C2simp), where (p =—-— / ——.
^ J j
55- y'L - Ц^у'х + a2(fxJf2n-2y = 0.
Solution:
У =
where Jm and Ym are Bessel functions.
f" ^ . b2(f'J
56.
57.
)Ух
f2 + a fx
Solution: y = C1(
Particular solution: yo = expl —— e x 1.
V Л
© 1995 by CRC Press, Inc.

58. V'L + (/ + aeXx)y'x + aeXx(f + X)y = 0.
Particular solution: yo = exp ( —— eXx J.
59- V'L + (« + be2Xx)fy'x + A[(o - be2Aa;)/ - A]» = 0.
Particular solution: y0 = beXx + ae~Xx.
во- v'L + 2fvL + (f2 + fL + ae2Xx + beXx + c)v = o-
The substitution и = yex.pl If dx J leads to an equation of the form 2.1.3.5:
(ae2Xx + beXx + c)u = 0.
61- V'ix + f sinh(ax)y'x — a[a + f cosh(oa;)]y = 0.
Particular solution: yg = sinh(aa;).
62- V'ix + f cosh(ax)y'x — o[o + / sinh(oa;)]y = 0.
Particular solution: yo = cosh(aa;).
63. y'ix - fxy'x + а2е2*у = 0.
\ / f dx I.
Solution: у = C\ sin I a / e* dx\ + C2 cos a /
64. yl - /i»i - o2e2^y = 0.
Solution: у = C\ exp I a J е-'' с?ж I + C2 exp I —a f e? dx\.
65. (aeXx + ЪУу'1х + (aeXx + b)fy'x + ceXx(f - ceXx + Xb)y = 0.
__c_
Particular solution: y0 = (aeXx + b) aX .
66. xiC + (l-fx In x)y'x + fy = 0.
Particular solution: yo = In ж.
67. жу^ + (/ + ax In x)y'x + a(f In ж + l)f/ = 0.
Particular solution: y0 = еоа:ж"о:г:.
68- v2y'L + 2жAпж + o)/y4 + [\ - (In ж + a + 2) fly = 0.
Particular solution: yo = -^(hirr + a).
69. x2yxx + ж(/ + о In ж)у^, + о(/ In ж — In ж — 1)у = 0.
Particular solution: yo = exp(—yaln x).
70- У^ + / з1п(ож)у^, + o[o - / соэ(ож)]у = 0.
Particular solution: yo = sin(aa;).
© 1995 by CRC Press, Inc.

71- У'хХ + / cos(ax)y'x + a[a + f sin(ax)]y = 0.
Particular solution: yo = cos(arr).
72- y'L + fvL + o[A + / tan(Aa;) + (A - o) tan2(Aa;)]y = 0.
Particular solution: y0 = [cos(Aa?)]°/A.
73- y'L + fvL + o[A - / cot(Aaj) + (A - o) cot2(Xx)]y = 0.
Particular solution: yo = [sin(Aa;)]o'A.
74- V'ix + (/ + ° sin x)v'x + °(/ sin ж + cos ж)у = О.
Particular solution: yo = ехр(асовж).
75- У^ + (/ + ° cos ж)у4 + °(/ cos ж - sin ж)у = О.
Particular solution: yo = exp(—a sin ж).
76- V'L + (/ + °tan ж)у4 + (о + 1) (/ tan ж + l)y = 0.
Particular solution: yo = cos°+1 x.
77- l/i» + (/ + acotx)y'x + (o - l)(/cotж - l)y = 0.
Particular solution: yo = sinl~0a:;-
78- V'L + tan ж (/ + о - l)y^ + [(о tan2 ж - 1)/ + 2o + 2]y = 0.
Particular solution: yo = sina;cos0a;.
79. y^ + (/ + a cos" ж)у^, + a cos" ж (/ cos x — n sin ж)у = 0.
Particular solution: yo = exp f —a / cos™ xdx).
80. y^ + (/ + asinn x)y'x + a sin" ж (/sin ж + псоэж)у = 0.
Particular solution: yo = exp(—a / sin™xdx).
81. sin xyxx+sinx (f + a)y'x +a(f — cosx)y = 0.
/ ж \ a
Particular solution: yo = (cot — j .
82. cos2 ж yxx + cos ж (о + f)y'x + o(/ + sin ж)у = 0.
[/ X 7Г \  а
cot ( — + — J .
© 1995 by CRC Press, Inc.

2.1.10. Some Transformations
Notation: f, g, and h are arbitrary composite functions of their argument which is
written in paretheses following the name of a function (the argument is a function
ofx).
2-
1 У
—, ад = —
х х
The transformation ? = —, ад = — leads to the equation w'Je + f(?,)w = 0.
О IT I n 1]
The transformation ? = -, w = г leads to a simpler equation: ад!', +
ex + d ex + d ««
2 = °> where A = ad - be.
3- ж2у1 + + x2nf(axn + Ь)\у = О.
та-1
The transformation ? = аж™ + 6, ад = ух 2 leads to a simpler equation: ад!', +
(an)-2f(Ow = 0.
b)y = 0, f = f(x),g = g(x).
The substitution у = ж^ад, where к is a root of the quadratic equation /с2 + (а — l)/c +
6 = 0, leads to the equation xw'^.x + (xf + a + 2k)w'x + (g + kf)w = 0.
5. xPn(x)y^x + Qn(x)y'x + Rn-1(x)y = 0,
wnere rn\X) — 2^m=0 am% > 4n\x) — 2^m=0 °rnx i J^n-lKX) — 2~/m=0 cmx •
The substitution у = xkw, where к = 1 , leads to an equation of the similar form:
xPn{x)w'lx + [Qn{x) + 2kPn(x)]w'x + [Rn-!(x) + Fn-!{x)]w = 0,
where i^_i(rr) = -[Qn(x) + (k - l)Pn(x)}.
X
6. x(x - l)Pn_1(x)y'^x + Qn(x)y'x + Rn-1(x)y = 0,
wVipre P i fri — Г" n rm О (чЛ — V™ h rm R i fri — Г" с rm
The transformation ? = , ад = x — l\~ky, where к is a root of the quadratic
x — 1
equation an-\k2 + (bn — an-\)k + cn_i = 0, leads to an equation of the similar form:
where
n — 1
m—0 m—0
n—1 ^
m^O
© 1995 by CRC Press, Inc.

7- y'L + \fXxf(ae^ + b) - -X^ у = 0.
The transformation ? = aeXx + b, w = yeXx^2 leads to a simpler equation: ад!', +
(аА)/(?)ад = 0.
The substitution z = eXx leads to the equation X2z2yzz + Xz[f(z) + X]y'z + g(z)y = 0.
9- VL + I"*2 + втЬ-4(Аж)/(^Ь(Аж))]у = О.
V
The transformation ? = со^(Аж), ад = t-tt—г leads to a simpler equation: ад!', +
втЦАж) ««
А/(?)ад = 0.
10. y'lx + [-A2 + со5Ь~4(Аж)/^апЬ(Аж))]у = 0.
V
The transformation ? = tanh(Aж), ад = т-тт—г- leads to a simpler equation: ад!', +
совЦАж) ss
А/(?)ад = 0.
I p2Xx / „^Ла;
ае _|_ ^ ye/
The transformation ? = —г , ад = —г leads to a simpler equation: w'
ceXx + d ceXx + d
= 0, where A = ad- be.
= 0, f = f(x), g = g(x), h = h(x).
The substitution и = у cosh ж leads to the simpler equation: fu'^x +gu'x + (h— f)u = 0.
= 0, f = f(x), g=g(x), h = h(x).
The substitution и = у sinh ж leads to a simpler equation fuxx + gu'x + (h — f)u = 0.
14. x2y'L + [/(o In ж + 6) + \\y = 0.
The transformation ? = a In ж + 6, ад = уж/2 leads to a simpler equation ад^', +
а/(?)ад = 0.
15. (a;2-lJy^, + /(ln )у = 0.
The transformation ? = In , ад = —, leads to a simpler equation:
16. ж2/Aпж)У;/а; + ждAпж)у4 + h(lnx)y = 0.
The substitution ? = In ж leads to the equation /(?)y^ + [#(?) — /(?)]y^ + ^(?)y = 0.
© 1995 by CRC Press, Inc.

• УхХ + 1л2 + зт~4(Аж)/(со<;(Аж))]у = 0.
У
The transformation ? = со^Аж), w = . ,.—г leads to a simpler equation: w'L +
вт(Аж) ««
X-2f(?)w = 0.
• V'L + [*2 + со3-4(Аж)/^ап(Аж))]у = О.
У
The transformation ? = tan(Aa;), w = leads to a simpler equation: w'J? +
сов(Аж)
= 0.
Г „
+ [A
= "¦
The transformation ? = -. -7-, w = 7т r^ leads to a simpler equation:
sin(Arr + b) sm(Arr + b)
w'^ + [Asin(> - a)]-2f(?)w = 0.
20- /»i/a, + (fl-2/tanaj)»i + (/»-fltanaj)» = 0, f = f(x), g = g(x), h = h(x).
The substitution u = у cos ж leads to a simpler equation: /w^. + gu'x + (/ + /i)w = 0.
0' f = f(x),g = g(x),h = h(x).
The substitution u = у sin ж leads to a simpler equation: /w^. + gu'x + (/ + /i)w = 0.
22. (ж2 + lJy^ + /(arctan ж + b)y = 0.
У
The transformation ? = arctan ж + b, w = —r^=^= leads to a simpler equation:
Vx2 + 1
<? + [/@ + i]w = 0.
23. (ж2 + lJy^ + /(arccot ж + b)y = 0.
The transformation ? = arccot ж + b, w = leads to a simpler equation:
w»t + [/(?) + l]w = 0.
24- »»» + /(«)» = 0-
The transformation ж = f(?), у — w* \<р'Л leads to an equation of the similar form
2.1.11. Asymptotic Solutions
This subsection presents asymptotic solutions, as e —>• 0 (e > 0), of some second-order
linear ordinary differential equations containing arbitrary functions (sufficiently smooth),
with the independent variable being a real number.
1. Consider the equation
e2y':x - f(x)y = 0 A)
on a closed interval a < ж < b.
© 1995 by CRC Press, Inc.

Case 1. With the condition / ф 0, the leading terms of the asymptotic expansions of
the fundamental system of solutions, as e —>• 0, are given by the formulae
yi = /-1/4exp(~ f\/fdx\ У2 = Г1/4ехр^ f \/f dx\ if / > 0,
Vi = ("/)/4 cos Г | У y/=fdx\ y2 = (-f)-1^SinfjJv^fdx\ if / < 0.
Case i?. Discuss the asymptotic solution of equation A) in the vicinity of the point
x = Xo where function f(x) vanishes (such a point is referred to as a transition point). We
assume that function / can be presented in the form
f(x) = (xo — х)ф(х), where ip(x) > 0.
In this case, the asymptotic solution, as e —> 0 (e > 0), is described by three different
formulae:
with x > Xo,
У = [-
with x <
У =
in the vicinity of the transition point x = Xo,
y = C1 Bi(z) + C2 Ai(z), z = е-2/3[-ф(хо)}1/3(хо - x),
where Ai(z) and Bi(z) are the Airy function of the first and second kind, respectively (see
equation 2.1.2.2).
Constants C\, C2 and C\, C2 ш the above asymptotic solutions are related by
г =
2. The two-term asymptotic expansions of the solution of equation A) with / > 0, as
e —>• 0, on a closed interval a < x < b has the form
where xo is an arbitrary number satisfying the inequality 1 < Xo < b.
© 1995 by CRC Press, Inc.

The asymptotic expansions of the fundamental system of solutions of equation A) with
/ < 0 are derived by separating the real and imaginary parts in either formula B).
3. Consider the equation
C)
on a closed interval a < x < b, where a < 0 and b > 0, under the condition that mis a
positive integer and f(x) ф 0. In this case, the leading term of the asymptotic solution, as
e —> 0, in the vicinity of the point x = 0 is expressed in terms of a simpler model equation
which results from substituting function f(x) in equation C) by constant f(xo) (the solution
of the model equation is expressed in terms of the Bessel functions of the order 1/m, see
equation 2.1.2.7).
We specify below the formulae by which the leading terms of the asymptotic expansions
of the fundamental system of solutions of equation C) with a < x < 0 and 0 < x < b
are related (excluding a small vicinity of the point x = xo)- Three different cases can be
extracted:
1°. Let m be an even integer, and f(x) > 0. Then,
[ДЖ)Г1/4ехр1-
If{x) dx
if rr<0,
2/2 =
*-1[/(а0Г1/4ехр[1 Г yff(x)dx\ if x>0;
Г 1 fx I I
exp / df{x)dx\ if x < 0,
L e Jo v J
' I
f(x) dx\ if ж > 0,
exp --
? ./o
where / = fix), к = sin( — ).
V m /
2°. Let m be an even integer, and f(x) < 0. Then,
2/1 =
2/2 =
where / = f(x), к = tan(
3°. Let m be an odd integer. Then,
2/1 =
x)\dx+ — | if ж > 0,
^ Hx<0,
© 1995 by CRC Press, Inc.

2/2 =
Г 1 fx I 1
/с[/(ж)]~1'4ехр / \/f(x)dx\ if x > 0,
L ? Jo v J
where / = /(ж), к = sin(-—J.
4. Consider an equation of the form
D)
on a closed interval a < ж < b under the condition that / ф 0. Assume that the asymptotic
relation holds
k=0
Then, the leading terms of the asymptotic expansions of the fundamental system of solutions
of equation D) are given by the formulae
2/1 =
«p f-| / jMx)dx + j f ^b dx] [i + 0(e)];
c+i /"^LdJfi + o^)].
2 J Vfo(x) J
exp [I
5. Consider an equation of the form
ey'L + g{x)y'x + f(x)y = о
on a closed interval 0 < x < 1. With g(x) > 0, the asymptotic solution of this equation,
satisfying the boundary conditions y@) = C\ and y(l) = C2, can be presented in the following
form:
[ / Щ- dx] + O(e),
9(x) J
2/ = (Ci — /сСг) ехр[—е 1(/@)ж] + С2 exp
Г г1 f(x)
where к = exp / —^—^- с?ж |.
|_Л> Р(ж)
6. Now let us take a look at an equation of the form
e2yxx + eg(x)y'x + f(x)y = 0 E)
on a closed interval a < x < b. Assume
D(x) = [g{x)f - 4/(ж) ф 0.
Then, the leading terms of the asymptotic expansions of the fundamental system of solutions
of equation E), as e —>• 0, are described by the formulae
2/1 =
2/2 =
g'x(x)
fflx)
dx [1
© 1995 by CRC Press, Inc.

7. The more general equation
e2y'x'x+eg(x,e)yx + f(x,e)y = 0. F)
is reducible, with the aid of the substitution у = w exp ( / gdx), to an equation of
the form C):
?2<, + (/ " T92 ~ W*)w = 0,
to which the asymptotic formulae given above in Paragraph 4 are applicable.
2.1.12. Series Solutions
Let us consider a homogeneous linear differential equation of the general form
ylx + f(x)y'x + g(x)y = 0. A)
Case 1. Assume that the functions f(x) and g(x) are representable, in the vicinity of
the point x = Xo, in the form
oo oo
f{x) = Y,Mx-xo)n, g{x) = Y,Bn{x-x0)n, B)
n=0 n=0
in the region \x — xo\ < R, where R stands for the minimum radius of convergence of the
two series in B). In this case, the point x = xq is referred to as an ordinary point, and
equation A) posesses two linearly independent solutions of the form
oo oo
2/i (ж) = ^2an(x - xo)n ,2/2 (ж) = ^2bn(x - xo)n. C)
n=0 n=0
Coefficients an and bn are determined by substituting the series B) into equation A) followed
by extracting the coefficients with respect to identical powers of (x — Xo)-
Case 2. Assume that the functions f(x) and g(x) are representable, in the vicinity of
the point x = Xo, in the form
oo oo
f(x)= ? An(x-x0)n, g(x)= ? Bn(x-x0)n, D)
n=-l n=-2
in the region \x — xo\ < R- In this case, the point x = xq is referred to as a regular singular
point. Let Ai and A2 be the roots of the quadratic equation
There are three cases, depending on the values of the exponents of the singularity.
1) If Ai ф А2 and Ai — A2 is not equal to an integer, equation A) has two linearly
independent solutions
2/i (ж) = |ж-ж
2/2(ж) = |ж - жо|Л2 11 + ^J &„(ж - хо)п
E)
OO
© 1995 by CRC Press, Inc.

It is clear that in the general case, coefficients an and bn of the series D) and E) will be
different.
2) If Ai = A2 = A, equation A) possesses two linearly independent solutions in the forms
2/1 (ж) =
L n=i
00
ro| + |ж-жо|А^&„(ж-жо)п.
n=0
3) If Ai = A2 + N, where N is an integer greater than 0, equation A) has two linearly
independent solutions in the forms
со
2/1 (х) = |rr-rro|Al |1 + У]ап(ж-ж0)п|,
2/2(ж) = %1(ж) In |ж - жо| + |ж - жо|Л2 У] Ьп(х - хо)п,
оо
п=0
where к may be equal to zero.
To construct the solution in each of the three cases, the following procedure is to be
done: Substitute the above expressions for y\ and 2/2 into the original equation A) and
equate the coefficients of (x — Xo)n and (x — жо)п1п \x — Xo\ for different values of n to
obtain recurrence relations for the unknown coefficients. From these recurrence relations
the solution sought can be found.
2.2. Autonomous Equations y"x = F(y, y'x)
Preliminary Comments. Equations of this form are often encountered in different
areas of mechanics, applied mathematics, physics, and chemical engineering science.
1. The substitution w(y) = y'x leads to the first order equation
w'y = w^Fiy, w) A)
(see Chapter 1).
2. The solution of the original autonomous equation can be represented in the implicit
form
where w = w(y, C\) is the solution of the first order equation A).
3. The solution of the original autonomous equation can be written in the parametric
form
'Л C)
x= / ~~(—7TTdT + C2' У = У(т, Ci), C)
J w(t, Ci)
where у = у(т, C\), w = w(t, Ci) is the parametric form of the solution of the first order
equation A).
Formula B) is a special case of formula C) with у = т.
© 1995 by CRC Press, Inc.

2.2.1. Equations of the Form y"x — y'x = f(y)
Preliminary Comments. Equations of this form are encountered in the theory of
combustion and the theory of chemical reactors.
1. The substitution w(y) = y'x leads to the Abel equation ww'y — w = f(y) which is
considered in Subsection 1.3.1 for some concrete functions /.
2. The solution of the original autonomous equation can be written in the parametric
form C), where у = у(т, Ci), w = w(t, Ci) is the parametric form of the solution to the
Abel equation of the second kind ww'y — w = f(y).
Solution in the parametric form:
\-m ГП-1 f d,T
J
У 1 L ГП + 3
то + 3 J VI ± т"Ч
2. у" — у' =
Solution in the parametric form:
Г /" 1 Г /" I
x = - In \d / ехр(±т2) dT + C2\, y = ad ехр(±т2) d / ехр(±т2) dr + C2\ ¦
L J J L J J
°* Ухх Ух — дУТд" У
Solution in the parametric form:
x = -31n{Ci ехр(-т) [ехр(Зт) + C2 sin(v/3T)] },
Г2ехрCт) - C2sin(v/3T) + v/3C2cos(v/3tI2
у = аехрBт) 5 .
[expCT)+C2sin(v/3T)]
4 о," — ,,' — 9 _i_ 9 8/3?.-5/3
4- Ужа; «a; — 100 " ± 100 а У
Solution in the parametric form:
x = -\ 1п[±(т4 - 6т2 + 4Git - 3)] + C2,
у = а(т3 - Зт + dK/2 [±(т4 - 6т2 + 4Cir - 3)] "9/8.
5- у1-у4 = -^У-^8/3У-5/3-
Solution in the parametric form:
? = Ci-21n[sinTCOsh(T+C2)+cosTsinh(T+C2)], у = a[tanT+tanh(r+C2)]/2.
© 1995 by CRC Press, Inc.

> In the solutions of equations 6-9, the following notation is used:
C\Jv{t) + C2Yv{t) for the upper sign,
t C\Iv(j) + C2Kv{t) for the lower sign,
where Jv and Yv are Bessel functions, Iv and Kv are modified Bessel functions.
6- v'L -vL = Ay-1/2.
Solution in the parametric form:
x = -2Jt-xZ-\tZ't + \Z) dr + C2, y = aT-A'zZ-\TZ'T + \Zf ± t2Z%
where v = -|-, A = =F-|-a3/2.
* **XX ** X **
Solution in the parametric form:
x = Tt [tZ2 [(tZ't + \ZJ ± t2Z2} -1 dr + C2,
у = 2ar^Z2 [(tZ't + \ZJ ± t2Z2] ~\
where v = ±, A = -36a3.
8- V'L ~VL = 2A2
Solution in the parametric form:
x = ±2 I' T-1{ZlT)-1{TZ±2ZlT)dT + C2, y = a{Z'T)-2(TZ±2Z'TJ,
where v = 0, A = a1/2.
9- v'L -vL = Av~1/2 + 2B2 + Bv1/2-
Solution in the parametric form:
x = -2 f T-1Z-1(TZ'T-Z)dT + C2, y = B2Z-2(TZ'T-ZJ,
diere A = A - v2)Bz.
> In the solutions of equations 10-Ц, the following notation is used: the function
p = р{т) is defined in the implicit form
-1
4/±Dp3 -
The upper sign in this formula corresponds to the classical elliptic Weirstrass func-
function p = р(т + C2, 0, 1).
© 1995 by CRC Press, Inc.

Solution in the parametric form:
х = Б1пт + С2, у = 5a(r2p=F -j), where A
Solution in the parametric form:
x = 5 In т + С2, у = Ъат p, where A = it
12. yxx — y'x = Ay2 + -§^y-
Solution in the parametric form:
x = 51пт + С2, у = 5а(т2р =F 1), where A -
I3- y'L -У'х
Solution in the parametric form:
where / = а/±Dр3 - 1), A =
Solution in the parametric form:
x = -f J(f ± 2rp2)(r/ + 2P) dr + C2, y = 2o(/ ± 2rp2K/2(r/
where / = 4/±Dp3 - 1), ^4 = _ma2BaJ/3.
> In the solutions of equations 15-18, the following notation is used:
+ 1
- 1)
is the incomplete elliptic integral of the second kind in the Weierstrass form,
= л/±Dт3 - 1),
- У'х =
Solution in the parametric form:
x = -7 f tR-1!'1 dr + C2, y = 7ат2ГА, where A
© 1995 by CRC Press, Inc.

16. V'L - y'x = 6y + Ay-4.
Solution in the parametric form:
x = —
\ It^R-1!'1 dr + C2, y = ат~3/51~2/5, where A = т150а5
V'L -VL = 2°У + Ay'1/2.
Solution in the parametric form:
x=± R-1I-1I2 dr + C2, y = al~4/3l%, where A = ±108a3/2.
Solution in the parametric form:
/'^-1/iDt/12t/22) dr+C2, у = а/11/2Dт/12т/22)/8. where A = ±j-c
x =
v'L -у'х = ау + ву~2 - в2у-3.
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.1.5:
ww'y -w = Ay + By'2 - B2y~3.
20. ylx — yx = —-^y + Ay-^-l6-
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.1.61:
ww,. — w = — -
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.1.62:
ww,. — w = — -
22. y'xx — y'x = —y -
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.1.14:
ww'y -w= -jf-y + 2Ay2 + 2A2y3.
23. yxx-y'x = Ay*-1 -kByk + kB2y2k~1.
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.1.6:
ww'y -w = Ay1"'1 - kByk + kB2y2k~1.
© 1995 by CRC Press, Inc.

• УхХ &Х I п ¦
Solution in the parametric form:
/" 1 1 9 9
ж = =F E~1F~1(F2 ±2E2)dr+ C2, V =
J K '
r
where E = / exp(=Fr2) dr + Cb F = 2tF ± exp(=Fr2).
25. С - у'х = A + Вехр(-2у/А).
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.1.8:
ww'y - w = A + Вещ>(-2у/А).
26- V'L -VL = а2Хе2ХУ - а(ЬХ + l)eA« + b.
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.1.73:
v'y - w = a2Xe2Xy - a(b\ + l)eXy + b.
wwy
27. y'lx -y'x = a2Xe2Xy + a\yeXy + beXy.
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.1.74:
ww'y -w = a2Xe2Xy + a\yeXy
2.2.2. Equations of the Form y%x + f{y)yfx + у = О
Preliminary Comments. Equation of this form are often encountered in the theory
of nonlinear oscillations.
1. The transformation
z = -\y2 + a, w = y'x
leads to the Abel equation
ww'z = g(z)w + 1, where g(z) = f(y)/y, y
whose special cases are outlined in Subsection 1.3.2.
2. For oscillatory systems with a weak nonlinearity
ylx + eF{y)y'x + у = О,
two leading terms of the asymptotic solution, as e —>• 0, are described by the formula
у = ^4cos(rr + B),
where functions A = A(?) and В = B(?) depend on the slow variable ? = ex; they are
determined from the autonomous system of the first order equations
A f27T 1 f27T
A'e = / F [A cos ip) sin2 ip dip, Bi = / F (A cos ip) sin ip cos ip dip.

The right-hand sides of these equations depend only on function A. The system is solved
consecutively starting from the first equation.
V'L + aVV'x + V = °
Solution in the parametric form:
x = -A f t-^Ci + 2A2 In 1-7-1 - 2At)~1/2 dr + C2, y = (d + 2A2 In \т\ - 2АтI/2,
where A = а~г.
2-
Van der Pol oscillator
Solution, as e —> 0:
у = a cos(rr - в) - -^ea3 sin[3(rr - в)} + O(e2),
where
a2 = ~ , в = \e\na — -?т-еа2 + -ттгЕ2х + C2.
1 + DС - 1)е-еа; 8 64 16
In applications, ж plays the role of time, C\ is the initial oscillation amplitude, and
C<i is the initial phase with e = 0.
As e —>• +oo, the periodic solution of the Van der Pol equation consits of intervals
with fast and slow vibrations and describes damping oscillations with the period
T= C-
3- Ухх + У(ау2 + b)y'x + у = 0.
The transformation z = — \y2, w = y'x leads to the Abel equation of the form 1.3.2.1:
ш!, = (—2az + b)w + 1.
Solution in the parametric form with a < 0:
и '/2
x= I
y= |±—'
where
Г CiЛ/з(т) + C2Y1/3(t) for the upper sign,
I CiIi/3(t) + С2ЛГ1/з(т) for the lower sign,
Ji/з and Yi/з are Bessel functions, Д/3 and ЛГ1/3 are modified Bessel functions.
4. у" + у {ay2 + b)~2y' + у = 0.
The transformation z = — -j2/2> w = y'x leads to the Abel equation of the form 1.3.2.2:
ш!, = (—2az + b)~2w + 1.
© 1995 by CRC Press, Inc.

The transformation z = — \y2, w = y'x leads to the Abel equation of the form 1.3.2.4:
ww'z = (-2az + b)~1/2w + 1.
Solution in the parametric form:
x = -aCi I аС{Е^ — \-C2, У =
j \ a J t2 — т + a
where E = exp
f TdT
~J т2-т + а
6-
The transformation z = у , w = y'x leads to the Abel equation of the form
2 2a
1.3.2.3:
ww'z = (A —Jw + 1, where A = — 2a.
The transformation z = —\y2, w = y'x leads to the Abel equation of the form 1.3.2.7:
ш!, = a exp(—2Xz)w + 1.
8- V'L + у[оехр(Лу2) + Ьехр(-Лу2)]у^, + у = 0.
The transformation z = —\y2, w = y'x leads to the Abel equation of the form 1.3.2.8:
ш!, = [6expBAz) + aexp(—2\z)]w + 1.
9- V'L + 2аУ exp[o(b - y2)]y'x + у = 0.
Solution in the parametric form:
(b — 4fc t — In \kE-r- |) dr + C2, у = (b — 4/c т — In
where a = =FxK~2> ET = f ещ>(тт2) dr + Ci.
10- v'L + Av cosH*y2)y'x + у = о.
This is a special case of equation 2.2.2.8 with a = b = \A.
n- y'L + Av sinh(Ay2)y4 + у = °
This is a special case of equation 2.2.2.8 with a = —b = \A.
12. y'lx + 2Ayy'x v/sinh2[A(JB-y2)]+2A-1 + у = 0.
Solution in the parametric form:
x = 2a {F2 + 2E2)G-1Q-1dT + C2, у = Q; A=\a~2,
where
E = f ещ>(-т2) dr + d, F = 2rE + exp(-T2), G = V'F2 - 2E2 + 8E2F2,
Q= JB- 4a2 Arsinh^^-1^2 - 2E2)}, Arsinhz = ln(z + \Jz2 + 1).
© 1995 by CRC Press, Inc.

13- V'L ~ 2AW'X\Jcosh2[A(y2 - В)] - 2A-1 + у = 0.
Solution in the parametric form:
x = 2a (F2-2E2)G-1Q-1dT + C2, у = Q; A= \a~\
where
E = fехр(т2) dr + Clt ^ = 2т?-ехр(т2), G= VF2 + 2E2 - 8E2F2,
Q = ^JB + 4a2 Arcosh[aE-1F-1(F2 + 2E2)}, Arcoshz = ±In(z + \/z2 - 1).
V'L + AV cos(u>y2)y'x +y = 0.
L
2/2
The transformation z = — -j2/2> w = Ух leads to the Abel equation of the form 1.3.2.11:
ww'z = AcosBuiz)w + 1.
V'L + AV sin(wy2)y4 + у = 0.
L
\y2
The transformation z = — \y2, w = y'x leads to the Abel equation of the form 1.3.2.12:
ww'z = —AsmBuiz)w + 1.
2.2.3. Lienard Equations y?x + f{y)y'x+g{y) = 0
V'L
Duffing equation.
Solution:
x = ±
The period of oscillations with the amplitude С is expressed in terms of the complete
elliptic integral of the first kind:
/ dt
T= , - g( "" ), where "' ^ '
K(m) = Г
Jo
Jo
The asymptotic solution, as a —>• 0, has the form
у = Сг cos[(l + f aC2)x + C2\ + ^-aCfcos[3(l + f aC2)x + 3C2] + O(a2),
where C\ and C2 are arbitrary constants. The corresponding asymptotics for the
period of oscillations with the aplitude С is described by the formula
Т = 2тгA-|аС2)+О(а2).
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.1:
ww'y = (ay + 3b)w + cy3 — aby2 — 2b2y.
© 1995 by CRC Press, Inc.

3- y'L = C°у + b)y'x - ° V - abv2 + cv-
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.2:
ww'y = (Say + b)w — a2y3 — aby2 + cy.
4- 2v'L = Goy + bb)y'x - 3a2y3 - Icy2 - 3b2y.
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.3:
2ww'y = {lay + 5b)w - 3a2y3 - 2cy2 - 3b2y.
5- V'L = У"^! + 2n)y + an]y'x - ny2n(y + о).
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.8:
ww'y = yn~x [A + 2n)y + an]w — ny2n(y + a).
6- y'L = а(У ~ rtb)yn~^y'x + c[y2 - Bn + l)by + n(n + I)b2]y2"-1.
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.9:
w'y = a(y - nb)yn~1w + c[y2 - Bn + l)by + n(n + l)^2]^2™.
y'L = [oBn + k)yk + b\yn-^y'x + (-a2ny2k - abyk + c)y2n-\
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.10:
ww'y = [aBn + k)yk + b]yn~1w + (-a2ny2k - abyk + c)y2n~l.
8. y'L = [aBm + k)y2k + bBm -
- (a2my4k + cy2k + b2m)y2m-2k-1.
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.11:
'y = [oBm + k)y2k + bBm - k)}ym-k-1w - [a2myAk + cy2k + b2m)y2
9- y'L = аехУу'х + ЬехУ.
Solution in the parametric form:
ж=-у fт-1(С1+А21п\т\-Ат)~1 dr+C2, y=y
where A = b/a.
10. y'L = (аеУ + b)y'x + се2У - аЬеУ - Ь2.
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.15:
'y = (aev + b)w + ce2y — abev — b2.
© 1995 by CRC Press, Inc.

11. y'lx = [OB/Lt + X)eXy + b]e»yy'x + {-а2це2Ху - abeXy + c)e2>xy.
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.16:
ww'y = [oB/i + X)eXy + b]e™w + {-a2\xe2Xy - abeXy + c)e2^.
17 •
12- V'L = (aeXv + Ь)У'Х + c[a2e2Xy + ab(Xy + l)eXy + Ъ2Ху].
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.17:
ww'y = (aeXy + b)w + c[a2e2Xy + ab(Xy + l)eXy + Ъ2\у].
13. y'lx = eXyBaXy + a + b)y'x - e2Xy(a2\y2 + aby + c).
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.18:
ww'y = eXyBaXy + a + b)w- e2Xy(a2\y2 + aby + c).
14- V'L = eayBay2 + 1y + b)y'x + e2ay(-ay4 - by2 + c).
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.19:
ww'y = eayBay2 + 2y + b)w + e2ay{-ayA' - by2 + c).
15- V'ix = (° cosh У + b)y'x- ab sinh у + с.
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.20:
'y = (a cosh у + b)w — ab sinh у + с.
У'хх = (о sinh у + b)y'x — ab cosh у + с.
The substitution w(y) = y'x leads to the Abel equation of the form 1.3.3.21:
ww'y = (a sinh у + b)w — ab cosh у + с.
This is the equation of oscillations of the mathematical pendulum.
Solution:
= ± f
With a > 0 and initial conditions y@) = С > 0, y'x@) = 0, where variable x takes
the role of the time, and у is the angle deviation from the equillibrium state, the
oscillations of the mathematical pendulum are described by the formula
sin — = msn(v/aa;), m = sin —,
Zi Zi
where sn = sn(z) is the Jacobi elliptic function defined implicitly as follows:
sn(z)=smp, z=
Jo
(parameter /3 is to be eliminated from these relations).
The period of oscillations of the mathematical pendulum is expressed in terms of
the complete elliptic integral of the second kind:
T =
K(m) = Г
Jo
/a Jo a/1 - т2 sin2 /3
At small amplitudes, as С —> 0, the following asymptotic formula holds for the period:
© 1995 by CRC Press, Inc.

'х + bsin(u>y) = 0.
Solution in the parametric form:
x = -A fт^2 - to2F2)~1/2 dr + C2, y= — arccos(-^-i^),
where A = b/a, F = At - A2 In \т\ + Съ
'x + bcos(wy) = 0.
The substitution toy = uiu + ¦& leads to an equation of the form 2.2.3.18:
u'xx ~ flsin(aiw)w^. — bsin(u)u) = 0.
2.2.4. Rayleigh Equations y?x + f{y'x) + g(y) = 0
Preliminary Comments. Equations of this form are encountered in the theory of
nonlinear oscillations.
1. Let us discuss the particular case g(y) = у which corresponds to the equation
+ у = о. (i)
Differentiating equation A) with respect to x and substituting z(x) = y'x, we obtain the
equation of nonlinear oscillations
4; + ФBL + z = 0, where Ф(г) = fz{z), B)
which is considered in Subsection 2.2.2.
The solution of equation A) can be written in the parametric form:
x = х(т, C1: C2), y=-f(z)-^,
xT
where x = х(т, С\, C2), z = z(t, C\, C2) is the parametric form of the solution of equa-
equation B).
2. The transformation
С=-|Ш2 + а, w = -y-f{y'x),
reduces equation A) to the Abel equation
[ = H(?)w + 1, where Я(?) = г~1Ф(г), z = ±y/2(a-C), C)
where function Ф = Ф(г) is defined above in equation B). Concrete equations of the form C)
are outlined in Subsection 1.3.2.
3. The equation of the special form
y'L + a(y'xJ + 9(y)=0 D)
© 1995 by CRC Press, Inc.

is reduced, with the aid of the substitution w(y) = (y'x) , to a first order linear equation
w'y + 2aw + 2g(y) = 0. Therefore, the solution of the equation D) can be written in the
implicit form
x = C2±J[C1e-2ay-G(y)]-1/2dy, where G{y) = 2e^ j'e2a*g{y)dy.
4. The equation of the special form
y/:x + <y/xf + b(y'xJ+g(y) = 0 E)
is reduced, with the aid of the substitution w(y) = (y'x) , to the Riccati equation w'y +
2aw2 + 2bw + 2g(y) = 0 which is outlined in Section 1.2.
5. For the oscillatory systems with a weak nonlinearity
Ухх + sF(Vx) + У = 0,
two leading terms of the asymptotic solution, as e —> 0, are described by the fromula
у = ^4cos(rr + B),
where functions A = A(?) and В = B(?) depend on the slow variable ? = ex and are defined
from the autonomous system of the first order equations
1 f2* 1 f2*
A'f = / F(—A sin tp) shop dtp, ABl = / F(—A sin tp) cos tp dtp.
4 2тг Jo « 2тг Jo
The right-hand sides of these equations depend only on function A. The system is solved
consecutively starting from the first equation.
v'L
This equation describes small oscillations when the drag force is proportional to the
speed squared.
Solution in the implicit form:
x = C2±a f[Cia2e-2av + b{\ - ay)} ~1/2 dy.
2-
Van der Pol equation.
Differentiating the equation with respect to x and passing on to a new variable
w(x) = y'x, we arrive at an equation of the form 2.2.2.2: wxx — e(l — w2)w'x + w = 0.
Solution, as e —>• 0:
2Ci V i
y = — cos ж -| sin ж + O(e2).
a/1 - C2e~ex л/1 - C2e~ex
© 1995 by CRC Press, Inc.

The transformation ? = — -j(y'xJ, w = —y — a(y'x)A — b(y'xJ leads to the Abel equation
of the form 1.3.2.1: юю[ = (-8a? + 2b)w + 1.
4- I&, + (У'хJ Hy'J + Ь] -1 + у = 0.
The transformation ? = — -jB/i) > w = —y — (y'x) [a(y'x) + 6] leads to the Abel
equation of the form 1.3.2.2: ш[ = 26F - 2a(,)~2w + 1.
5- tC + А ехР[Л(у4J] + В + у = 0.
Differentiating the equation with respect to x and passing on to a new variable
w(x) = y'x, we arrive at an equation of the form 2.2.2.7:
wxx + 2AXw exp(\w2)w'x + w = 0.
Differentiating the equation with respect to x and passing on to a new variable
w(x) = yx, we arrive at an equation of the form 2.2.2.11:
wxx + 2,4A«;sinh(A«;2)«4 + w = 0.
y'L + A sinh[*«J] + в + у = о.
Differentiating the equation with respect to x and passing on to a new variable
w(x) = yx, we arrive at an equation of the form 2.2.2.10:
wxx + 2AXw cosh(A«;2)«4 + w = 0.
This equation describes the oscillations of the mathematical pendulum when the drag
force is proportional to the speed squared.
Solution in the implicit form:
Л 2b 1/2
Cie-2ay + 4a2 + 1 (cosy - 2asiny) dy.
Differentiating the equation with respect to x and passing on to a new variable
w(x) = yx, we arrive at an equation of the form 2.2.2.15:
wxx — 2^4A«; sin(Xw )w'x + w = 0.
ю- y'L + A sHMy'J2} + B + y = o.
Differentiating the equation with respect to x and passing on to a new variable
w(x) = y'x, we arrive at an equation of the form 2.2.2.14:
wxx + 2AXw cos(Xw2)w'x + w = 0.
© 1995 by CRC Press, Inc.

2.3. Emden—Fowler Equation y" = Axny
m
2.3.1. Exact Solutions
Preliminary comments. The value of the insignificant parameter A is in many cases
defined in the form of a function of two (one) auxiliary coefficients a and b:
A = tp(a, b)
and the corresponding solutions are represented in the parametric form
x = Л(т, Ci, C2,a), у = /2(т, Ci, C2,b),
A)
B)
where т is a parameter, C\ and C2 are arbitrary constants, Д and f2 are some functions.
Having fixed the auxiliary coefficient sign a > 0 (or b > 0) in A), the coefficient b should
be expressed in terms of both A and a with the help of
b = ф{А, а).
Substituting this formula into B), we obtain a solution of the equation under consideration
(where the concrete numerical value of the coefficient a may be chosen arbitrarily). The
case a < 0 (or b < 0), which may lead to the branch of the solution or to a different domain
of determining the variables x and у in B), should be considered in a similar manner.
One may also use a different approach by setting one of the auxiliary coefficients (e.g., a)
equal to ±1 in A) and B); then the other coefficients will be identically expressed in terms
of A by means of A).
Table 2.7 represents all solvable Emden—Fowler equations whose solutions are outlined
in Subsection 2.3.1. The one-parameter families (in the space of parameters n, то) and iso-
isolated points are represented in a consecutive fashion. Equations are arranged in accordance
with the growth of то and the growth of n (for identical то). The number of the equation
sought is indicated in the last column in this table.
1.
Solution:
У =
2)
Arf.n+2
\X + C2 ltnf —1; —2,
h if n = -2,
A I In \x dx + dx + C2 if n = -1.
2-
V'L = A
Solution:
x =
2A
,-1/2
± /l^1— yn+1 + d) ' dy + C2 ifTO^-1,
TO+1
± fBAln\y\+d)~1/2dy
ifm = -l.
© 1995 by CRC Press, Inc.

TABLE 2.7
Solvable Cases of the Emden—Fowler Equation y'? = Axnyr
No
1
2
3
4
5
6
7
8
9
10
11
m
n
Equation
One-parameter families
arbitrary
arbitrary
arbitrary
0
1
0
—m — 3
-i(m + 3)
arbitrary
arbitrary
Isolated points
-7
-7
-5/2
-2
-2
-5/3
1
3
-1/2
-2
1
-10/3
2.3.1.2
2.3.1.3
2.3.1.4
2.3.1.1
2.3.1.5
2.3.1.15
2.3.1.16
2.3.1.22
2.3.1.28
2.3.1.27
2.3.1.10
No
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
m
-5/3
-5/3
-5/3
-5/3
-5/3
-7/5
-7/5
-1/2
-1/2
-1/2
-1/2
-1/2
-1/2
-1/2
2
2
2
n
-7/3
-5/6
-1/2
1
2
-13/5
1
-7/2
-5/2
-2
-4/3
-7/6
-1/2
1
-5
-20/7
-15/7
Equation
2.3.1.8
2.3.1.23
2.3.1.24
2.3.1.7
2.3.1.9
2.3.1.14
2.3.1.13
2.3.1.12
2.3.1.6
2.3.1.26
2.3.1.17
2.3.1.18
2.3.1.25
2.3.1.11
2.3.1.19
2.3.1.21
2.3.1.20
Special cases:
1°. In the case m = —1/2, the solution can be written in the parametric form
у = ЬС\{т2 - IJ, where A = ±^-a~2b3/2.
ж = аС3(т3-Зт + С2
2°. In the case m = —4, the solution can be written in the parametric form
т dr
2т
R
where R = у^т3 - 1), A = тба65.
3°. In the case m = 2, the solution can be written in the parametric form
x = аС^т, y = bC2p, where A = ±6a~2b~1,
the function p of the parameter т is defined in the implicit form:
т =
\/±Dp3 -
-C2.
The upper sign in this formula corresponds to the classical elliptic Weirstrass function
р=р(т + С2, 0, 1).
4°. In the case m = —5/2, the solution can be written in the parametric form
x = aC\p~2 [a/±Dp3 - 1) ± 2тр2], у = ЪС\р~2, where A = тЗа67/2.
The function p of the parameter т is defined in the previous case.
© 1995 by CRC Press, Inc.

Vxx — AX У ¦
1°. Solution in the parametric form with m ф —1:
-1/2dT+d\ , у = ЬС™+1т\ I (l±Tm+L)-L/2dT+d
where A = ± am+1b1~m.
2°. Solution in the parametric form with m = — 1:
\ f 1 -1
ж = CA j exp(=Fr2) dr + C2\ , у = 6exp(=Fr2) | / exp(=Fr2) dr + C2 \ ,
where A = =f262.
тгг+З
1°. Solution in the parametric form with m ф —1:
x = aC22 exp [ f(^—Tm+1 + \т2 + dY^ dr],
\J \ m +1 4 / J
m—1
. ( a \
where A = I —
V
2°. Solution in the parametric form with m = — 1:
1 \ -1/2
21п|т| + —T2 + CiJ
/f21|| 2 C)" dr\,
.2
where ^4 = 62/a.
5. y^ = Axny.
For n ^ —2, see equation 2.1.2.7. For n = —2, see equation 2.1.2.118.
6. y'lx y
Solution in the parametric form:
x = аС-3(т3 - Зт + C2)~\ у = bd(T2 - 1J(т3 - Зт + d)~\
where A = ±±а112Ъг12.
7. у»я = Аху-5'3.
Solution in the parametric form:
x = ±аС1{тА - 6т2 + 4С2т - 3), у = ЬС^т3 - Зт + C2K/2,
where A = ±-§Ia~3bs/3.
© 1995 by CRC Press, Inc.

8. y'lx у
Solution in the parametric form:
аСГ8 ЬСг(т3 - Зт + C2f'2
t4 - 6т2 + 4С2т - 3 ' У ~ t4 - 6т2 + 4С2т - 3 '
where A = ±-§Ia1/3bs/3.
9. y'lx = у
1°. Solution in the parametric form with A < 0:
ж = aC\ cos TCOsh(T + C2) [tan r + tanh(r + C2)], у = 6Cf [cosTCOsh(r + C2)]3/2,
where A=-^-a68/3.
2°. Solution in the parametric form with A > 0:
ж = aC2[sinhr + cos(t + C2)], у = 6C3[coshr - sin(r + C2)]3/2,
where A = f a
10. y'lx = у
1°. Solution in the parametric form with A < 0:
ж = aCj [cost cosh (т + C2)]~1[tanr + tanh(r + C2)]~1,
у = bCi [cost cosh (t + C2)]1/2[tanr +tanh(r + Сг)],
where A=-^-
2°. Solution in the parametric form with A > 0:
ж = aCj[sinhr + cos(t + Сг)],
у = 6Ci[coshr - sin(T + C2)]3/2[sinhT + cos(t
where A = f a4/368/3.
11. y'^
Solution in the parametric form:
x = aC\ exp(—т) [ехр(Зт) + C2 sin(v/3r)],
у = 6С2ехр(-2т)[2ехр(Зт) - C2sin(v/3T) + v/3C2cos(v/3t)]2,
where A = 16a~3b3/2.
© 1995 by CRC Press, Inc.

> In the solutions of equations 12-14, the following notation is used:
Si = ехр(Зт) + C2sin(v/3T), S2 = 2ехрCт) - СгвЦл/Зт) + л/3 C2cos(v/3t),
s3 = 2s1(s2yT-(s1yTS2-s1s2.
12. y»m =
Solution in the parametric form:
x = aCi1exp(r)Si1, у = 6Ciexp(-T)S'-1S'|) where A = 16(abK/2.
Solution in the parametric form:
x = aC$exp(-2T)S3, у = 6С15ехр(-|тM'15/2) where A =
14. yl = Ажу
Solution in the parametric form:
x = aC~A expBT)Sr3) у = ЬСг ехр^т)^572^1, where A =
> In the solutions of equations 15-18, the following notation is used:
R = ^/±Dт3 - 1), / = 2т/(т) + С2т Т R,
where 1(т) = /тД dr is the incomplete elliptic integral of the second kind in the
form of Weierstrass.
Solution in the parametric form:
Ж = аС?[4т/2тт-2(/Д-1J], y = bCff1'2, where A = ±^a
16. y':x
Solution in the parametric form:
x = aCf[4r/2 т T~2{fR - lJ], у = bC^f^rf2 T т(/Д - IJ],
where A = ±-^a68.
17. vZ<B y
Solution in the parametric form:
x = aClf, y = bCtT-2{fR-lJ, where A = ±|a/363/2.
18. y»m =
Solution in the parametric form:
x = aCff-3, y = bClT-2rz{fR-lJ, where A = ±|а
© 1995 by CRC Press, Inc.

> In the solutions of equations 19-24, the function p of the parameter т is defined
in the implicit form
-I
\/±Dp3 -
-C2.
The upper sign in this formula corresponds to the classical elliptic Weirstrass func-
function p = р(т + C2, 0, 1).
Solution in the parametric form:
x = aCiT~1, y = bCfr~1p, where A = ±6a3b~
20. yxx = Ax
Solution in the parametric form:
х = аС\т7, y = 6Cir(T2pTl), where A = ±-^a
21. y'lx -
Solution in the parametric form:
х = аС\т~7, у = 6С^г-6(т2рт1), where A = ±-^a6/7b~1.
Solution in the parametric form:
x = aC7p2 [\/±Dp3 - 1) ± 2тр2] ~\ у = 6С13[ч/±Dр3-1)±2тр2]"
where A = тЗа/267/2.
23. y'^x -
Solution in the parametric form:
l3/2
aCf
r? == i nj ——
[тч/±Dр3-1)+2р]2' [тч/±Dр3-1) + 2р]2
where ^4 = — 4-а~
24. у»я = Ax-^y-5'3.
Solution in the parametric form:
ж =
where А=-^
© 1995 by CRC Press, Inc.

> In the solutions of equations 25-28, the following notation is used:
( СаЛ/зС^) + C2Y1/3(t) for the upper sign,
\ Ci/i/3(t) + С2-К/з(т) for the lower sign,
where Ji/% andY1/?, are Bessel functions, I1/3 andK1/3 are modified Bessel functions.
25. y»m =
Solution in the parametric form:
x = ar2'3Z2, y =
where A = T ±(T1
26. y'lx = у
Solution in the parametric form:
where A = T-b3/2.
27- V'L = Axy~2.
Solution in the parametric form:
where a = -|(A
28.
Solution in the parametric form:
where A = -f-63.
2.3.2. First Integrals (Conservations Laws)
In this subsection, first integrals of the form
к
>
L ^
a=0
fa(x,y)(y'x)a = C, where /0 = 2,3,4,5,
for the Emden—Fowler equation y'J.x = Axnym are given.
First integrals with к = 2
1. For n = 0 and arbitrary m (m ф —1),
© 1995 by CRC Press, Inc.

2. For n = — and arbitrary m (m ф —1),
x{y'xf - yy'x -
3. For n = —m — 3 and arbitrary m (m ^ — 1),
= С.
2(^J - 2xyy'x + y2 -
= С
4. For n = —^-, m = 2,
2 - у = С.
5. For n = -Ц-, m = 2,
First integrals with к = 3
1. For n = 0, m = —I",
(y'xf -
2. For n = 1, m = —I",
3. For n = —y,m = —-j>
4. For n = —-|, m = —-j>
^J + (у2 -
5. For п = — ^г, т = — -j,
2(у;K - 2Жу(у^J + (у2
6. For п = —-Jr, m = —\,
x\y'xf - ix2y{y'xf + 3(жу2 -
© 1995 by CRC Press, Inc.
= С,
= С.
= С.
= с
- у3
2 = С,
^-1 = с.
+ 9А2х2'3 = С.
у; - у3
= С.

First integrals with к = 4
1. For n = 1, m = -J-,
2 - 18Ау1/3^ + QA^y-^ = C,
x(y'xf - y{y'xf + 6Ax2y-2/3(y'xJ - 27 Axy^3y'x + f- V/3 + S^Vy/3 = С.
2. For n = 2, m = —§-,
(y^L + 6Аж2у-2/3(^J - ЗбАгу1^ + 9AVy/3 = С.
3. For n = 0, m = -J-,
^iL - уЮ3 + б^у-2/3(у;J - э v'V* + 9А2жу-4/3 = с,
Ж2(^L - 2Жу(^K + (у2 + 6AcV2/3)(^J - lSAry^Vx + 12V/3 + S^Vy/3 = С.
4. For n = — y,m = — -|-,
<y'xf ~ V(y'j + 6Ax^y-^(y'J + 9A2y-4/3 = C.
5. For n = — y,m = — -|-,
Ж2(у;L - 2Жу(у;K + (у2 + 6Ar2/3y-2/3)(y;J + ЬАх-^'Ч + ^х-^у~т = С,
x\y'xf - 3x2y(y'xf + My2 + 2Ax2/3y-2/3)(y'xf
- (у3 + SAx2^3)y'x - ЗАх-1^3 + 9А2х^3у-^3 = С.
6. For n = --j, m = -J-,
- (у3 - 15^Ж-1/3у1/3)у; - f Ar-4/3y4/3 + 9А2Ж/Зу-4/3 = С,
x\y'xf - ±x3y(y'xf + 6x2(y2 + Ах-^у-^ШJ
-2хBу3 - SAx-^3y^3)y'x + у4 - 12Аж-1/3у4/3 + 9А2Ж-2/3у/3 = С.
7. For п = --§-, m = -¦§-,
- (у3 + 12Аж7/6у1/3)у; + бАх1^3 + 9А2х^3у~^3 = С.
8. Forn = -^-, m = —§-,
Ж4(у^L - 4Ж3у(у^K + 6Ж2(у2 + Аг-4/3у-2/3)(у;J
- 4ж(у3 - 6^Ж-4/3у1/3)у; + у4 - 30Аг-4/3у4/3 + 9А2Ж-8/3у/3 = С.
9. For п = 1, т = —7,
x(y'xf - y(y'xf + iAx2y-\y'xJ - \AxyS'x ~ ^Ay~^ + ±А2х3у~12 = С.
10. For n = 3, m = -7,
Ж3(у^L - 3Ж2у(у^K + 3Ж(у2 + ±Ax5y-6)(yxJ
- у(у2 + Ах5у-6)у'х + iАг4у + ^А2Ж9у-12 = С.
It should be noted that in the case к = 4 we omitted the first integrals of the form
aF2 + 0F + 7 = C,
where function F = F(x,y,y'x) is the left-hand side of the above intergals for к = 2, and a,
C, and 7 are some constants.
© 1995 by CRC Press, Inc.

First integrals with к = 5
1. For n = 0, m = -f,
(y'xf - lbAy^(y'xf + ^A2y2'3y'x ~ ^Asx = С
2. For n = —-|-, m = —-|-,
Ж5(^M - bx^y{y'xf + 5x3yBy - iAx-^yf
- bx2y2By - QA
\Ах-213уъ'3 - \А2х~1) = С.
2.3.3. Some Formulas and Transformations
1. With m/1, the Emden—Fowler equation has the particular solution
та+2
у = Xx 1~m , where A =
¦2)(n-
A(m - IJ
2. The transformation у = w/t, x = 1/t leads to the Emden—Fowler equation with the
independent variable to a different power:
w't = At-n-m-3wm.
3. Some more complicated transformations leading to the Emden—Fowler equation are
outlined in Subsection 2.5.3 (see Fig. 3).
4. With m/1 and m ф —In — 3, the transformation
2n + m + 3 "+2 Ji+2- / n + 2
С = —, x ™-i y, u = x ™-i [xyx H —— у
leads to the Abel equation:
(n + 2)(n + m + l)
uu> u
whose special cases are given in Subsection 1.3.1.
5. Some more complicated transformations leading to other Abel equations are outlined
in Subsection 2.5.3.
2.4. Equations of the Form y%x = A1xniymi + А2хП2ут2
See Section 2.3 for the special cases Аг = 0 and A^ = 0.
2.4.1. Classification Table
Table 2.8 represents all solvable equations of the form y%x = A1xnxymx + A2xn2ym2
whose solutions are outlined in Subsection 2.4.2. The two-parameter families (in the space
of parameters mi, m,2, n\, and H2), the one-parameter families, and isolated points are
represented in a consecutive fashion. Equations are arranged in accordance with the growth
of mi, the growth of m^ (for identical mi), the growth of щ (for identical mi and m.2),
and the growth of n<i (for identical mi, m.2, and n\). The number of the equation sought is
indicated in the last column in this table.
© 1995 by CRC Press, Inc.

TABLE 2.8
Solvable equations of the form
= A1xniymi + A2xn2ym2
No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
nil
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
1
1
1
-7
-5
-3
-3
-3
-3
-2
-2
-2
5
3
5
3
5
3
5
3
5
3
5_
3
5
3
3
2
3
2
7
5
4
3
4
3
ГП2
arbitrary
arbitrary
arbitrary
0
0
arbitrary
arbitrary
-3
-7
-5
-7
-7
-4
-4
-3
-3
-2
5
3
5
3
5
3
5
3
5
3
5_
3
5
3
-2
-2
7
5
5_
3
5
3
ГЦ
0
—nil — 3
-|(mi + 3)
0
—mi — 3
2
2
arbitrary
(m ^ -2)
4
2
0
0
0
0
-2
1
-1
7
3
4
3
4
3
2
3
0
2
2
3
2
0
8
5
5_
3
0
П2
0
—m<i — 3
-|(m2 + 3)
0
-3
2
—Ш2 — 1
0
3
0
1
3
0
1
0
0
-2
10
3
10
3
7
3
4
3
2
0
1
-2
1
13
5
7
3
1
Ax
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
2(m2 + 1)
(m2 + 3J
2(m2 +1)
(m2 + 3J
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
A2
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
Equation
2.4.2.1
2.4.2.2
2.4.2.3
2.4.2.19
2.4.2.20
2.4.2.4
2.4.2.5
2.4.2.83
2.4.2.39
2.4.2.16
2.4.2.42
2.4.2.43
2.4.2.17
2.4.2.18
2.4.2.88
2.4.2.87
2.4.2.28
2.4.2.48
2.4.2.49
2.4.2.24
2.4.2.90
2.4.2.89
2.4.2.47
2.4.2.46
2.4.2.81
2.4.2.80
2.4.2.25
2.4.2.102
2.4.2.101
© 1995 by CRC Press, Inc.

TABLE 2.8 Continued
Solvable equations of the form y'? = А1хПгутг + А2хП2ут2
No
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
mi
3
5
3
5
l
2
1
3
1
3
1
3
1
3
1
3
1
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
TO2
7
5
7
5
1
2
5
3
5
3
5
3
5
3
5_
3
5
3
-2
-2
-1
-1
2
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
Hi
12
5
0
5
2
8
3
8
3
8
3
0
0
0
-3
0
-3
0
-3
0
-4
-3
-3
-3
-3
5
3
3
2
3
2
3
2
3
2
4
3
0
0
0
0
П2
13
5
1
7
2
10
3
7
3
4
3
0
1
2
-2
1
-2
0
7
3
0
5
2
7
2
5
2
-2
l
2
7
6
5
2
-2
l
2
0
4
3
-2
l
2
0
1
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
A2
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
Equation
2.4.2.53
2.4.2.52
2.4.2.23
2.4.2.55
2.4.2.59
2.4.2.57
2.4.2.56
2.4.2.58
2.4.2.54
2.4.2.108
2.4.2.107
2.4.2.22
2.4.2.21
2.4.2.73
2.4.2.72
2.4.2.96
2.4.2.51
2.4.2.45
2.4.2.106
2.4.2.85
2.4.2.41
2.4.2.100
2.4.2.79
2.4.2.78
2.4.2.99
2.4.2.40
2.4.2.86
2.4.2.105
2.4.2.44
2.4.2.50
© 1995 by CRC Press, Inc.

TABLE 2.8 Continued
Solvable equations of the form y" = AiX^y™1 + A2xn2ym2
No
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
mi
0
l
3
l
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
5
3
5
3
-7
-7
-4
-4
-3
-3
5
2
5
2
-2
-2
5
3
5
3
_5_
3
5
3
5
3
5
3
5
3
_5_
3
7
5
7
5
1
2
1
2
1
2
1
2
1
2
1
2
0
Hi
1
10
3
0
-2
-2
-2
-2
-5
1
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-5
П2
0
7
3
1
-2
6
-2
3
0
0
-2
3
2
-2
1
-2
2
3
-2
2
-2
2
3
-2
2
?
-2
2
T
-2
l
2
-2
l
2
-2
l
2
-3
arbitrary
arbitrary
arbitrary
15
4
15
4
6
6
arbitrary
arbitrary
12
12
2
2
3
16
3
16
9
100
9
100
3
4
3
4
63
4
63
4
5
36
5
36
2
~~ 9
2
4
25
4
25
20
20
arbitrary
A2
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
Equation
2.4.2.95
2.4.2.98
2.4.2.97
2.4.2.35
2.4.2.36
2.4.2.31
2.4.2.32
2.4.2.84
2.4.2.82
2.4.2.64
2.4.2.65
2.4.2.6
2.4.2.7
2.4.2.26
2.4.2.27
2.4.2.10
2.4.2.11
2.4.2.12
2.4.2.13
2.4.2.66
2.4.2.67
2.4.2.29
2.4.2.30
2.4.2.14
2.4.2.15
2.4.2.8
2.4.2.9
2.4.2.33
2.4.2.34
2.4.2.77
© 1995 by CRC Press, Inc.

TABLE 2.8 Continued
Solvable equations of the form y^x = A1xn^ym^ + A2xn2ym2
No
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
mi
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
m2
0
l
2
1
2
0
0
0
0
0
0
0
0
1
1
1
1
1
1
2
2
Hi
1
-2
-2
-5
-5
20
7
20
7
15
7
15
7
0
0
-3
-3
-2
-2
-6
0
18
5
12
5
n2
0
-2
3
2
-4
-3
13
7
12
7
9
7
8
7
0
1
-2
-2
-2
-2
-5
1
14
5
11
5
arbitrary
12
49
12
49
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
A2
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
arbitrary
6
25
6
25
6
25
6
25
arbitrary
arbitrary
arbitrary
arbitrary
Equation
2.4.2.76
2.4.2.37
2.4.2.38
2.4.2.92
2.4.2.69
2.4.2.94
2.4.2.71
2.4.2.70
2.4.2.93
2.4.2.68
2.4.2.91
2.4.2.61
2.4.2.63
2.4.2.60
2.4.2.62
2.4.2.104
2.4.2.103
2.4.2.74
2.4.2.75
2.4.2. Exact Solutions
l.
V'L =
i + A2ym2, mi ф -1, m2 ф -1.
1°. Solution in the parametric form:
x = a
dr + C2
y = Ьт,
where Аг = ^a^b1'^^ + 1), A2 = ±\a
2°. Solution in the parametric form:
+ 1).
x =
a/,
= Ът,
where A1 = —\a 261 mi(mi + l), A2 = ±\a
© 1995 by CRC Press, Inc.
^(m2 + 1).

2. y^x = AlX-m^-3ym^ + A2x-m2-3ym2, mi ф -1, m2 ф -1.
1°. Solution in the parametric form:
= a I / (Ci + rmi+1 ± rm2+1) 1/2 <2т + C2 I ,
-l
у = Ьт\ I (Ci+Tmi+1±Tm2+1) 4Ut + C2}
where Ax = \ах+тхЬх~тх{гпх + 1), A2 = ±^a1+m2b1~m2(m2 + 1).
2°. Solution in the parametric form:
x = a\f(C1- rmi+1 ± rm2+1)/2 dr + C2] ,
у = Ьт\[(С1- rmi+1 ± rm2+1)/2 dr + C2j ,
where Аг = -{й1^1!I-™1^! + 1), A2 = ±±a1+m2b1-m2(m2 + 1).
rrai+3 тгг2+3
1°. Solution in the parametric form with mi ф —1 and m2 ф —1:
.-V2
mi + 1 m2 +
I2JV 4
2°. Solution in the parametric form with mi ф —1 and m2 = —1:
x = Cl exp \[(C2 + \t2 + -^—Tmi+1 + 2A2 In |r|)/2 dr],
U V 4 mi +1 / J
(m + 3J
Solution in the parametric form:
m+3 _
x = C1\l(l±Tm+1)-irUT + C2\ , y = br\l(l±Tm+1)-irUr + C2-m~1
where A =
2(m
© 1995 by CRC Press, Inc.

~1ym, m ф —3, m ф —1.
Solution in the parametric form:
ж = (
m+3
«* —1
>zdT+C2\ , y = l
1/2 dr+C2
m+1
m—1
6. y^ = 2x~2y -
Solution in the parametric form:
¦Cs
-ln(v/f+v/TTT) +C2'
1-1/3
J >
-2/3
where A = -f-63.
= 2ж~2
- ln(v/f+
Axy~2.
Solution in the parametric form:
У =
where A = -f-63.
// A О
xX — 25 X У
Solution in the parametric form:
x = d(r3 - Зт
where A = ±^-63/2.
»," — Ё_ -2.. I 4-.-1/2..-1/2
Ухх — 25 " ^ "
Solution in the parametric form:
x = Ci(t3 - 3t + С2M/3, у =
where A = ±^-6
-1/3
, у = Ъ(т2 - 1J(т3 - Зт + С2)
/3
-i)V-3T + c2r;
I/»» = -T5o Ж
Solution in the parametric form:
ж = Ci [±(r4 - 6r2 + 4C2r - 3)] /4,
у = Ь(т3 - Зт + C2K/2 [±(т4 - 6т2 + 4С2т - 3)
where A = ±
© 1995 by CRC Press, Inc.

Solution in the parametric form:
ж = С1[±(т4-6т2+4С2т-3)]5/4, у = 6С1(т3-Зт+С2K/2[±(т4-6т2+4С2т-3)]1/8)
where A = ±^b
Solution in the parametric form:
x = d(r3 ± 3r + C2)~1/2, у = Ь(т2 ± lK/2(r3 ± 3r + C2)/4,
where A = ±j-bs/3.
Solution in the parametric form:
x = d(r3 ± 3r + C2I/2, у = 6d(r2 ± lK/2(r3 ± 3r + C2)/4,
where A = ±j-bs/3.
tC = —1х-2У +
Solution in the parametric form:
x = Cx{Cxe2kT + C2e-kTsmu>y3, u> =
у = Wc2(Cie2feT + C2e-kT ainu)~2[2C1e2kT + С2е~кт(лДcosw - si
where A = ^§-bk3.
Solution in the parametric form:
sin u>f, u> =
2кт + С2е~кт
у = Ък2Сг{Сге2кт + С2е~кт sinw) [2Сге2кт + С2
where A = ^§-bk3.
I6- V'L = Aix2y~5 + A2y~5.
Solution in the parametric form:
= <V{cos[/(c,
© 1995 by CRC Press, Inc.

y%a = А1У~3 + A2y~4.
1°. Solution in the parametric form:
x = a\ I (Ci+t-3±t) 1/2dT + C2\, у = Ьт,
where Аг = т«~2&4, M = —|a65.
2°. Solution in the parametric form:
x = a\ / (Ci - t ± t)
where Л = т«&4, Л = 4а65.
1°. Solution in the parametric form:
i — 1 г /* ~\ — 1
X = ( "-¦-¦- ^-i ¦ "
where A1 = =Fa 2bi, A2 = — -|« 3&5-
2°. Solution in the parametric form:
\[]
where Аг = T«&4, ^2 = ^a
+ А2, тф -1.
1°. Solution in the parametric form:
x = a\ I (Ci + rm+1 ± t) 1/2 dr + C2 I, у = Ьт,
where Ai = ^a61-m(m + 1), A2 = ±\а~2Ъ
2°. Solution in the parametric form:
= а\({С1-тт+1±т) 1/2dr + C2\ у = Ът,
where Ax = -ia61-m(m + 1), A2 = ±\a~2b.
3°. See equation 2.4.2.21 for the case m = —1.
© 1995 by CRC Press, Inc.

20. y'lx = A1x-m-3ym + A2x~3, m ф -1.
1°. Solution in the parametric form:
= a A) 2
where Ax = \a1+mb1-m{m + 1), A2 = ±\ab.
2°. Solution in the parametric form:
x = a\ f (С1-тт+1±т)~1/2с1т + С2~\ ,
where Ax = -^-a1+m61-m(m + 1), A2 = ±\ab.
21. у?а = Ai + A2y-\
r
Solution: x= (C1+2A1y + 2A2ln\y\y1/2dy
22. y'lx = AlX~3 + А2х~2у-1.
Solution in the parametric form:
\
x = i / [Сг + 2AlT + 2A2ln |t|)/2 dr + C2~\ ,
i -i
2AlT + 2A2In \t\)-^2 dr + C
23. y'lx = Aioj-5/2»-1/2 + A2x-r/2y-1/2.
Solution in the parametric form:
2kT + C2e~kT
=—, у = —{2de2kT + C2e
F F
where F = Cie2kT + С2е~кт sin(wT) - 41, M = 16fc3, w =
A2
Solution in the parametric form:
У = \~qA2t + 3Cit + C2) (-og-^427" ~\~ С\т + С2т + С3) ,
where 9dC2 = Аг + A2C3.
© 1995 by CRC Press, Inc.

25. y'lx = AlX-8^y-7^ + A2x-
Solution in the parametric form:
X =
where S = C1e2kT + C2e-kTsm(V3kT), F = (S'TJ-2SS?T, ^2 = -
26. y»m = —5r
1°. Solution in the parametric form with A < 0:
x = Ci[cosh(T + C2)cosT]~2[tanh(T + C2) + tanr]~2,
у = 6[tanh(r + C2) + tanr]/2,
where A = --^bs/3.
2°. Solution in the parametric form with A > 0:
x = Ci[sinhT + cos(t + C2)]~2,
у = b[coshr - sin(T + C2)]3/2[sinhT + cos(t + C2)]/2,
where A = ^-&
27. y'lx = -^r
1°. Solution in the parametric form with A < 0:
x = Ci[cosh(r + C2) cosT]2[tanh(Y + C2) + tanr]2,
у = 6Ci[cosh(r + C2) cosr]2[tanh(T + C2) + tanr]1/2,
where A = --^bs/3.
2°. Solution in the parametric form with A > 0:
x = Ci[sinhT + cos(t + C2)]2,
у = 6Ci[coshT - sin(T + C2)]3/2[sinhT + cos(t + C2)]1/2,
where A = ^
28. у»я = A^y-2 + А2х~2у-2.
Solution in the parametric form:
x = {aClT-^ [(rZ; + \Zf ± r2Z2] -
]
у = ЬСгт^г^аСгт2^ [(rZ'T + \Zf ± r2Z2] - A
where
Г CiJ1/3(t) + C2Y1/3(t) for the upper sign,
I CiIi/s(t) + С2ЛГ1/з(т) for the lower sign,
Ji/з and Y1/2, are Bessel functions, Ix/3 and Kx/3 are modified Bessel functions;
A2 = -|a-363.
© 1995 by CRC Press, Inc.

> In the solutions of equations 29-30, the following notation is used:
Sx = de2kT + C2e-kT sin(v/3/cT),
S2 = 2kde2kT + kC2e-
S3=S%- 2S1(S2)'T.
29. y»m = -^
Solution in the parametric form:
r, c-3/2 iri5/2c-5/4
x = dS3 , у = bS^ S3
where A = -^12^6
30. y'L = —kx~2y +
Solution in the parametric form:
x = ClSl'\ у =
where A = -^12^
> In the solutions of equations 31-39, the following notation is used:
R = у^т3 - 1), I=JTR~1dT, F1 = 2tI + C2ttR, F2 = t^^ - 1),
where I = I(t) is the incomplete elliptic integral of the second kind in the Weierstrass
form.
31- v'L = 6x~2v + лх~2у-4.
Solution in the parametric form:
x = dr/^5, у = b
where A = =f15065.
32- V'L = 6х~2У + Лх3у~4.
Solution in the parametric form:
x = dr^F-1'5, у = bC
where A = =f15065.
33. y»m = 20x~2y + Ax-^y-1/2.
Solution in the parametric form:
x = CxF\l\ y = bF
where A = ±10863/2.
© 1995 by CRC Press, Inc.

34. y'lx = 20x~2y
Solution in the parametric form:
x = dF-1/3, у
where A = ±10863/2.
35. y'L = ^x-2y + y
Solution in the parametric form:
x = C^tF? t Fif\ у =
where A = ±-|&8.
Solution in the parametric form:
x = Ui(^t±<{ =F *2) > У =
where A = ±-jb8.
О I * (J iX/
Solution in the parametric form:
where A = ±^|-
00 ,.// _ 12 -:
Solution in the parametric form:
where A = zb-Ц-б1/2.
39. y^ = AlX4y-r + A2x3y-7.
Solution in the parametric form:
Л I
Г ^Я/ 9 9ч All о 1/2 Г Я, 9 9ч ^1 1 '
ж = aCf D:tF2 =f -^2) г" , У = bC3F,f аС?DтК2 Т F%) -
L A2 \ L А2 \
where А2 = ±-^а~368.
© 1995 by CRC Press, Inc.

> In the solutions of equations 40~43, the following notation is used:
Rl = (d + г ± T'2f/2, R2 = (Сг - г ± гI72,
E1= f R-1 dr + C2, E2= f R-1 dr + C2,
F1=t-R1E1, F2 = t-R2E2,
Hx = 3t3F? + 3A ± r)El, H2 = Зт3^| + 3(-l ± r)E
40. y'lx = AlX-4/3 + А2х~4/Зу-1/2.
Solutions in the parametric form:
x = ат-3Е3к, у = bFl
where Ax = т|а/36, А2 = \а-2'Ч312{-1)к; к = 1 and к = 2.
41. y»m = AlX~5/3 + А2ж-7/бу-1/2.
Solutions in the parametric form:
x = ат3Е~3, у = bT3E~3F2,
where Аг = ±%a~1/3b, A2 = \a-b/%3/2(-l)k+1; к = 1 and к = 2.
42. y'lx = Aiy~3 + A2xy-r.
Solutions in the parametric form:
where A\ = T^a 264, A2 = —-^a 3bs; к = 1 and к = 2.
Vxx = -A-iy + A2X у
Solutions in the parametric form:
r _ nT3 tr-l , _ г,т5/2 Rl/2 rr-1
where A\ = ±-^-a~2b4, A2 = —-^-a~5bs; к = 1 and к = 2.
> In ifte solutions of equations 44~45, the following notation is used:
CxekT + C2e~kT - ^j-т if Ax > 0,
A2
C\ sin(/cr) + C2 cos(/ct) — т if A\ < 0,
A\
k(CiekT - C2e~kT) - 41 */^i>0,
fc[Ci sin(A:T) - C2 cos(fcr)] - -?- if Ax < 0,
where к = J\\A\\.
© 1995 by CRC Press, Inc.

44. у':я = А1 + А2у-1/*.
Solution in the parametric form:
= /1, У = /I-
45. y»m = AlX~3 + A2x-5/2y~1/2
Solution in the parametric form:
r-l
> In the solutions of equations 46-^7, the following notation is used:
For Ax > 0,
Ti = CiefeT + С2е~кт + C3 вш(кт), к = (|
T2 = k(dekT - C2e-kT) + kC3 cos(/ct);
For Аг < 0,
Tx = eST[ClSin(sT) + C2coS{st)} + C3e~ST sin(sT), s = (-|
T2 = seST[(Ci - C2) sin(sr) + (d + C2) cos(st)] - sC3e-ST[sm(sT) - cos(sr)].
46. y^ = А1Ж2у-5/3 + A2xy-5/3.
Solution in the parametric form:
where arbitrary constants C\, C2, and C3 are related by
4dC2 + C\ = \A~2A\ if Ax > 0,
dCs = -^A-2A\ if A! < 0.
47. !&, = А1Ж2у-5/3 + A2y-5/3.
Solution in the parametric form:
where arbitrary constants C\, C2, and C3 are related by
4dC2 + C\ = -\AlxA2 if Ai > 0,
dC3 = -\А-гА2 if Ax < 0.
© 1995 by CRC Press, Inc.

> In the solutions of equations 48~49, the following notation is used:
For A2 > 0,
Ti = dekT + С2е~кт + C3 sin(jfcT), к = (|^2I/4,
T2 = k(dekT - С2е~кт) + kC3 cos(/ct);
For A2 < 0,
Тг = eST[Cisin(sT) + C2cos(st)} + C3e~ST sin(sT), s = {-\A2I//k,
T2 = seST[(Ci - C2) sin(sT) + (Ci + C2) cos(st)} - sC3e-ST[sm(sT) - cos(sr)].
48. y'lx = AlX-7^y-b^ + А2ж-
Solution in the parametric form:
where arbitrary constants C\, C2, and C3 are related by
4dC2 + C7| = l^^sT2 if A2 > 0,
C1C3 = ТвЛ^А-2 if A2 < 0.
49. y^ = А1Ж-4/3у-5/3 + А2ж-
Solution in the parametric form:
where arbitrary constants C\, C2, and C3 are related by
4CiC2 + Cl = -\AXA~Y if A2 > 0,
C1C3 = -i^i^1 if ^2 < 0.
> In the solutions of equations 50-53, the following notation is used:
Rx = ClTkl + С2тк2 + С3ткз,
R2 = (d + С2т)ект + С3ешт,
R3 = dekT + eST(d sin сот + C3 cos сот),
Qx = dklTkl + dk2Tk2 + С3к3ткз,
Q2 = (kd +d + kdr)ekT + соС3ешт,
Q3 = kCxekT + eST[(sd - шС3) sincor + (sC3 + cod) cos сот],
Si = t(Qi)'t, S2 = (Q2)'T, S3 = (Q3)'T,
where k\, k2, and k3 (real numbers) or к and s ± ico (one real and two complex
numbers) are the roots of the cubic equation A3 —jB2X —\B\ = 0. Subscripts of
© 1995 by CRC Press, Inc.

functions Rm, Qm, and Sm (m = 1, 2, 3) are selected depending on the sign of the
following expression:
{> 0 subscript 1,
= 0 subscript 2,
< 0 subscript 3;
if 2B\ = 27B\ (subscript 2), then
\ w = -2(|B2I/2 if Bl<0,
k = -{±rB2yl\ lo = 2{±B2I/2 if Bl>0.
Remark. The expressions for Rm, Qm contain three constants C\, C<i, and C3.
One of them may be arbitrarily fixed to set it equal to any nonzero number (for ex-
example, we may set C3 = ±1), and the other constant may be arbitrary.
50- V'L = M +
Solution in the parametric form:
= Rm, y = Q2m, A1 = B2, A2 =
» = AlX~3
51. y»m = AlX
Solution in the parametric form:
52 ?/" ^^ /4 11—*^/^ —i— A tii—^/^
Solution in the parametric form:
x = aBQ2m - ARmSm +
where Аг = -ab~^bA2B2, A2 = -^а'Ч
Solution in the parametric form:
x = aBQ2m - ±RmSm + B2R2m)~\ у = ЪнЦ2B<32т - ±RmSm + B^J'1,
where Аг = -^2-a2/5bs/5B~2B2, A2 = --^2-a3/5b12/5B~2.
> In the solutions of equations 5^-55, the following notation is used:
1°. For A2 > 0, Аг Ф 0:
Тг = Сгект + С2е~кт + С3 sin lot,
Т2 = к(Сгект - С2е~кт) + и>С3 cos lot,
where к = {\[{А\ + ЪА2у/2 + Аг}}1/2, и = {\[{А\ + ЪА2у/2 - Аг}}1/2; arbitrary
constants Сг, С2, and C3 are related by the constraint 4к2СгС2 + lo2C% = 0.
© 1995 by CRC Press, Inc.

2°. For -A2 < ЪА2 < 0, Аг> О:
Ti = CiTfel + C2T~kl + С3тк2 + С4т~к2
T2 = /ci(CiTfel - C2T-kl) + к2(С3тк2 -
where h = {-З-^ + ^ + ЗАзI/2]}1^, k2 = {^[Аг-(А2+ ЗА2)^2}}^2; arbitrary
constants C\, C2, and C3 are related by the constraint (C\C2 + Cs
(СгС2 - СгС^Аг = 0.
3°. For -A\ < ЪА2 < 0, Ax< 0:
T\ = C\ sintJiT + C2 costJiT + Cs sinter,
T2 = u)\(Ci costJiT — C2 sinwir) + ui2Cz cosco2t,
where Ш1 = {-|[A1 + (^ + 3)]} {j[{ )
arbitrary constants C\, C2, and C3 are related by ш\(С\ + Cf) — u)\C\ = 0.
4°. For A\ + 3A2 =0, Аг> О:
Ti = (d + С2т)ект + (C3 + САт)е~кт,
T2 = {кСг + С2 + кС2т)ект - (кС3 -О± + кС±т)е~кт,
where k = (-jAiI/2; arbitrary constants C\, C2, and C3 are related by the constraint
(C1C4 - C2C3)(|AiI/2 + 2C2C4 = 0.
5°. ForA\ + iA2 =0, Ax <0:
T\ = (Ci + C2t) sintJT + C3T cos lot,
T2 = (ojCi + C3 + ojC2t) cos cot + (C2 — ojC^t) sin cot,
where со = (—^AiI/2; arbitrary constants C\, C2, and C3 are related by the con-
constraint CiC3(—|ЛI/2 + C| + C| = 0.
6°. For ЪА2 < -А2:
Ti = ekT(Ci sincoT + C2 coscot) + С3е~кт sincoT,
T2 = ekT[(kC2 + cod)coscot + (ВД -coC2)siucot)}
+ Сзе~кт (со coscot — ksincor),
where к = {±[Аг + {-ЗА^1/2}}1/2, со = {^[-Аг + {-ЗА^1/2}}1/2; arbitrary con-
constants Ci, С2, and С3 are related by C2AX + Ci(-A2 - 3A2I/2 = 0.
54. y»m = Агу-1/3 + А2х2у~5/3.
Solution in the parametric form:
T.
55. y'lx = А1Х-8/3у-г/3 + A2xy
Solution in the parametric form:
x — j1 , у —
© 1995 by CRC Press, Inc.

> In the solutions of equations 56-59, the following notation is used:
COT i /"* —COT i /"* ' ¦? Л ~~~^ f\
[в +C2e + O3T if Ax > 0,
1 \ С\ sin сот + С2 cos сот + С3т if Ах < 0;
т = Г со(Сгешт - С2е~шт) + С3 if Аг > 0,
2 \ co(Ci cos сот — С2 sin сот) + С3 if Ai < 0;
where со = |/2
56. !&, = Аху-^з + А2у-*/3.
Solution in the parametric form:
where arbitrary constants C\, C2, and C3 are related by
3(AiCf + A2) + 16AldC2 = 0 if Ax > 0,
ЦАгС! + A2) + 4A\(Cl + Cl) =0 if Ax < 0.
57. y'lx = А1Х-8/3у-г/3 + А2х-*/3у-5/3.
Solution in the parametric form:
r — T-1 „ _ rp-lrp3/2
where arbitrary constants C\, C2, and C3 are related by
iCf + A2) + 1&А\СгС2 = 0 if Аг > 0,
С1 + А2) + AA\{Cl + Cl) =0 if Аг < 0.
58. y'lx = AlV-^3 + A2xy~5/3.
Solution in the parametric form:
A2 2 ( A2 \3/2
X = Tl-JA~T' y=V2-^A-T) '
where arbitrary constants C\, C2, and C3 are related by
3^iC| + 1&A\C1C2 + -^A~2A\ = 0 if Ax > 0,
Solution in the parametric form:
,-1 /_ А-2 .л-1/^ А2
-T
where arbitrary constants C\, C2, and C3 are related by
iAxCl + l&Aldd + -^A~2Al = 0 if Аг > 0,
3^iC| + 4^(C2 + Cl) + -^A~2A\ = 0 if Аг < 0.
© 1995 by CRC Press, Inc.

> In the solutions of equations 60-67, the following notation is used:
I o f dp
f = л/±Dр3 — 1), т = / — — C2.
V J v/±(^p3 - 1)
Function p = р(т) is defined implicitly in terms of the above elliptic integral of the first
kind. For the upper sign, function p coincides with the classical elliptic Weirstrass
function p = р(т + С2, 0, 1). In the solution given below, one can take p as the
parameter instead of т and use the explicit dependence т = т(р).
60. у»х у4у
Solution in the parametric form:
х = С1т5, у = Ьт2р, where A = ±-^b~
61. y'lx -
Solution in the parametric form:
x = ClT-5, y = bClT-3p, where A = ±4rb~1.
62. y»x y + 4y
Solution in the parametric form:
х = С1т5, у = 6(т2рт1), where
63. у»х у + 4у
Solution in the parametric form:
x = ClT-5, у = 6С1т-5(т2рт1), where A = ±-^b~1.
64. y'lx = 12x~2y + Ax~2y-5/2.
Solution in the parametric form:
x = Clf?l\f ± 2rp2)-1/7, у = bp-^(f ± 24/7
where A = =f14767/2.
65. y'lx = 12x~2y
Solution in the parametric form:
x = C.p-^if ± 2rp2I/7, у = ЬСгр-^if ± 2rp2)-3/7,
where A = =f14767/2.
66. y»m = Щ-х~
Solution in the parametric form:
x = d(r/ + 2p)/4, у = b(rf + 2p)"9/8(/ ± 2rp2K/2,
where А=-Щ-
67. y" = —x-
Solution in the parametric form:
x = d(r/ + 2PI/4, у = бСДт/ + 2p)~T\f ± 2rp2f,
where A = —Щ-1
© 1995 by CRC Press, Inc.

> In the solutions of equations 68-73, the following notation is used:
З _ 2pi -C2, т=
l±Ap\ - 2pi - C2
and
f 2p2 - C2
where functions p\ = pi(r) and p2 = р2(т) are the inverse functions for the above
elliptic integrals. For the upper signs, they are the classical Weierstrass functions
px = р(т + i, 2, C2) and p2 = р(т + ь -2, C2).
68. yxx = Axy2 + A2.
Solutions in the parametric form:
x = ат, у = bpk,
where AY = ±6a~2b-1, A2 = a~2b(-l)k, к = 1 and к = 2.
69. y'lx = А1х~ъу2 + A2x~3.
Solutions in the parametric form:
х = ат~х, у = Ьт~1рк,
where Ai = ±6a3b~1, A2 = ab(—l)k, к = 1 and к = 2.
7П »," А т. —15/7„.2 I л т.—9/7
Solutions in the parametric form:
x = ат7, у = Ът(т2рк т 1),
where Аг = ±-^a1/7b~1, A2 = -^а-ъ/7Ъ{-1)к, к = 1 and к = 2.
71. у'1х = А1Ж-2°/V + А2х-12'7.
Solutions in the parametric form:
where Ax = ±-^a6/7b~1, A2 = ^a~2/7b(-l)k, к = 1 and к = 2.
Solutions in the parametric form:
x = a[fk-(-l)kT], y = bp\,
where A\ = ±\a~2b, A2 = ¦j2-a~2bb/z(—l)k, к = 1 and к = 2.
Solutions in the parametric form:
x = a[fk - (-1)кт}-\ у = bp\[fk - (-1)кт}-\
where Ax = ±\ab, A2 = ¦j2-a1/3b5/3(-l)k, к = 1 and к = 2.
© 1995 by CRC Press, Inc.

> In the solutions of equations 7^-75, the following notation is used:
E= I A ± т4)/2 dr + C2, k2 = ±1.
Function E can be expressed in terms of elliptic integrals or lemniscate functions.
74. y'lx = AlXy + 2y
Solutions in the parametric form:
x = aC\E~b, у = ЪС$Е~\тЕ - к),
where Ax = ±-^a^5b~2, A2 = ±-^a^5b~1k.
75. y»m = AlX
Solutions in the parametric form:
x = aCfE5, у = bdE(TE - k),
where A1 = ±^a2lbb~2, A2 = ±^а1^Ь~1к.
> In the solutions of equations 76-81, the following notation is used:
Г Л/з(т) for the upper sign (Bessel function),
I I\/z (T) for the lower sign (modified Bessel function),
( Y1/^(t) for the upper sign (Bessel function),
\ ATi/3 (t) for the lower sign (modified Bessel function),
i n i о f f ? j с f j \ I "T71" f°r the upper sign,
+ C2g + 0u[g fdr-f gdr), u> = < 2 J w у>
\ J J J [—1 for the lower sign.
76. y'xx = A^xy + A2.
Solutions in the parametric form:
x = ат , у =
where A, = Tf a, A2 = \а~2Ц.
77. y'ix = AlX-5y + A2x~3.
Solutions in the parametric form:
x = ат~2/3, у = т~1/3Н,
where A\ = =F-|-a3, A2 = \afi.
© 1995 by CRC Press, Inc.

78. у%а = Агх~3/2 + A2x~1/2y-1
Solutions in the parametric form:
where Аг = -\а~1/2ЪC, А2 = ^\а~3/2Ъ3/2.
70 »," _ Д ~-3/2 i д -2 -1/2
Solutions in the parametric form:
Ж = ат/3Я-2, у = Ът-^3Н-2(тН'т + \Н
where ^i = --^а~1/2&/3, ^2 = т\Ь3/2.
80. y^ = ^1У~3/2 + A2xy~2.
Solutions in the parametric form:
x = ат~2/3 [тт2Н2 + 2/ЗтЯ - (тН'т + ^-ЯJ], у =
where Аг = -ab~xl2fiA2, A2 = \а~3Ь3.
Solutions in the parametric form:
x = ат2'3 [тт2Н2 + 2/ЗтЯ - (тН'т + |ЯJ] "\
у = Ьт^Н2 [тт2Н2 + 2/ЗтЯ - (тН'т + \Н)
where Аг = -\a~xl2bbl2fi, А2 = |б3.
2] ~\
> In the solutions of equations 82-88, the following notation is used:
j.j. _ j C\Jv{t) for the upper sign (Bessel function),
v ~ \ C\Iv(t) for the lower sign (modified Bessel function),
У _ j C2Yv{t) for the upper sign (Bessel function),
v ~ \ C2Kv(t) for the lower sign (modified Bessel function),
Zv = aiUv
\ZVXV forA = -(a1p2-a2p1J;
Г ZVGV + XVFV forA = -(a1C2-a2C1J,
\ tN' + 2vN for A = 4«7 - (i2,
=\^ for the upper sign,
2 = N2±At2N2+u:2A, co=\
[—1 for the lower sign.
Prime denotes differentiation with respect to т.
© 1995 by CRC Press, Inc.

82. y'lx = AlXy + A2y~3.
Solutions in the parametric form:
x = ат2'3, у = Ътг'3Мг'2,
where v=\, Аг = =pf a, A2 = ^-a64w2A.
83- V'L = А^хПУ + А2У-3, n ф -2.
Solutions in the parametric form:
2\
x = ат2\ у = b^
where v = ——, Ax = Т——a"™, A2 =
n + 2 Av1
, Ax = Тa, A2 = T
84- V'L = Aix~by + А2у~3.
Solutions in the parametric form:
x = ат-21\ у = Ъ
where v=\, Ax = т|a3, A2 = ^-a64w2A.
85. y»m = AlX~3 + Aix-^y-1'2.
Solutions in the parametric form:
x = aT2'3N, у = br
where v=\, A1= -2abu2A, A2 = Tf a/263/2.
86. !&, = Аг + Aix-^y-1^.
Solutions in the parametric form:
where v=\, Аг = -2а6ш2А, А2 = =pf &3/2.
87- y'L = Amy~2 + My3-
Solutions in the parametric form:
x = ut-2/3N-1N2, у = br2/3N,
where v=\, Ax = --^a~zbz, A2 = -^a64w2A.
88- y'L = AlX-2y~2 + А2у~3.
Solutions in the parametric form:
where v=\, Аг = -^b3, A2 = -^
© 1995 by CRC Press, Inc.

> In the solutions of equations 89-90, the following notation is used:
Д = С2-2СЬ R= C6A + 54Bt-2t3I/2, z = 3 /t^R'1 dr,
-ff ifA<0;
? «/A>0;
if A = 0, C2
= 0, C2
W(z) =
tanh(=FvAz) +
1 Г~2~
ClZ
1
\Ci\
2
89. y^ = Aiy-5/3 + A2x-2/3y-5
Solutions in the parametric form:
x =
у =
- 2C2W
- 2GW
90.
where Аг = 24а-268/3Сь А2 = -36а/368/3В.
Solutions in the parametric form:
dW2 - 2GW + 2)/2,
W2 - 2C2W + 2)-3/4
x =
у =
where Ax = -36а-4/368/3В, А2 =
W - 6C2 т RK/2,
- 6C2 т Rf/2,
> In the solutions of equations 91-102, the following notation is used: functions Pi
and P2 are the general solution of four modifications of the first Painleve equation:
P'-l = ±6P2 + т, P2' = ±6P22 - т
(in the case of the upper sign, the equation for Pi is the canonical form of the first
Painleve equation, see Subsection 2.8.2). In addition,
Qi = ±&Pi + т, Q2= ±6P| - т,
Ti = T2Pi т 1, T2= т2Р2 т 1,
Ui = (P{J - 2PiQi ± 8P3, U2 = (P2J - 2P2Q2 ± 8P23,
V = P'O' + P' — O2 V = P'O' — P' — O2
where primes denote differentiation with respect to т.
© 1995 by CRC Press, Inc.

91. у»я = AlV2 + А2х.
Solutions in the parametric form:
x = ат, y = bPk,
where Аг = ±6a~2b-1, A2 = a-3b(-l)k+1; к = 1 and к = 2.
92. y^x = AlX-5y2 + A2x~4.
Solutions in the parametric form:
х = ат~1, у = Ьт~1Рк,
where Аг = ±6a3b~1, A2 = a2b(-l)k+1; к = 1 and к = 2.
93. y»m = AlX
Solutions in the parametric form:
x = ат7, у = ЬтТк,
whereA1 = ±-^a1'7b-1, A2 = -La-6/7&(-l)fe+1; к = 1 and к = 2.
Q Л it — А <т — -«и / 7п|Л | Л ™
*у^* У'г'г — -^-Ч "^ if \^ -Гя-З**-1
Solutions in the parametric form:
x = ат~7, у = Ьт~6Тк,
where A1 = ±-^a6/7b~1, A2 = -^a-1/7b(-l)k+1; к = 1 and к = 2.
Solutions in the parametric form:
x = aPk, у = b(P'kJ,
where Ax = ±24a6, A2 = 2a-2bz'2{-l)k+1; к = 1 and к = 2.
nc ».// Д ~ —4 I л ~ —5/2,.—1/2
Solutions in the parametric form:
ж — aifc > У — ork Угк) >
where Л = ±24a26, A2 = 2a}'2bz'2{-l)k+1; к = 1 and к = 2.
Solutions in the parametric form:
x = aUk, у = ЬРк/2,
where Ax = ^f8ab~2A2, A2 = -^-а~368/3; к = 1 and к = 2.
© 1995 by CRC Press, Inc.

98. y'lx = А1Жу + 2
Solutions in the parametric form:
where Ax = T&ab~2A2, A2 = -^a}'zb&'z; к = 1 and к = 2.
99. у?а = AlX-3/2 + Azy-1/2.
Solutions in the parametric form:
x = a(Ptf, у = bQ2k,
where Аг = \a-1l%{-l)k, A2 = ±6a63/2; к = 1 and к = 2.
100. y»m = AlX~3/2 + A2x-5/2y~1/2.
Solutions in the parametric form:
where Ax = \а-г'2Ь{-1)к, A2 = ±&a}'2bz'2; к = 1 and к = 2.
101. y'lx = А1У-*/3 + A2xy-5/3.
Solutions in the parametric form:
x = aVk, у = b(Ptf,
where Аг = ab-^3A2(-l)k, A2 = ^-a68/3; к = 1 and к = 2.
102. y»m = AlX-^3y~^3 + А2ж-7/зу-5/з
Solutions in the parametric form:
x = aV-\ y = b{P'kfV-\
where Ax = -La-1/367/3(-l)fe, A2 = -La1/3?,8/3; к = 1 and к = 2
> In the solutions of equations 103-108, the following notation is used: functions P\
and P2 are the general solution of four modifications of the second Painleve equation
(with parameter a = 0):
P" = тРг ± 2Pf, P^ = -rP2±2P|.
where primes denote differentiation with respect tor. In the case of the upper sign, the
equation for P\ is the canonical form of the second Painleve equation (with parameter
a = 0, see Subsection 2.8.2).
© 1995 by CRC Press, Inc.

v'L =
Solutions in the parametric form:
х = ат, y = bPk,
where Аг = ±2a~2b-2, A2 = a3(-l)fe+1; к = 1 and к = 2.
104. у?а = AlX-ey3 + A2x~5y.
Solutions in the parametric form:
х = ат~1, у = Ьт~1Рк,
where Аг = ±2aAb~2, A2 = a3(-l)fe+1; к = 1 and к = 2.
105. y»m = Аг + А2х~1/2у-1/2.
Solutions in the parametric form:
x = aP2k, y = b(Pk-f, Р^ = (РкУт,
where Ax = ±2a~2b, A2 = ±а~3/2Ь3/2(-1)к+1; к = 1 and к = 2.
106. y'lx = AlX~3 + A2x~2y-1/2.
Solutions in the parametric form:
x = aP~2, у = ЬР-2{Р'кJ, P'k = {Pk)'T,
where Ax = ±2ab, A2 = ±Ь3/2(-1)к+1; к = 1 and к = 2.
ЮГ. y'lx = Аг + A2xy~2.
Solutions in the parametric form:
х = а[тРк2±Р*-(Р>J}, у = ЬР2, P^ = (Pk)'T,
where Л = T2a-26(-l)fe, A2 = 2a-3b3(-l)k+1; к = 1 and к = 2.
108. y'lx = AlX-3 + A2x~2y-2.
Solutions in the parametric form:
r_nrTp2_|_p4 (-PM21-1 hp2\ p2 _i_ pi (-PM21-1
x — a\rrk it rk — \rk) j , у — ork \тгк ±rt- \rk) j ,
where Ax = ±2ab, A2 = 2b3; k = l and к = 2.
2.5. Generalized Emden—Fowler Equation
V'L = Axny™(y'J
2.5.1. The Classification Table
The case / = 0 corresponding to the classical Emden—Fowler equation is outlined in
Section 2.3. In this section, the case / ф 0 is considered.
Table 2.9 represents all solvable equations of the form y'^x = Axnym(y'x) whose solutions
are outlined in Subsection 2.5.2. The two-parameter families (in the space of parameters
n, то, and /), one-parameter families, and isolated points are represented in a consecutive
fashion. Equations are arranged in accordance with the growth of /, the growth of то (for
identical /), and the growth of n (for identical то and /). The number of the equation sought
is indicated in the last column in this table.
© 1995 by CRC Press, Inc.

TABLE 2.9
Solvable cases of the generalized Emden—Fowler equation yxx = Axnym(y'x)
No
/
m
n
Equation
Two-parameter families
1
2
3
arbitrary
arbitrary
2n + m + 3
n + m + 2
arbitrary
0
arbitrary
(m ^ -1)
0
arbitrary
arbitrary
(n ^ -1)
2.5.2.1
2.5.2.2
2.5.2.3
One-parameter families
4
5
6
7
8
9
10
11
12
13
14
arbitrary
(/^1,2)
arbitrary
3m + 5
2m+ 3
3m + 5
2m+ 3
3n + 4
2n + 3
3n + 4
2n + 3
3n + 4
2n + 3
1
2
3
3
-1
l
2
arbitrary
(m^-|)
arbitrary
(m^-|)
l
2
1
n 3
arbitrary
(m^-1,0)
-1
arbitrary
(m ф -2)
-n-3
-1
l
2
1
2
1
arbitrary
(n^-i)
arbitrary
(n^-f)
arbitrary
(n^-i)
-i
arbitrary
(n^-1,0)
1
arbitrary
2.5.2.6
2.5.2.97
2.5.2.13
2.5.2.10
2.5.2.11
2.5.2.12
2.5.2.107
2.5.2.5
2.5.2.4
2.5.2.96
2.5.2.9
Isolated points
15
16
17
18
19
20
21
l
2
1
2
1
2
1
2
1
2
2
?
4
5
l
2
1
1
1
1
1
2
5
2
5
2
15
8
20
13
5
4
0
7
6
1
2
2.5.2.33
2.5.2.83
2.5.2.86
2.5.2.80
2.5.2.78
2.5.2.53
2.5.2.76
© 1995 by CRC Press, Inc.

TABLE 2.9 Continued
Solvable cases of the generalized Emden—Fowler equation y'? = Axnym(y'xI
No
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
/
1
1
8
7
8
7
6
5
5
4
5
4
9
7
9
7
13
10
27
20
18
13
7
5
7
5
7
5
7
5
7
5
7
5
7
5
10
7
22
15
3
2
3
2
3
2
3
2
3
2
3
2
3
2
23
15
11
7
m
-2
-1
1
1
l
2
1
1
13
8
1
2
1
2
1
2
1
2
7
4
10
7
2
3
1
2
1
1
5
l
2
1
2
-2
-2
l
2
1
2
1
2
1
1
2
5
2
n
1
-1
3
4
1
2
2
1
2
0
1
1
5
2
2
7
2
1
1
1
1
0
1
1
5
2
2
3
1
2
1
-2
l
2
1
-2
l
2
1
2
1
2
Equation
2.5.2.14
2.5.2.8
2.5.2.54
2.5.2.52
2.5.2.68
2.5.2.58
2.5.2.56
2.5.2.39
2.5.2.38
2.5.2.47
2.5.2.72
2.5.2.40
2.5.2.18
2.5.2.46
2.5.2.32
2.5.2.17
2.5.2.89
2.5.2.91
2.5.2.75
2.5.2.19
2.5.2.70
2.5.2.106
2.5.2.99
2.5.2.100
2.5.2.29
2.5.2.98
2.5.2.105
2.5.2.103
2.5.2.84
2.5.2.27
© 1995 by CRC Press, Inc.

TABLE 2.9 Continued
Solvable cases of the generalized Emden—Fowler equation y'? = Axnym(y'xI
No
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
/
8
5
8
5
8
5
8
5
8
5
8
5
8
5
21
13
33
20
17
10
12
7
12
7
7
4
7
4
9
6
13
7
13
7
2
2
li
5
7
3
5
2
5
2
5
2
5
2
5
2
3
3
3
3
m
0
1
1
1
1
1
1
7
2
2
3
5
2
1
1
1
2
0
2
3
3
4
1
2
-1
1
1
2
7
6
5
2
15
8
20
13
5
4
0
-5
7
2
10
3
20
7
n
1
7
4
10
7
2
1
2
1
5
l
2
1
2
1
2
13
8
1
2
1
1
1
2
1
1
-1
-2
5
2
1
2
1
2
1
1
1
1
2
l
2
5
3
2
Equation
2.5.2.73
2.5.2.26
2.5.2.48
2.5.2.35
2.5.2.24
2.5.2.74
2.5.2.94
2.5.2.45
2.5.2.87
2.5.2.49
2.5.2.44
2.5.2.42
2.5.2.51
2.5.2.50
2.5.2.81
2.5.2.65
2.5.2.61
2.5.2.7
2.5.2.16
2.5.2.95
2.5.2.64
2.5.2.36
2.5.2.69
2.5.2.71
2.5.2.67
2.5.2.66
2.5.2.79
2.5.2.31
2.5.2.37
2.5.2.85
© 1995 by CRC Press, Inc.

TABLE 2.9 Continued
Solvable cases of the generalized Emden—Fowler equation y'? = Axnym(y'xI
No
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
/
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
m
5
2
13
5
7
3
15
7
-2
-2
-2
-2
4
3
7
6
5
6
1
2
1
2
0
0
0
0
1
1
1
1
1
1
1
2
3
n
l
2
7
5
5
3
2
-2
-1
l
2
1
1
2
1
2
5
3
5
2
5
3
-4
5
2
1
2
2
-7
-4
-2
5
3
7
5
1
2
0
5
3
-7
Equation
2.5.2.22
2.5.2.43
2.5.2.25
2.5.2.82
2.5.2.104
2.5.2.15
2.5.2.101
2.5.2.28
2.5.2.59
2.5.2.60
2.5.2.93
2.5.2.90
2.5.2.92
2.5.2.55
2.5.2.88
2.5.2.20
2.5.2.77
2.5.2.62
2.5.2.57
2.5.2.102
2.5.2.23
2.5.2.41
2.5.2.30
2.5.2.21
2.5.2.34
2.5.2.63
© 1995 by CRC Press, Inc.

2.5.2. Exact Solutions
1- V'L = Aym(y'J.
1°. Solution in the parametric form with m ф —1, l ф 2:
f i
ж = aG, / A±t )' ат + C2,
у =
where A =
2°. Solution in the parametric form with m = —1, l ф 2:
/i
т i~2 exp(=FT2) dr + C2, y = ЬСХ exp(=Fr2),
x =
3°. Solution in the parametric form with m ф —1, / = 2:
1 — m
x = aC\ I т !+m exp(=FT ) dr + C2, у = Ът m+1 ,
where A = ±(m + l)^-1-™.
4°. Solution with m = -1, / = 2:
(ClX + C2) i-A ИАф
C2 exp(Cirr) UA =
2. y'lx -
1°. Solution in the parametric form with n ф —1, / ф 1:
x = аС\-1т, у = bCl+n~l f A ± Tn+1I^r dT + c2,
where A = ± al~n~2b1~l.
2°. Solution in the parametric form with n = —1, / ф 1:
/3-1
т i-* exp(=FT2) <ir + C2,
b \3-i
3°. Solution in the parametric form with n ф —1, / = 1:
2 /• 1-та
x = ат n+1 , у = ЬС\ I т 1+п exp(=Fr2) dr + С2,
where А = т(п + 1)Ь1-п.
4°. Solution with n = —1, / = 1:
' ClXA+1 + C2 ИАф1,
У ' d In \x +C2 ifA = 1.
© 1995 by CRC Press, Inc.

2тг+тгг+3
3. y'lx = АхПут{у'х) гь+т+2 _
Solution in the parametric form with n ф — 1, то ф —1:
Г Г dT ] Г n +1 /• dr n + 1^1
rr = exp / ——-+C2 \, у = техр —- / —— ~TCA^
[J /(т) J [ m + 1 J /(t) m +1 J
where function / = /(т) is defined impicitly by the formula
[/ + (l)](/ + )i C +
u ' y" rsyj ' w>' "^ ' п + то + 2' "- m+1-
See equation 2.5.2.5 for the case n = —1. See equation 2.5.2.4 for the case m = —1.
4. !&, »^)
Solution in the parametric form with n Ф —1,
i Г Р i iin 1 Л
1 / 1 — 71 / yi 7l-\-1 1 \ — 1
ж = ат п у = ± exp / т « ( —r « + nr « + Ci 1 dr + C2 ,
U Vn+1 У J
where ^4 = —a~n.
See equation 2.5.2.7 for the case n = —1. See equation 2.5.2.1 for the case n = 0.
5. J&, »»;
Solution in the parametric form with m ф —1, то т^ 0:
[ f 1~m ( то m+i j_ \-i 1 J_
X = ± exp / T m I т m + TOT m + Ci 1 dr + Сг , У = (Ат) "» .
[j v то +1 / j
See equation 2.5.2.8 for the case то = —1. See equation 2.5.2.2 for the case то = 0.
6. y'L ^\J
Solution in the parametric form with / ф 1, / ф 2:
where the function / = /(т) is defined implicitly by the formula
т А j
See equation 2.5.2.7 for the case / = 2. See equation 2.5.2.2 for the case / = 1.
7. y'lx ^\f
Solution in the parametric form:
x = ±ет, у = С2(тАт + eT + d) exp \±A I'(тАт + eT + d)'1 dr~\.
© 1995 by CRC Press, Inc.

8. у%а yyx
Solution in the parametric form:
\ r 1
x = C2(±At + eT + Ci)exp ^A I (±At + eT + Ci) dr\, y = ±eT.
L *f J
9- V'L y(yxf
Solution in the parametric form with n ф —1:
i — 1 г p ~\ — 1
where A =
2
See equation 2.5.2.15 for the case n = —1.
Solution in the parametric form with m ф —3/2:
x = аСГа{A ± r^f2 [J A ± т»+Г1/2 dr + Ca] - r},
у = 6c^+1)(^-2) / A ±r^+1)-1/2 dr + C2\
l
2m + 3 Lib >x+2 ,
where и = , A = —; — ± -
И m+1 ' (u + 2)r '
11. yf:ai = y (»i)
Solution in the parametric form with n 7^ —3/2:
^ п+1 а(// + 1) V Ь
Зтг+4
12. у'1х
Solution in the parametric form with n ф —3/2:
\
where, = -
© 1995 by CRC Press, Inc.

1 Зтгг+5
v'L = Ах~^утЮ2т+3 ¦
Solution in the parametric form with m ф —3/2:
x = aCt{(l ± r^f2 У A ± r^1)-172 dr + Ca] - г}',
у = 6Cf+1) V+1 \f A ± r^+1)-1/2 dr + C2] "M,
m + 1 b(// + 1)
Solution in the parametric form:
x =
[ f ехр(тт2) dr + C2] ± ехр(тт2)|, y = bC1\f ехр(тт2) dr + C2],
where A = ^\a~2b
15. V'ix = y(yxf
Solution in the parametric form:
If \ If
x = a ехр(тт2) / ехр(тт2) dr + C2\ , У = СЛ1 ехр(тт2) dr + C2
where A = ±2a2.
16.
Solution in the parametric form:
x = aC1\jeMTT2)dT + C2\ у = bd f.2r\ /*ехр(тт2) dr + C2~\ ±ехР(тт2)|,
where A = ±a2b~2.
17. y'ix =
Solution in the parametric form:
x = ±аС1{т2 - 1)(т3 - 3t + C2)/2, у = ЪС\*{т* - 6т2 + 4С2т - ЗJ,
±л
2/5
166
18. y'ix =
Solution in the parametric form:
x = ±аС27(т3 - Зт + С2)~1/2(т6 - 15т4 + 20С2т3 - 45т2 + 12С2т + 27 - 8С|),
а \2/5
© 1995 by CRC Press, Inc.

19.
Solution in the parametric form:
x = aC'^T3 - Зт + Cy'V4 - 6t2 + 4С2т - 3J/3,
у = ЬС27(т3 - Зт + С2)~1(т6 - 15т4 + 20С2т3 - 45т2 + 12С2т + 27 - 8С|J,
where A = 28a(abI/2/ п Л
20. I&, =
Solution in the parametric form:
ж = аС4(т2 - lJ,
where A = ±|а3/26-2.
21. y':x =
Solution in the parametric form:
ж = аС3(т3-Зт + С2), у = ЬСгт, where A = -6ab~3.
22. y':x = Ах-
Solution in the parametric form:
x = аСг{т2 - 1J(т3 - 3t + C2)~\ у = ЬС~3(т3 - Зт + C2)~\
where A = T-f-
23. y^
Solution in the parametric form:
x = aCf(T3 - Зт + C2K/2, у = ±6Cf(T4 - 6т2 + 4С2т - 3),
where A = т^
24. y'lx
Solution in the parametric form:
x = аС16(т* - 6т2 + 4С2т - ЗJ, у = ±6Ci(t2 - 1)(т3 - Зт + C2)/2,
25. у'1х = Ах~5/
Solution in the parametric form:
ж = ±аС1(т3-Зт+С2K/2(т4-6т2+4С2т-3)-1, у = ±6С'-8(т4-6т2+4С2т-3)-1)
where A = ^-^-с
© 1995 by CRC Press, Inc.

26. y'^
Solution in the parametric form:
X =
у = ±ЬС27(т3 - Зт + С2) 1/2(т6 - 15т4 + 20С2т3 - 45т2 + 12С2т + 27 - 8С|),
27. y'ix
Solution in the parametric form:
x = aCf(r3 - Зт + C2)~1(t6 - 15t4 + 20С2т3 - 45т2 + 12С2т + 27
у = ЪС-\т* - Зт + Cy'V " 6т2 + 4С2т - 3J/3,
where А = -2&Ъ(аЪI/2^f'
28. у'1х -
1°. Solution in the parametric form with A < -j-:
x = t(Citv + Cit~v), у = т2, where v = y/l - A A.
2°. Solution in the parametric form with A = ^-:
ж = т(С11п|т| + С2)) у = т2.
3°. Solution in the parametric form with A > -j-:
x = tC\ sin(jvlnT + C2), у = т2, where jv = y/AA — 1.
29. y^
Solution in the parametric form:
x = ±t2{C1tv + C2t~v)\ у = \т-\1 + v)ClTv + A - u)C2t-v],
where A = т/с2, v = /c(/c4 + AI/2.
30. y'L
Solution in the parametric form:
x = aCl ехр(-2т) [2ехрCт) - C2 sin(v/3T) + уДС2 cos(v/3t)]2,
у = ЬСХ ехр(-т) [ехр(Зт) + С2 вш
where A = -16а3/26.
© 1995 by CRC Press, Inc.

31. y':x
Solution in the parametric form:
aC\ exp(—т) [2ехрCт) — C2 вш(\/3т) + л/3 С2 cos
x =
У =
where A = -l&(abf'2.
32. v" -
1°. Solution in the parametric form with A < 0:
x = aCi[cosh(r + C2) cosT]1/2[tanh(Y + C2) - tanr],
у = bCf cosh3(r + C2) cos3 т [tanh(r + C2) + tanr]3,
v2/5
2°. Solution in the parametric form with A > 0:
ж = aCi[coshr — sin(r + C2)]~1^2[sinhT — cos(t + C2)],
у = 6Cf [sinhr + cos(t + C2)]3,
33. y^ = Ах-*/2у-1/2(ухI/2.
Solution in the parametric form:
x = aC^1 [cosh(r + C2)cost]~1,
у = bCi cosh(r + C2) cost [tanh(r + C2) - tanr]2,
where A = —Aab.
34. y'lx = Ax-^y\y'xf.
1°. Solution in the parametric form with A > 0:
x = aCl [cosh(r + C2) cost]3/2,
у = ЬС\ cosh(r + C2) cos т [tanh(r + C2) + tan r],
where A = ^
2°. Solution in the parametric form with A < 0:
ж = aC3[coshт - sin(r + C2)]3/2,y = 6C2[sinhr + cos(t + C2)],
where A = -fa8/36.
© 1995 by CRC Press, Inc.

35. у»я
1°. Solution in the parametric form with A > 0:
x = aC\ cosh3(r + C2) cos3 r[tanh(r + C2) + tanr]3,
у = 6Ci[cosh(r + C2) cosr]1/2[tanh(T + C2) - tanr],
12a
2°. Solution in the parametric form with A < 0:
ж = aCf [sinhr + cos(t + C2)]3,
у = 6Ci[coshr - sin(Y + C2)]/2[sinhr - cos(t + C2)],
5a6f)
V 6a /
36. y^ WW
Solution in the parametric form:
x = aC\ cosh(r + C2) cost [tanh(r + C2) — tanr]2,
у = ЪС'1 [cosh(T + C2) cos t}-\
where A = Aab.
37. J&, =
1°. Solution in the parametric form with A > 0:
ж = aCi[cosh(r + C2)cosT]1/2[tanh(r + C2) + tanr],
у = 6Cj[cosh(T + C2) cosrj'^tanh^ + C2) + tanr],
where A = ^-a8/364/3.
2°. Solution in the parametric form with A < 0:
ж = aCi[coshr - sin(r + C2)]3/2[sinhr + cos(t + Сг)],
У = 6C1[sinhr + cos(t + C2)]~\
where A=-f8/34/3
> In the solutions of equations 38-45, the following notation is used:
.Е = ехрCт), Si = E + C2sin(V^T), S2 = 2E - C2sin(V^r) + V3C2 cos(v/3t),
q OC/'CV (Q V Q Q Q Q ОС /С V Kf Q \^ Q _i_ С С
O3 — ^O]^ IO2J-7- I *J 1 /-7- *^2 — *^1 *^2; *^ 4 — *^l\*^3/-r — 1*^1) т 3 "г *^1 *^3 ¦
38. y'lx = Аху-У\у'х)9П'¦
Solution in the parametric form:
x = aC1E~1/6S~1/2S2, у = bCf.
where A = 7a~2b1'2
646
© 1995 by CRC Press, Inc.

39. y'^ = Axy-™/*(y'x)9/r.
Solution in the parametric form:
x = aClbEb/&S1/2SA, у =
10. й'„
Solution in the parametric form:
x = aC^E^S^Sl'*, у =
where A =-208a5/V/2(^
41. y'L =
Solution in the parametric form:
x = aClE-^Sl12, у =
where A=-T^Ia1
42. y'lx
Solution in the parametric form:
x = aCfE-^S2, у = bC1E~1/6S;1/2S2,
43. y'ix =
Solution in the parametric form:
x = aCiE-^S^S-1, у
where A=-T^Ia12/5b3/5.
44. y»m = Ax-13/sy(y'xI2/r.
Solution in the parametric form:
x = aC32E-16^sl/5, у = b
256 \2/7
*» •<= -^'"ЬЛШ
45. y'ix = Ах-^У
Solution in the parametric form:
x = aC^E-^S-'S2, у =
© 1995 by CRC Press, Inc.

> In the solutions of equations 46-49, the following notation is used:
Ti = cosh(r + C2) cosт, Q\ = coshт — sin(r + C2),
T2 = tanh(r + C2) + tan r, 02 = sinh т + cos(t + C2),
T3 = tanh(r + C2) — tanr, #3 = sinhr — cos(t + C2),
T4 = 3T2T3 - 4; 04 = 3(92<93 - 20?.
46. !&,
1°. Solution in the parametric form with A < 0:
Jb — UL/1 J. -I -Z. 4 5 V — IA-m J. -I J. л , W licit;
2°. Solution in the parametric form with A > 0:
я = аС?0Г1/204, У = bCf?27/3, where
146 )
47. y" =
Solution in the parametric form:
= аС-1Т~1/3Т*/3, y = bC\T*Tl where A = -20а(а6I/2(^)
а \3/10
48. y'lx -
1°. Solution in the parametric form with A > 0:
96
x = aCi Lx ±2 i y = 6G1i1 14, where A=—a
2°. Solution in the parametric form with A < 0:
/-<14/i7/3 i./-<9/i—1/2/) 1 л 5
ж = аСг v2 > y = bL>\v1 ' #4, where Л= а
28a )
49. y^, = Aa;~1/Jy
Solution in the parametric form:
х = аСЩт1 у = ЬС-1Т~1/3Т*/3, where A = 2Qb{abIl2(^f'W.
> In the solutions of equations 50-65, the following notation is used:
/1
——— is the incomplete elliptic integral of the second kind in the
К
Weierstrass form.
© 1995 by CRC Press, Inc.

50. у»я = Ax(yx)r/4.
Solution in the parametric form:
= aC~3R, y = bC51T-1F1, where A = T|a-2(T-^
51. y'^ =
Solution in the parametric form:
= aC~1F2, y = bClT2F-\ where A =
52. V'ix =
Solution in the parametric form:
x = aC-^Fl y = bClF-3'2F2, where A = TXa-
53. fa = Ах-^у-^(у'хJ/3-
Solution in the parametric form:
x = aClF-zFl y = bC1F-3(F2F3-8F2J, where A =
54. y'L = Ax-V*y{yx)8/7.
Solution in the parametric form:
x = aC~32F-\ у = bC3F^3/2(F2F3 - 8
i/7
55. y'L = \yxf
Solution in the parametric form:
х = аС2т~1, y = bClr~1F1, where A = ±6a5b~2.
56. y'L = Ay(y'xM/4.
Solution in the parametric form:
x = aCblT-1F1, y = bC~3R, where A = ±^&(т —
57. y'L = Ax-*y(y'xf.
Solution in the parametric form:
x = aCfF'1, у = bClrF'1, where A = ±6a5b~3.
© 1995 by CRC Press, Inc.

58. у»я
Solution in the parametric form:
K/*
x = clCIt2F-2, y = bC~1F2, where A = ±W'2b-2(±—)
59. y?m
Solution in the parametric form:
= bCfF?, where A = ^
60. y?m
Solution in the parametric form:
= bCfF-3, where A = т~ a3/2b~5/6.
61. fa
Solution in the parametric form:
= aC*F-3/2F2, y = bC-™Fi, where A = ±2-а-1Ь-1'2
62. yr:x = Ax
Solution in the parametric form:
= aC3Fl/2, y = bCfF3, where A = Т—-a86.
63. y?m = Ax' V(»i)S-
Solution in the parametric form:
= aC\Fl/2F-1, y = bCfF-\ where A = T-^as
64. jC = Ax-i/2y
Solution in the parametric form:
x = aCiF-3(F2F3 - &F?J, y = bC^F-3F35, where A = ±4a3/V5/6(—)
65.
Solution in the parametric form:
-,3 77.-3/2, р р от?2\ л,_г,/-г-32т7.-4 ¦ ¦ ? _i. _i/и / 32Ь
x = aC3F~3/2(F2F3-8F2), y = bC~32F^\ where A = ±^-
3a )
© 1995 by CRC Press, Inc.

> In the solutions of equations 66-95, the following notation is used:
f =
_ i).
Function p = р(т) is defined implicitly. The upper sign in the formulae corresponds
to the classical elliptic Weierstrass function p= р{т + С2, О, 1). The solutions given
below are written in the parametric form. One can assume as the parameter either т,
hence p = р(т), or p, hence т = т(р).
Solution in the parametric form:
x = aC~3f, y = bClT, where A
1/2
abJ
67.
Solution in the parametric form:
= aC-\Tf-p), у = ЬС\т\ where A = -I
V2
68 xiff ^ Ax—^'**ij
Solution in the parametric form:
x = aC9lT-3p3, у = ЪС$(т/ - pJ, where
1/5
46 J
69. y'ix =
Solution in the parametric form:
= ЪС$т-\ where A = i-
70.
Solution in the parametric form:
x =
if, У = bCfT-12(T3f + Зт2
if,
46 /
71. y'ix =
Solution in the parametric form:
x = aC1T(r3f-Ar2p±6), у = ЬС\3т13, where A = T Aa-i
© 1995 by CRC Press, Inc.

72. V'ix
Solution in the parametric form:
x = aC-9T-ls(T2pTlf, У = bClT2(T3f - \т2р ± 6J,
, . 20 _i/3,i/2/. a \V
where A =—a i/6bi/2[±—)
3 V 4b J
73. y'^ = Ax(y'x)s/5.
Solution in the parametric form:
x = aC~3f, y = bClp-2(f±2rp2), where A = T|a(^
74. y'L =
Solution in the parametric form:
x = aC-\rf + 2p), у = bC7Mf ±
where A = -—a b (——)
6 V о J
75. y'L = Axyb{y'JI\
Solution in the parametric form:
x = aC-27(r2p т 1)(/ ± 2тр2)-1/2, у = bCt{rf
76. y'ix
Solution in the parametric form:
x = аС27(т2р т 1J(/ ± 2TP2), у = bCl{f ±
4)
2b I
Solution in the parametric form:
x = aC2p, у = ЬС^т, where A = ^ба^.
78- V'L = Ay(y'xI/2.
Solution in the parametric form:
о 2 / 6 \l/2
~3f, where A = ±—(±—)
© 1995 by CRC Press, Inc.
о 2 /
= aClT, y = bC~3f, where A = ±—(±)
a V ab J

79. V'ix =
Solution in the parametric form:
х = аС3т~1р, y = bC1T~1, where А
80. y'L = Ax-^y(y'xI/2.
Solution in the parametric form:
x = aC\r\ y = bC-1(Tf-p), where A= ^-
81. у" = Ах~1/2у
Solution in the parametric form:
x = aC\(jj - pf, у = bCfT-3p3, where
82. y?m = Ах2у
Solution in the parametric form:
x = clCit{t2p T 1), У = bClr7, where A = =p—с
83. y'lx =
Solution in the parametric form:
111 ' 8 V a
Solution in the parametric form:
X — <M^iT \T J -f- 6T p -\- I) , у — OO-l T [T p+L) ,
, / b \7/15
w ere - (^ — J
Solution in the parametric form:
x = aCfr (т p T 1)) V = ЬС\т . where A = =p—a
49
86. y'ix =
Solution in the parametric form:
х = аС±3т13, у = ЬС1т(т3/-4т2р±6)) where A = ±4
13
© 1995 by CRC Press, Inc.

87. у" = Аж-1/2у-2/3(у'K3/20.
Solution in the parametric form:
x = aClT2{r3f - Ат2р t бJ, у = bC-\-l8(r2p T if,
88. y»m =
Solution in the parametric form:
x = aCfp~2, y = bClp-2(f±2rp2), where A = ±3a7/V2.
89. y'L = Ay(y'xO/5.
Solution in the parametric form:
x = aClp-2(f±2Tp2), y = bC~3f, where A = ±A&-2(—
6 V a
90. yl = Ax-^y
Solution in the parametric form:
ж = аС13(/±2тр2), у = ЪС71р2(/±2тр2)~\ where A = ±3a7/V3/2.
91. y'J,x =
Solution in the parametric form:
x = aClp{f±2Tp2)~1/2, y = bC-\rf + 2p), where A = -—a^b^f—Y^.
о V (X J
92. J&,
Solution in the parametric form:
x = aCf(f±2Tp2f/2, y = bCl6(Tf + 2pJ, where A= \a
93. y'J,x =
Solution in the parametric form:
x = aCl(f ± 2rp2fl2(TJ + 2p)~2, у = bC\\rf + 2p)~2,
where A = -g-a8/36~7/6.
94. у»я = Ахьу{у'х)8/Ь.
Solution in the parametric form:
x = aC*{rf + 2РГ1'3, у = bC-27(r2p T 1)(/ ± 2rp2y1/2,
where A = 10a6 (—
95. у^ = А*-*/2»
Solution in the parametric form:
x = aCl(f ± 2тр2у\т/ + 2pf3, у = ЬС27{т2р т lf(f ± 2тр2
~\
© 1995 by CRC Press, Inc.

> In the solutions of equations 96-97, the following notation is used:
C\Jv{t) + C2Yv{t) for the upper sign,
Z =
C\Iv(t) + C2Kv{t) for the lower sign,
where Jv and Yv are Bessel functions, Iv and Kv are modified Bessel functions.
96. y'L = Axym(y'xf.
Solution in the parametric form with m ф —2:
x = tvZ, y = br2v, where v = -, A = ±( x
m + 2' V 26
See 2.5.2.28 for the case m = -2.
97. jC
Solution in the parametric form with / ф 2>/2:
x = aT2vZ2, у = bT-2v{TZ'T + vZJ,
where v = ^ ~l , A = X (т—) 2 • See 2.5.2.29 for the case / = 3/2.
tj — ЛЬ tj — ЛЬ \ (X J
> In the solutions of equations 98-106, the following notation is used:
Ji/3{t) + C2Yi/3(t) for the upper sign,
Z =
CA() + C'2K1/3(t) for the lower sign,
where J1/3 and У1/3 are Bessel functions, /1/3 andKijz are modified Bessel functions;
f/i = tZ't + \Z, U2 = Ul ± t2Z2, U3 = ±^t2Z3 - 2f/if/2.
98. y'lx
Solution in the parametric form:
= aT~2/3Z-1U1, y = bT-^U2, where A = --(T-
a V a
99. y':x
Solution in the parametric form:
¦ = ат-^ъг-хиъ, у = Ът-2'3и2, where A = -Аг
az
100. y'lx
Solution in the parametric form:
x = aT-^3Z-2U2, y = bT-&l3Z-2Ul where A = ±±а3/2.
© 1995 by CRC Press, Inc.

101. y'L = y\yxf
Solution in the parametric form:
x = ar-A/3Z-2Ul y = bT~2/3Z-2, where A = ±|a3/2.
102. у»я =
Solution in the parametric form:
9 /a
= clt2'3Z2, у = Ьт~2/Зи2, where A = - (±) .
2 V b I
103. y'lx = Ax-i^yiy'J3'2.
Solution in the parametric form:
1/2
х = ат-А'3и2, y = br-2l3Z-1U1, where A=±-U^
b V b
104. y'lx = y\yxf
Solution in the parametric form:
-\ y = bT2'3Z-1U-\ where A = fa3.
105. yxx = Ax
Solution in the parametric form:
x = aT-2'3U2, y = bT~^3Z-1Uz, where A = A(±3a&I/2.
106 ii — д np / ft (ii i
Solution in the parametric form:
x = aT~s'3Z-2Ul y = bT-^3Z~2U2, where A = т\Ь3'2.
Зтг+4
107. y'lx = Ахпу-П-З(у'х) 2«+з .
In the books by Zaitsev & Polyanin A993, 1994), it was shown that this equation is
reducible to the Riccati equation whose solution is expressed in terms of associated
Legendre functions.
2.5.3. Some Formulae and Transformations
For the sake of visualization, we use the symbolic notation
{n, m, /}
to denote the generalized Emden—Fowler equation
Vxx = Axnym{y'J-
© 1995 by CRC Press, Inc.

Hereinafter we omit the insignificant parameter A (which can be reduced to ±1 by scaling
the variables in accordance with the rule x —>• ax, у —>• by, selecting appropriate constants
a and b).
1. With m + / ф 1, the generalized Emden—Fowler equation has a particular solution:
п + 2-г \ m+;_i Г n + m + 1 ]
where В = —
Vl-m-7 L^(l-m-)J
у = Bx i-m-i
2. Assuming у as the independent variable and x as the dependent one, we obtain the
generalized Emden—Fowler equation for function x = x(y) with the parameters changed:
<„ = -Aymxn(x'yf-1.
Denote this transformation as J- and represent it as follows:
{n, m, 1} <— — — — —>• {m, n, 3 — /} transformation T.
Twofold transformation J- yields the original equation.
3. With m ф 0, n ф — 1, and l ф1, the transformation
leads to the generalized Emden—Fowler equation for function w = w(t) with the parameters
changed:
2m+l
w"t = Bt i-* u; «+1 (w't) m ,
l
m Г A(l — /) 1 m
where В = — — . Denote this transformation as Q and represent it as
n+1 [ n+1 J P
follows:
r ;i Г Х n 2m+ 1 -I
¦{n, m, U i > < -, —, > transformation y.
LI—/ n+1 m )
Threefold transformation Q yields the original equation.
When obtained the solution of the transformed equation in the form w = w(t), the
solution of the original equation can be written in the parametric form as
x = w n+1 , у = k(w't)
l
Г П+1 1
where k = [ {
Different compositions of transformations J- and Q generate six different generalized
Emden—Fowler equations whose parameters are shown in Figure 1.
4. In the particular case / = 0, the transformation у = w/t, x = 1/t leads to the Emden—
Fowler equation with the independent variable to a different power:
w't't =
© 1995 by CRC Press, Inc.

го
го + 1
{го, п, 3 — /}.
{п, го, /}
FIGURE 1
Denote this transformation as Q and represent it as follows:
{n, m, 0} < > {—n — m — 3, m, 0} transformation Ti.
With / = 0, different compositions of transformations JF, Q, and 7i generate twelve
different generalized Emden—Fowler equations whose parameters are shown in Figure 2.
With / = 0 and n = 1, different compositions of transformations J-, Q, and 7i generate
twenty four different generalized Emden—Fowler equations whose parameters are presented
in Figure 3.
5. The substitution
= -y'x,
У
reduces the generalized Emden—Fowler equation to the equation
(zlv - z2 + z)v'z = [(m + l-l)z + n-l + 2}v.
Furthermore, using the substitution
we obtain the Abel equation
?? = [(m + 2/ - 3)z + n- 2/ + 3]2"гС +[(m + l- l)z2 + (n-m-2l + 3)z-n + l- 2}z1~21.
2.6. Equations of the Form
y'?x = AlXniymi(yfxI1 + А2хп*ут*(у'хУ2
2.6.1. Modified Emden—Fowler Equation y1' = A\x~xy' + A2Xnym
See Section 2.3 for the case A\ = 0.
For the sake of clearness, below in this subsection we use the convetional notation
хУхх — ky'x = Axn+1ym for the modified Emden—Fowler equation.
The classification Table 2.10 represents all solvable equations whose solutions are out-
outlined in Subsection 2.6.1. Equations are arranged in accordance with the growth of pa-
parameter m. The number of the equation sought is indicated in the last column in this
table.
© 1995 by CRC Press, Inc.

n 2ro+ 1
1 n-l
I !L_
I n+l '
{n, m, 0}
{—ro — n — 3, ro, 0}
{m, —m — n — 3, 3}\
m + n + 2
m 2m + 2n + 5
2 ro+1
1, -
ro+n+3 2ro+l
m -\- n -\- 2 m
{-
ro + 1 ' Y' ro + n + 3
j}
FIGURE 2
1.
n
2
Solution in the parametric form:
Г f
x = aC\~m \ I A ± T"»+1)-1/2 ^T _)_ с1,
where A = ±4-(m + l)(n + 2Ja-™-2b1-m.
n+2
2.
-1, пф-1.
Solution in the parametric form:
x = aC\-m I"/" A ± rm+1)/2 dr + C2j
m+l
n+2
© 1995 by CRC Press, Inc.

га + 4 га-
гага -\- 3 m
4
1 m + 5
m + 1 ' 2 ' m + 4
} \ {-
Г 1 m 2m + 7 
I 2 ' m + 1 ' m + 4 J
¦j m, —m — 4, 3 J-
{-ro-4, m, o}-^
m + 4 2m+ 1
m + 3
H
Г m + 7 5m + 7~l
I ' 4 ' 3m+ 5 J
J" m + 7 4(m + 2) \
I 4 ' ' 3m+ 5 J
m + 3 m + 3
•¦}
{-7-
I m + 1 2 J
Г 1 5m+ 7 
I ' 2 ' 3m+ 5 J
П
Г m + 7 3m+ 5 ~l
I m + 3 ' m + 3 ' J
m + 3
1
J
3m+ 5
FIGURE 3
where A = ±
2(m + 1)
2n + m + 3 _ +1_-m
У xx ' i Ух — -^*-Л
m — 1
Solution in the parametric form:
1 — m ,
V 1 ГП Г ~ L1
x = exp
n + 2
+
4 m + 1
—*¦
-1/2
-1/2
where A =
4(n + 2J_
(m - IJ
4- жу! - —У; = Axn^y-\ n ф -2.
Solution in the parametric form:
Г f
x = aCl I ехр(тт2) dr + C2
where A = т\{п + 2Ja-n~2b2.
© 1995 by CRC Press, Inc.
n+2

TABLE 2.10
Solvable cases of the modified Emden—Fowler equations
*y'L ~ ky' = Ахп+гут
No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
m
arbitrary
(m ф -1)
arbitrary
(m ф -1)
arbitrary
(m ф -1)
arbitrary
(m ^ -1)
-7
-7
-4
-4
-4
-2
-2
5
2
5
2
5
2
5
3
5
3
5
3
5
3
5
3
5
3
n
arbitrary
(n ф -2)
arbitrary
(n ф -2)
arbitrary
(n ф -2)
-2
arbitrary
(n ф -2)
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
-2
arbitrary
(n ^ -2)
-2
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
-2
arbitrary
(n ^ -2)
arbitrary
(n ф -2)
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
/c
in
n + m + 3
m + 1
2n + m + 3
1 — m
-1
l(n-l)
l(n-3)
|n
l(n-l)
-1
l(n-l)
arbitrary
(/с ф -1)
In
|Bn+l)
-1
-3n-7
\n
iCn + 4)
l(n-l)
|Bn+l)
ТС")
Equation
2.6.1.1
2.6.1.2
2.6.1.3
2.6.1.6
2.6.1.45
2.6.1.46
2.6.1.40
2.6.1.42
2.6.1.41
2.6.1.28
2.6.1.29
2.6.1.35
2.6.1.37
2.6.1.36
2.6.1.14
2.6.1.8
2.6.1.9
2.6.1.13
2.6.1.38
2.6.1.18
© 1995 by CRC Press, Inc.

TABLE 2.10 Continued
Solvable cases of the modified Emden—Fowler equation
XVxx КУх — ЛХ У
No
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
m
5
3
5
3
5
3
7
5
7
5
-1
-1
-1
-1
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
n
arbitrary
(n ^ -2)
arbitrary
(n ф -2)
-2
arbitrary
(n ф -2)
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
-2
-2
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
arbitrary
(n ф -2)
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
-2
-2
arbitrary
(n ^ -2)
arbitrary
(n ^ -2)
-2
к
—LCn + 10)
|Fn + 5)
-1
l(n-l)
—i-En + 13)
n + 1
In
arbitrary
(fc ф -1)
-1
-2n-5
in
iCn + 4)
l(n-i)
|Bn+l)
-|Bn + 7)
|Fn + 7)
arbitrary
(к ф -1)
-1
\n
-|Bn + 7)
-1
Equation
2.6.1.19
2.6.1.39
2.6.1.22
2.6.1.24
2.6.1.25
2.6.1.5
2.6.1.4
2.6.1.7
2.6.1.20
2.6.1.12
2.6.1.11
2.6.1.43
2.6.1.16
2.6.1.26
2.6.1.17
2.6.1.44
2.6.1.27
2.6.1.21
2.6.1.15
2.6.1.10
2.6.1.23
© 1995 by CRC Press, Inc.

TABLE 2.10 Continued
Solvable cases of the modified Emden—Fowler equation
No
42
43
44
45
46
m
2
2
2
2
2
n
arbitrary
(n + -2)
arbitrary
(n + -2)
arbitrary
(n + -2)
arbitrary
(n ^ -2)
-2
-7n - 15
\n
-iGn + 20)
-1
Equation
2.6.1.33
2.6.1.30
2.6.1.32
2.6.1.34
2.6.1.31
6-
7-
Solution in the parametric form:
X =
у = т ехр
n ф -2.
' 4 ' (n + 2J
1 2 2A
TT + (n + 2J П
т ф -1.
Solution in the parametric form:
where A = ±\b1-m{m + 1).
*y'L ~ кУ'х = Ах^у-1, к ф -1.
Solution in the parametric form:
x =
У =
2A
(k
2A
lnr
(fc
1 -5/3
У ' П
-1/2
= Ът,
dr + Co
fe+i
72
Solution in the parametric form:
x = aCf(T3±3T +
у =
± 1K/2,
where A = ±^a
n + 2J.
© 1995 by CRC Press, Inc.

9- *У1 - ^^VL = Ax^y-s/*, n ф -2.
Solution in the parametric form:
K/2(т3 ± Зт + C2)~\
x = аС*(т3 ± Зт + C2)~ з^+б } у = bCfn+6(r2 ± 1K/2(т3 ± Зт + C2)~\
where A = ±f a"n-268/3(n + 2J.
Solution in the parametric form:
з
-1
x=ac1\ -j з +c2\ , у=ъс(п+'кт*\ i ,,;,; ^-+c2\ ,
where A h—a~n~2b1/2(n -A- 2J
^ y-V*, пф-1.
Solution in the parametric form:
x = aCl{rz - Зт + С2)"?*, У = ЬС2п+\т2 - if,
where A = ±\(n + 2Ja-n~2bz'2.
12. xy^x + Bn + 5)y'x = Ax^y-1/2, n ф -2.
Solution in the parametric form:
x = аСЦт3 - Зт + C2)^+4, у = ЬС2п+4(т2 - 1J(т3 - Зт + С2)~\
where A = ±-f-(n + 2Ja"n-263/2.
— 2
ЖУа;а;^ Ух АХ У
о
Solution in the parametric form:
x = aCf [±(t4 - 6t2 + 4С2т - S)]~*+* , y = 6С3п+6(т3 - Зт + C2K/2,
where A = ±-^(n + 2Ja"n-268/3.
14- xVx-x + Cn + 7)y4 = Ахп+^у-ъ1ъ, п ф -2.
Solution in the parametric form:
x = aCf [±(т4 - 6т2 + 4С2т - 3)] з^+б }
у = ЪС1п+\т* - Зт + C2K/2 [±(т4 - 6т2 + 4С2т - 3)] -1,
where A = ±|i(n + 2Ja"n-268/3.
© 1995 by CRC Press, Inc.

~».» VLh' — A<rn+1vi1/2 r> =? —•>
ХУХХ „ Ух — AX У •> n г= z-
Solution in the parametric form:
where A = ±3a,-n-2b1/2(n + 2J.
^ ^y-V*, пф-1.
Solution in the parametric form:
x= [Cie2sT + C2e-ST s
y= {2Cise2sT + C2se-ST[v/3cos(v/3sT) - sin(v/3 st)] }
where A = ^§-s3(n + 2J.
Solution in the parametric form:
x = [Gie °' +O2e °' sin(_
{2Cise2sT + C2se"ST [v/3cos(v/3 st) - sin(v/3 st)] }2
У ~ Cie2sT + C2e"ST sin(v/3 st)
where A = -^s3fn + 2J.
^ 2
ХУХХ . Ух — AX У i n г= z-
1°. Solution in the parametric form with A < 0:
x = aCf [cosh(T + C2) cost] ™+2 [tanh(r + C2) + tanr]'
у = ЬС3п+6 [cosh(T + C2) cos t] 3/2,
тдгЬрге /4 — 3—n~n~2h^/3(r> -I- 9^2
2°. Solution in the parametric form with A > 0:
4
ж = aCf [sinhT + cos(t + C2)J n+'2 ,
- sin(r + C2)J ,
-дтЬот-р A — _3_/J-«-2?i8/3('r? _i_ OY
© 1995 by CRC Press, Inc.

1°. Solution in the parametric form with A < 0:
4
x = aCf [cosh(Y + C2) cos т] 3™+6 [tanh(r + C2) + tan r] 3™+6 ,
у = 6С3п+6 [cosh(T + C2) cost] 1/2 [tanh(r + C2) + tanт] ~\
where A = -^-a"n-268/3(n + 2J.
2°. Solution in the parametric form with A > 0:
4
ж = aCf [sinhr + cos(t + C2)] 3n+6 ;
у = 6Cfn+6 [coshr - sin(T + C2)]3/2 [sinhr + cos(t + C2)] "\
where A = |^a""-268/3(n + 2J.
20. xy'L + y'x 11
Solution in the parametric form:
Г f
ж = С2ехр BА1п\т\+С1)-1/2<1т\, у = т.
Solution in the parametric form:
rr = exp(±T3-3CiT + C2), у = 6(±т2 - CiJ, where A = ±±b3/2.
Solution in the parametric form:
rr = exp(CiT3±3T + C2), y =
23. xy'lx + y'x = Ах-гуг/2.
Solution in the parametric form:
_, Г /" tcLt , 1 .,
x = Ci exp / — , у = от ,
where A = ±12b1/2.
> In the solutions of equations 2^-25, the following notation is used:
51 = C1e2sT + C2e"STsin(v/3sT),
52 = 2ClSe2sT + C2se-'
© 1995 by CRC Press, Inc.

24- жу! - ^-У'х = Ах^у-7'5, пф-1.
Solution in the parametric form:
з
x = aS3n+2, y = bS\'2, where A = - ^a-n-2b12'bS-\n + 2J.
25- xy':x + ^-^-y'x = Ax^y-V*, пф-1.
Solution in the parametric form:
^Ш \/2S-\ where A = ^а-п-2Ь12'ъ s~\
x = aS3bn+1\ y = bSl'zS-\ where A = -^a-n~2b12lb s~\n + 2J
> In the solutions of equations 26-29, the following notation is used:
iJi/s(t) + C2Y1/2,{t) for the upper sign,
Z =
CA() + ?2-^1/3 (У) for the lower sign,
where J1/3 andYi/3 are Bessel functions, I1/3 andKi^ are modified Bessel functions.
26- xy':x - ^±iy; = A^+V172, пф-1.
о
Solution in the parametric form:
^
x = aCfr^+z Z^+2", у = bC2n+AT-2/z{TZ'T + \ZJ,
where A = т^а-п-2Ь^2(п + 2J.
27. xy'^x - ky'x = Ax^y-1/2, к ф -1.
Solution in the parametric form:
x = C^r^Zf^, у = bT-^3Z-2(TZ'T + \ZJ,
where A = т^Ь3/2(к+ 1J.
28- xyZ. - ^-p-v'x = Axn+1y-2, пф-1.
Solution in the parametric form:
' + \ZJ
x = aC3r n+2[{TZ>T + ±ZJ±T2Z2}n+2, у = ЬСГ-
where A = -\a-n-2b3{n + 2J.
29. xy'^x - ky'x = Ах-гу-2, к ф -1.
Solution in the parametric form:
x = dr 3FF5- [(tZ't + \ZJ±t2Z2} ~"*+Г, у = br^3Z2 [(tZ't + \ZJ±t2Z2} ~\
where A=-%b3(k + lJ.
© 1995 by CRC Press, Inc.

> In the solutions of equations 30-39, function p is defined implicitly:
The upper sign in the formulae corresponds to the classical elliptic Weierstrass func-
function p = р(т + С2, 0, 1). The solutions outlined below are written in the parametric
form—one can assume bothr as the parameter, hence р=р(т), andp, hence т = т(р).
30. жу! - Ц-у'х = Axn+1y2, пф-1.
Solution in the parametric form:
х = аС-1т^+2, у = ЬС?+2р, where A = ifa""?)-1^ + 2J.
Solution in the parametric form:
x = С2ет, у = Ър(т, 0, d),
where A = ±6Ь~г, and the elliptic Weierstrass function p = р(т, 0, C\) is defined
fP -1/2
implicitly by the integral т = / Dz3 — C\) dz.
J 00
y AX у
Xyxx \yx
Solution in the parametric form:
= ЬС?+2т~1р, where A = ±|a"n-26-1(n + 2J.
33- *y'L + Gn + 15)vL = Axn+^y2, n ф -2.
Solution in the parametric form:
1 "+2т(т2
= 6С"+2т(т2рт1), where A = ±&a-n-2b~1{n + 2J.
34. xy':x+ 7П + 20у'х = Ахп+1у2, пф-1.
6
Solution in the parametric form:
6
x = aC'^T 7("+2) , у = ЪС?+2т-6(т2р т 1),
where A = ±\a-n-2b-1{n + 2J.
-»." VLh' — A<rn+1vi-5/2 r> =? —9
Xyxx Vx — AX у , П ^ Z.
Solution in the parametric form:
x = aClp-^{f ± 2тр2)^, у =
where A = =p-|a"n-267/2(n + 2J.
© 1995 by CRC Press, Inc.

36- *y'L + V'X y
Solution in the parametric form:
Ж = С2ехр[р-2(/±2тр2)]) у = Ьр-\
where A = =f367/2, and the elliptic Weierstrass function p = р(т, О, С\) is defined
implicitly by the integral т = / Dz3 — Сг) dz.
</oo
xVXx о Уж АЖ У
Solution in the parametric form:
2
where A = т|а"п&7/2(п + 2J.
3
Solution in the parametric form:
x = aCi (r/ + 2p) n+2 , у = bCfn+b (/ ±
where A = --^=ra~n~2b8/3(n + 2J.
39-
Solution in the parametric form:
where A = --^a-n-2bs/3(n + 2J.
> In the solutions of equations 40-46, the following notation is used:
where I(t) = j тВг1 dr is the incomplete elliptic integral of the second kind in the
Weierstrass form.
40- *V'L - ^-Vi = Axn+1y-\ пф-1.
Solution in the parametric form:
x = aCi(T-lF1)n+2 , у = ЪС™+лт-\ where A = т|«"пЬ5(п + 2)г
© 1995 by CRC Press, Inc.

y'x =
Solution in the parametric form:
, пф-1.
Solution in the parametric form:
where A = Tfa"n&5(n +2J.
43. xy':x - ^^y'x = Ax^y-^, пф-1.
Solution in the parametric form:
x = aC*F1n+2, y = bC2n+iF2, where A = ±3a"n-263/2(n + 2J.
44- *y'L ~ ^IT-y'* = Ax^y-1'2, пф-1.
О
Solution in the parametric form:
= bC*n+4F~3Fl where A = ±||a-"-263/2(n + 2J
45- жу! - ^-^-У'х = Axn+1y-r, пф-1.
Solution in the parametric form:
where A = ±т^
46. xyxx y'x = Axn+1y~7, n ф -1.
Solution in the parametric form:
5
x = aUi {vrt{ =F ft) "^ , У = b<J{ ' F1
where A = ±T§wa,-n-2bs(n + 2J.
2.6.2. Equations of the Form y%x = (Aia;nit/Wl + А2хП2ут2)(у'хI
See Section 2.4 for the case / = 0.
Table 2.11 represents all solvable equations whose solutions are outlined in Subsection
2.6.2. Equations are arranged in accordance with the growth of /, the growth of mi (for
identical /), the growth of m.2 (for identical / and mi, mi > m.2), the growth of n\ (for
identical /, mi, and m.2), and the growth of П2 (for identical /, mi, m.2, and n\). The
number of the equation sought is indicated in the last column in this table.
© 1995 by CRC Press, Inc.

TABLE 2.11
Solvable cases of the equation у„я, = (A
+A2xn2ym2)(y'SBI
I
Any
(?2)
mi+2m + 3
mi+ni+2
Any
(^1)
Any
(?2)
Any
(?2)
3mi+5
2mi+3
mi + 5
mi + 3
Зщ+4
Hi + 1
2(щ + 2)
ni+3
Any
(/^1,2)
l
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
1
1
1
3
2
mi
Any
(mi^-1)
Any
0
Any
(mi^-1)
0
Any
Any
1
1
1
l
4
1
1
1
1
1
1
1
1
0
0
0
1
Any
m.2
Any
(TO2^-1)
Any
0
-1
0
-mi-2
mi —1
2
0
0
0
7
4
0
0
0
0
0
0
0
0
0
0
-2
0
Any
Hi
0
Any
Any
(ni^-1)
0
Any
(ni^-1)
1
1
Any
Any
0
0
15
8
15
8
20
13
20
13
5
4
5
4
0
0
Any
(ni^-1)
Any
(ni^-1)
0
0
mi
П2
0
m.2(ni + l) — mi+ni
mi + 1
Any
(na^-1)
0
-1
0
0
-ni-2
Hi —1
2
1
1
7
4
13
8
15
13
14
13
3
4
1
2
1
2
Any
(na^-1)
-1
1
1
m.2
Ai
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
A2
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Equation
2.6.2.1
2.6.2.98
2.6.2.5
2.6.2.2
2.6.2.6
2.6.2.21
2.6.2.94
2.6.2.22
2.6.2.95
2.6.2.20
2.6.2.71
2.6.2.81
2.6.2.66
2.6.2.68
2.6.2.84
2.6.2.64
2.6.2.78
2.6.2.62
2.6.2.75
2.6.2.7
2.6.2.8
2.6.2.25
2.6.2.23
2.6.2.97
© 1995 by CRC Press, Inc.

TABLE 2.11 Continued
Solvable cases of the equation y'? = {А1хПгутг +А2хП2ут2)(у'хI
I
3
2
3
2
3
2
3
2
2
2
2
2
5
2
5
2
5
2
5
2
5
2
5
2
5
2
5
2
5
2
3
3
3
3
3
3
3
3
3
3
3
mi
0
0
1
1
Any
(mi^-1)
Any
(mi^-1)
1
1
7
4
13
8
15
13
14
13
3
4
1
2
1
1
2
Any
Any
Any
(mi ^ -2)
Any
Any
(mi ^ -2)
-5
-4
-3
-3
-3
14
5
TO2
-2
l
2
0
0
Any
(TO2^-1)
-1
0
0
15
8
15
8
20
13
20
13
5
4
5
4
0
0
0
Any
Any
Any
-3
0
-6
-5
-5
-5
7
2
18
5
Hi
0
0
-2
l
2
0
0
-2
0
0
0
0
0
0
0
7
4
0
0
—mi —3
-2mi-3
1
—mi —3
1
1
0
0
0
0
2
П2
1
1
0
0
0
0
0
1
1
1
1
1
1
1
1
4
1
1
—m.2 — 3
—2m.2 — 3
0
0
-3
3
2
1
2
l
2
3
A!
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
A2
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Equation
2.6.2.107
2.6.2.105
2.6.2.108
2.6.2.106
2.6.2.3
2.6.2.4
2.6.2.26
2.6.2.24
2.6.2.80
2.6.2.65
2.6.2.67
2.6.2.83
2.6.2.63
2.6.2.77
2.6.2.72
2.6.2.61
2.6.2.74
2.6.2.9
2.6.2.93
2.6.2.49
2.6.2.19
2.6.2.51
2.6.2.100
2.6.2.76
2.6.2.44
2.6.2.58
2.6.2.28
2.6.2.112
© 1995 by CRC Press, Inc.

TABLE 2.11 Continued
Solvable cases of the equation y^x = (AiX^y
+A2xn2ym2)(y'xI
I
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
mi
8
3
5
2
5
2
5
2
12
5
7
3
7
3
7
3
7
3
11
5
2
-2
-2
-2
-2
-2
2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
TO2
10
3
-4
7
2
-3
13
5
10
3
10
3
-3
8
3
12
5
—П2 —1
-3
-3
-3
-3
-3
2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
Hi
l
3
l
2
1
2
1
2
3
5
5
3
5
3
2
3
5
3
2
1
-2
1
1
-1
l
2
1
1
1
1
1
1
1
1
1
1
1
1
1
П2
5
3
0
1
2
0
7
5
5
3
1
3
0
1
3
3
Any
0
2
2
0
0
Any
-7
-4
5
2
-2
5
3
5
3
5
3
5
3
7
5
1
2
1
2
1
2
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
2(n2 + l)
П2 + 3
Any
6
25
6
25
Any
Any
2(n2 + l)
П2 + 3
15
4
-6
-12
-2
63
4
3
4
3
16
9
100
5
36
-20
4
25
2
9
A2
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Equation
2.6.2.38
2.6.2.86
2.6.2.13
2.6.2.32
2.6.2.30
2.6.2.34
2.6.2.88
2.6.2.70
2.6.2.42
2.6.2.113
2.6.2.132
2.6.2.104
2.6.2.145
2.6.2.144
2.6.2.12
2.6.2.102
2.6.2.116
2.6.2.117
2.6.2.118
2.6.2.119
2.6.2.120
2.6.2.124
2.6.2.123
2.6.2.121
2.6.2.122
2.6.2.125
2.6.2.128
2.6.2.127
2.6.2.126
© 1995 by CRC Press, Inc.

TABLE 2.11 Continued
Solvable cases of the equation y'? = {А1хПгутг +А2хП2ут2)(у'хI
I
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
mi
-2
-2
-2
13
7
12
7
5
3
8
5
3
2
3
2
3
2
3
2
4
3
4
3
4
3
4
3
9
7
7
6
8
7
-1
2
1
2
1
2
1
2
1
2
1
2
0
0
0
0
0
-2
-2
-2
20
7
20
7
7
3
13
5
5
2
-2
-2
-2
10
3
8
3
7
3
4
3
15
7
5
3
15
7
-2
4
3
-3
-2
-2
-2
3
2
-5
-2
-2
3
2
2
3
Hi
1
2
2
0
0
4
3
7
5
0
3
2
0
1
2
5
3
5
3
5
3
0
0
1
2
0
-2
5
3
1
2
1
2
1
2
1
2
1
2
-3
-3
0
1
2
5
3
П2
1
2
1
1
2
2
5
3
7
5
1
2
-2
l
2
1
5
3
1
3
5
3
1
2
2
0
2
-2
5
3
0
1
1
1
0
1
-2
l
2
0
5
3
M
12
49
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
A2
Any
6
25
_6_
25
Any
Any
Any
Any
Any
Any
Any
12
49
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
-20
4
25
2
9
Any
Any
Any
Any
Any
Any
Equation
2.6.2.129
2.6.2.131
2.6.2.130
2.6.2.82
2.6.2.60
2.6.2.92
2.6.2.109
2.6.2.90
2.6.2.48
2.6.2.46
2.6.2.143
2.6.2.36
2.6.2.40
2.6.2.14
2.6.2.15
2.6.2.59
2.6.2.16
2.6.2.79
2.6.2.110
2.6.2.115
2.6.2.53
2.6.2.141
2.6.2.134
2.6.2.137
2.6.2.45
2.6.2.52
2.6.2.56
2.6.2.54
2.6.2.89
2.6.2.114
© 1995 by CRC Press, Inc.

TABLE 2.11 Continued
Solvable cases of the equation y'? = {А1хПгутг +А2хП2ут2)(у'хI
I
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
mi
0
0
0
0
0
0
2
7Г
2
3
2
3
2
3
2
?
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
3
2
2
2
TO2
1
2
0
0
0
0
0
-2
-2
-2
-2
-2
-2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-2
0
0
ГЦ
0
l
3
0
0
0
2
7
5
5
3
5
3
5
3
5
3
-2
-7
-4
-2
-2
-2
5
3
5
3
5
3
7
5
1
2
0
0
1
1
1
5
2
-5
5
3
П2
1
2
5
3
-1
2
1
2
0
1
1
1
1
1
1
-3
-3
-3
3
2
0
4
3
1
3
1
3
3
5
0
l
2
2
-3
0
3
1
-5
5
3
м
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
A2
Any
Any
Any
Any
Any
Any
5
36
63
4
3
4
9
100
3
16
-2
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
Any
-12
Any
Any
Equation
2.6.2.101
2.6.2.39
2.6.2.11
2.6.2.69
2.6.2.31
2.6.2.57
2.6.2.139
2.6.2.147
2.6.2.136
2.6.2.135
2.6.2.138
2.6.2.133
2.6.2.17
2.6.2.10
2.6.2.55
2.6.2.47
2.6.2.103
2.6.2.91
2.6.2.41
2.6.2.87
2.6.2.29
2.6.2.27
2.6.2.85
2.6.2.73
2.6.2.50
2.6.2.43
2.6.2.99
2.6.2.146
2.6.2.96
2.6.2.35
© 1995 by CRC Press, Inc.

TABLE 2.11 Continued
Solvable cases of the equation y'? = {А1хПгутг +А2хП2ут2)(у'хI
I
3
3
3
3
3
3
2
2
3
3
4
6
1П2
0
1
-2
0
3
-2
Hi
5
3
5
3
-4
-7
-7
—7
П2
1
3
1
3
1
-3
-7
1
AX
Any
Any
Any
Any
Any
Any
A2
Any
Any
-6
Any
Any
Any
Equation
2.6.2.37
2.6.2.33
2.6.2.140
2.6.2.18
2.6.2.111
2.6.2.142
1.
1°. Solution in the parametric form:
x = a
mi+1
± rm2
where ^ =
^-1, A2 =
2-1
2°. Solution in the parametric form:
x = a
where ^ =
2-1
— 1, m2 Ф —1.
C2) y = Ът,
C2, y = Ьт,
2-
3.
Solution:
Solution:
x=
X = L
4. yxx = {A1ym +
Solution:
• +1
exp -
mi
m2
тф -1.
dy + C2.
^
C2
© 1995 by CRC Press, Inc.

5- V'L = (Aixni + А2хп^(у'хI, 1ф1, щф -1, п2 ф -1.
1°. Solution in the parametric form:
х = ат, у = Ь [(С1+тП1+1±тП2+1) i-'
where Al = ?±±±al-^-4i-1, A2 = ±?l±±al-^-4i-
2°. Solution in the parametric form:
х = ат, y = b / (Ci - тП1+1 ± тП2+1) i-' dr + C2)
where A1 = —i a'"™1^1"', A2 = ±—-—— al~n'2~'1b1~l.
6. Ухх ^ (Aixn -\- A2x~ )(Уа,) ? ^ Ф 1? "п Ф —!•
Solution:
l
У = / Ci H ——-^-жп+1 + A — /)А21пж dx + C2.
V'L = (Aixni + A2xn2)y'x, m ф -1, n2 ф -1.
Solution:
= G! /expf—^-^
7 Vnl + 1
8- V'L = (Aixn + A2x-^)y'x, n ф -1.
Solution:
f xM expf -^— xn+1) dx + C2.
J \ n + 1 J
9- t/i» = (Aia;-mi-3ymi + A2x-m*-zym*)(y'xf, mi ф -2, m2 ф -2.
1°. Solution in the parametric form:
i -l
x _ aT '" "-- ¦ "-- ч "
y = b\f(C1+ T"mi ±
where Ax = -i-ami+4&-mi-2(mi + 2), A2 = ±iam2+46-m2-2(m2 + 2).
2°. Solution in the parametric form:
i -i
x = ат
y = b^(C1-r-"*-a±r-"*"-a)-^A- + <
where Ax = -\ami+4~mi~2{m1 + 2), A2 = ±^-am2+46"m2(m2 + 2).
© 1995 by CRC Press, Inc.

y'lx = (AlX-4y + A2x-3)(y'xK.
1°. Solution in the parametric form:
i -i
where Ax = |a56, A2 = ±a4~2.
2°. Solution in the parametric form:
x = атI / 1C1 -т~6±т~2) ' dr + C2
-l
where Ai = — -|«5& 3, A2 =
Solution:
У =
12.
Solution in the parametric form:
i -l
I f / v^-ч i , i \ 1 /О ч ^-^ I
Г /" "I
= t\ J{C1 - 2А11пт - 2А2т)~1/2dr + C2\
y=\ f(C1-2A1lnT-2A2T)-1/2dT +
Solution in the parametric form:
x = h_[2Cie2kT + C2e~kT [V3cos(wt) - sin(wT)] }, у = -^,
where F = de2kT + С2е~кт sin(a)T) - ^-, A2 = -16/c3, u> =
A2
14. у»я = (А1Ж-5/3у-4/3 + А2х~53
Solution in the parametric form:
x = {^А2тА + ClT3 + С2т2 + Сзт)^^3 + 3Cit2 + 2С2т + С3K/\
у = (^42т4 + dr3 + С2т2 + C3t)~\
where Аг = 9dC3 - 3C|.
© 1995 by CRC Press, Inc.

> In the solutions of equations 15-18, the following notation is used:
Д1 = (С1+т-3±т-2I/2, Д2=(С1-т-3±т-2I/2)
E1= f R-1 dr + C2, E2= f Щ1 dT + C2,
F\ = т — R\E\, F2 = т — R2E2,
Нг = 3t3F? + 3A ± т)Е1, H2 = Зт3^| + 3(-l ± т)Е$.
15. y»m = (А1У-*/3 + A2x-1/2y
Solution in the parametric form:
x = aFl, у = Ьт-3Е3к,
where A1 = ±|а6/3, A2 = l«3/2&/3(-l)fe+1; * = 1 and к = 2.
16. y'L = (AiX-V*y-7'« + А2у-*/3)(у'хK.
Solution in the parametric form:
x = aT3E-3F%, у = Ьт3Е~3,
where Аг = ^а^Ч-5^(-1)к, А2 = т|-а&/3; к = 1 and к = 2.
17. y'lx = (AlX-7y + А2х-3)(у'хK.
Solution in the parametric form:
х = ат-112Е1к/\ y = br-3Hk,
where Ax = -^аЧ'3, A2 = ±^a4b~2; к = 1 and к = 2.
18- yr:x = (Aix-7y3 + A2x-3)(yxf.
Solution in the parametric form:
x = ar^2El/2H-\ у = Ьт3Н-\
where Ax = -^a8b~5, A2 = =F^-«4&; k = l and к = 2.
> In the solutions of equations 19-22, the following notation is used:
R1 = (Ci ± r^+1 + rI/2, R2 = (Ci ± r^+1 - rI/2,
-1 dT + C2, E2= IR-1 dT + C2,
Ei, G2 = 2i?2 H~ -^2?
#! = 4(r - RtFi.) + El H2 = 4(r - R2F2) - E\.
© 1995 by CRC Press, Inc.

19- V'L = (А1Х-т-3ут + А2у-*)(у'хK, т ф -2.
Solution in the parametric form:
х = атЕ~1, у = ЬЕ~1, 7=-m-3,
where Ax = ±\am+%-m-'1(m + 2), A2 = \ab{-l)k; к = 1 and к = 2.
20. yL
Solution in the parametric form:
1
x = arGk, у = ЬЕк, 7 =
1-2'
ry Г (л, I lUl1/
b'1 A2(-l)k+1A2 =r ±K \ '
where Аг = ab'1 A2(-l)k+1, A2 = '— \± w \ ' \ ; к = 1 and к = 2.
zao
Зтгг+5
21- V'L = (Aixym + A2y-m-2)(y'x) 2m+3 .
Solution in the parametric form:
4-2 2m + 3
= aHk, y =
where Al = ^JL-a-^^T [т^^] \ A2 = ab~^Al{-lf ¦ к = 1 and
к = 2.
\
22. y^ = (А1Я!"» + A2x-n
Solution in the parametric form:
n-Y+2 , rr 2n + 3
x = aEl , y = bHk, 7= ——,
it \~ -L
and к = 2.
> In the solutions of equations 23-26, the following notation is used:
R1 = (C1+t± ЬтI/2, R2 = (d - т ± 1птI/2,
f -i f -i
E\= I Rx dr + C2, E2= R2 dr + C2,
J J
1 — ZiXi — -t^l? Lt2 — ^-П-2 ~r -^2;
Я1 = 4(t - RxFx) + El, H2 = 4(t - R2F2) - E\,
© 1995 by CRC Press, Inc.

23. y'L = (А1У + A2x)y'x.
Solution in the parametric form:
x = aGk, у = ЬЕк,
where Аг = ab'1 А2(-1)к, A2 = ±\a~2; к = 1 and к = 2.
24. у^
Solution in the parametric form:
x = аЕк, у = bGk,
where A-± = т}Ь, A2 = a-1bAi(-l)k; к = 1 and к = 2.
25. y'L = (Аг + A2xy-2)y'x.
Solution in the parametric form:
x = aHk, у = ЪЕк,
where Ax = ab'2A2(-l)k, A2 = i-i-a"^; к = 1 and к = 2.
26. У1 =
Solution in the parametric form:
x = aEk, у = ЬНк,
where Ax = T^a2b~2, A2 = a-2bA1(-l)k; к = 1 and к = 2.
> In the solutions of equations 27-30, the following notation is used:
Rx = ClTkl + С2тк2 + С3ткз,
Д3 = dekT + eST(C2 sin сот + C3 coswt),
Q1 = Ci/ciTfel + С2к2тк2 + С3к3ткз,
Q2 = (кСг +С2 + кС2т)ект + соС3ешт,
Q3 = кСгект + eST[(sC2 - coC3) sin сот + (sC3 + coC2) cos сот],
S^riQM, S2 = (Q2)'T, S3 = (Q3)'T,
where k\, k2, and k3 (real numbers) or к and s ± ico (one real and two complex
numbers) are the roots of the cubic equation A3 — \B2X — \B\ = 0. Subscripts of
functions Rm, Qm, and Sm (m = 1, 2, 3) are selected depending on the sign of the
following expression:
{> 0 subscript 1,
= 0 subscript 2,
< 0 subscript 3;
if 2B\ = 27B2 (subscript 2), then
со =-2{\B2f'2 г/ Вг<0,
\2, co = 2{\B2f/2 г/ Вг>0.
Remark. The expressions for Rm, and Qm contain three constants C\, C2,
and C3. One of them may be arbitrarily fixed to let it be any nonzero number (for
instance, we may set C3 = ±1), while the other constants remain arbitrary.
© 1995 by CRC Press, Inc.

27. v':a> = (A1x-1'*v +
Solution in the parametric form:
x = Q2m, y = Rm, A1 = -B1, A2 = -B2
28. y'L = (AlV-* + A2x/3
Solution in the parametric form
29. у»я = (AlX-7/5y + A2x
Solution in the parametric form:
where Ax = -^а^Ч^В-1, А2 = -a-A/5bA1B2.
30. y»m = (А1Ж-3/5у
Solution in the parametric form:
x = aRU2BQ2m - ±RmSm + B2R2m)~\ у = bBQ2m - ARmSm + B2R2m)~2,
where Аг = —-^as'5b2'5B1 B2, A2 = -^a12
> In the solutions of equations 31-32, the following notation is used:
CiefeT + C2e~kT - ^-т if B2 > 0,
h= { B
\ sin(fcr) + C2 cos(/ct) - —-T if B2 < 0,
B2
k(CiekT - C2e~kT) -^- if B2>0,
Вл
k[d cos(fcT) - C2 sin(/cT)] - —3- if B2 < 0,
B2
where к = W-j
31. 1&в = (А1+А2ха
Solution in the parametric form:
Аг = -В2, А2 = -В1.
32. !&, = (А^-^ау-в/а + A2y~3)(y'xK.
Solution in the parametric form:
x = fi1fl V = fi\ Аг = -Въ А2 = -В2.
© 1995 by CRC Press, Inc.

> In the solutions of equations 33-36, the following notation is used:
For B1 > 0:
Ti = dekT + de~kT + C3sin(/cT), к = (-§-BiI/4,
T2 = k(dekT - de~kT) + kd cos(/ct);
For Bx < 0:
Ti = eST[dsm(sT) + dcos(sT)} + de~ST sin(sT), s= (-^-BiI/4,
T2 = seST[(d - d) sin(sr) + (d + d) cos(st)] - sde~ST[sm(sT) - cos(sr)].
°°- Ухх — \-r*-l<f' У ~Т~ S^2A-
Solution in the parametric form:
•3/9 Ao
x = i2 , 2/ = Jl"^47'
where B\ = —A\, B2 = —A2; arbitrary constants C\, C2, and C3 are related by
CiC3 = -^A~2Al if Ax > 0,
4CiC2 + Cl = \A~2A\ if Ai < 0.
34. y»m = (А1Ж-5/3у-7/з + A2x-5/3y
Solution in the parametric form:

where B\ = —A2, B2 = —A\; arbitrary constants C\, C2, and C3 are related by
dC3 = -^A\A~2 if A2 > 0,
4CiC2 + Cl = \A\A~2 if A2 < 0.
35. y»m = (AlX-5/3y2 + A2x
Solution in the parametric form:
where B\ = — A\, B2 = —A2; arbitrary constants C\, d, and C3 are related by
СгС3 = -\А~гА2 if A1 > 0,
4CiC2 + Cl = -\A~1A2 if Ax < 0.
36. y'lx = (AlX-5/3y-4/3 + A2x-5/3y-lo/3)(y'xK.
Solution in the parametric form:
_ T-iT3/2 _ T_i
a, — ±x ±2 , у — ±x ,
where B\ = —A2, B2 = — A\; arbitrary constants C\, C2, and C3 are related by
СгС3 = -\A1A~X if A2 > 0,
4CiC2 + Cl = -\AxA21 if A2 < 0.
© 1995 by CRC Press, Inc.

> In the solutions of equations 37-38, the following notation is used:
1. For B1 > О, B2 ф 0,
Ti = dekT + C2e~kr + C3 sin wt,
T2 = k(CiekT - C2e-kT) + coC3 cos lot,
where
к = { f[(B2 + 3B0172 + B2}}1/2, со = { f[(B2 + 3B0172 " B2}}1/2,
4/c2C1C2+w2C32 = 0.
2. For -B\ < 3Bi < 0, B2 > 0,
Ti = CiTfel + C2T~kl + С3тк2 + O±T-k2,
T2 = /ci(CiTfel - C2T~kl) + к2{С3тк2 - C4T-fe2),
where
) ( )b2 = o.
3. For -Bl < 3Bi <0, B2< 0,
T\ = C\ sintJir + C2 cosooxt + C3 sinter,
T2 = coi(Ci costJiT — C2 sintJiT) + to2C3 cosui2t,
where
, 1 / 2[R_i_f'R2_i_QR W2"!!1/2 , , S 2 TR /n2 i qd
C0\ — ^—g- \D2 -f- \-D2 + д?>\) \j , C02 = ^—з"[- — (,-С*2 + "J-t>
^. Jbr B| + 3Bi = 0, B2 > 0,
Ti = (Ci + C2T)efeT + (C3 + C4T)e-feT,
T2 = (/cCi + C2 + кС2т)ект - (kC3 - C4 + кО±т)е-кт,
where
5. For Bl + 3Bi = 0, B2 < 0,
2\ = (Ci + C2t) sinoir + С3т cos сот,
T2 = (coC\ + C3 + coC2t) cos cot + (C2 — соС3т) sin сот,
where
© 1995 by CRC Press, Inc.

6. For 3Bi < -B\,
Tx = ekT(Ci sinuT + C2 coswt) + С3е~кт sinuir,
T2 = ekT[(kC2 + wCi)cosujt + (ВД -u>C2)sinwr)
+ C3e~kT{ui cosujt — ksinuir),
where
k={\[B2 + {-Шг)^}}1'2, с = {\[-B2 + {-ШгУ/
С2В2 + Сг{-В1 - 3BiI/2 = 0.
37. y'L = (А.х-^у2 + Aix
Solution in the parametric form:
ж = Т23/2, у = Т1, B1 = -A1, B2 = -A2.
Solution in the parametric form:
x = T~1T*/2, у = Т~\ B1 = -A2, B2 = -Ax.
> In the solutions of equations 39-42, the following notation is used:
Г CieWT + С2е~шт + С3т if В > О,
\ Сг sin uit + C2 cos uit + С3т if В < 0;
(С\ешт — С2е~шт) + C3 if В > 0,
{C\ cos lot — C2 sin сот) + C3 if В < 0;
where to =
39. !&, = (Ахж-1/3 + A2x
Solution in the parametric form:
x = T23/2, у = Ть
where В = —A\, arbitrary constants C\, C2, and C3 are related by
1 + A2) - 4A\(Cl + Cl) =0 if Аг > 0,
| + A2) - 1&А\СгС2 = 0 if Аг < 0.
40. !&, = (AlX-^3y-^3 + A2x-1/3y-s/3)(y'xf.
Solution in the parametric form:
where В = —A2, arbitrary constants C\, C2, and C3 are related by
2 ^2 + C22) =0 if A2 > 0,
ldC2 = 0 if A2 < 0.
© 1995 by CRC Press, Inc.

41. y'lx = (AlX-5/3y + А2х-1/3)(у'хK.
Solution in the parametric form:
Ax \3/2
where В = — A2, arbitrary constants C\, C2, and C3 are related by
3A2Cf - 4^(C2 + C\) - -^A\A^2 = 0 if A2 > 0,
| ^iC2 - ^-^i^2 = 0 if A2 < 0.
42. !&, = (А1Ж-5/3у-7/з + А2ж-
Solution in the parametric form:
where В = —A2, arbitrary constants Ci, C2, and C3 are related by
ЪА2С1 - 4A\(Cl + C2) - -jqAIA^2 = 0 if A2 > 0,
3A2C| - 16A22dC2 - -^A\A~2 = 0 if A2 < 0.
> /n the solutions of equations 43-48, the following notation is used:
, _ Г Л/з(т) /or ifte upper sign (Bessel function),
\ I\/z (T) /or ^e lower sign (modified Bessel function),
( Y1/^(t) for the upper sign (Bessel function),
у Кх/3 (т) for the lower sign (modified Bessel function),
H = Clf + C2g
ffdr-f fgdr), u, = { ?* for the upper 8гдп,
J J I \ — 1 for the lower sign.
43. у':х
Solution in the parametric form:
x = т^Н, у = Ът2'\
where Аг = ±f b~3, A2 = -\b~2fi.
44. у»я = (AlV-3 + A2xy-5)(y'xK.
Solution in the parametric form:
x = т-1'3!!, у = Ьт~21\
where Аг = -f 6/3, A2 = ±f b3.
© 1995 by CRC Press, Inc.

45. y'L = (Ане-1/2»-1/2 + A2y~3/2)(y'xf.
Solution in the parametric form:
х = ат-2'3{тН'т + \НJ, у = Ът2'3Н2,
where A1 = ±±а3/2Ь~3/2, А2 = \ab~A
Solution in the parametric form:
x = ат^3Н2(тН'т + \Hf, у =
where Аг = ^аб/2/?, А2 = ±±a3/2.
47. y'L = (AlX-2y + A2x
Solution in the parametric form:
x = ат2'3Н2, у = Ьт-2'3 [Тт2Н2 + 2/ЗтЯ - (тН'т + \НJ],
where Ах = -\а3Ь~3, А2 = -агх12Ъ$Ах.
48. у'^х = (А1Ж-3/2у-3/2 + А2х~2у-2)(ухK.
Solution in the parametric form:
x = ат^'3Н2 [Тт2Н2 + 2/ЗтЯ - (тН'т + ±НJ} ~\
у = Ьт2'3 [Тт2Н2 + 2/ЗтЯ - (тН'т + ±
where Ах = ^'Ч-1'2^, А2 = -|а3.
~\
49- y'xx = (Aixymi + A2ym2)(y'x) , mi ф —2.
Solution in the parametric form:
v =
mi+ 2'
where
2m2 — m\ — 1
k=
, _ Г Jv(t) for the upper sign (Bessel function),
\ Ivij) for the lower sign (modified Bessel function),
_ Г Yv (t) for the upper sign (Bessel function),
\ Kv(t) for the lower sign (modified Bessel function),
Al=±i(m1+2Jb-^-2, A2 = -b-m*/3, со=Нж'
[ — 1
=Нж' for the upper sign,
tor the lower sign.
© 1995 by CRC Press, Inc.

> In the solutions of equations 50-56, the following notation is used:
_ \ C\Jv{t) for the upper sign (Bessel functions),
\ C\Iv{t) for the lower sign (modified Bessel functions),
У _ j C2Yv[f) for the upper sign (Bessel functions),
v \ C2Kv(t) for the lower sign (modified Bessel functions),
N=
N = I ZvGv + XvFv f°V A = ~("-ч^ ~ '
1 \tN' + 2vN for A = 4a7 - (i2,
.2 Ar2 , , ,2 л ,,_jf f°r the
\ZVXV forA = -(a1f32-a2[31J;
' — 1 /or me lower sign,
where prime stands for differentiation with respect to т.
50- V'L = (А1*У + Mx-3)(y'xf.
Solution in the parametric form:
x = атг'3Кг'2, у = Ът2'3,
where v=±, Ax= ±f b~3, A2 = --^а%~2и2А.
51. y'^ = (A1xym + A2x-3)(y'xK, тф-2.
Solution in the parametric form:
x = ат^Ы1'2, у = Ьт2\
where v =—Ц-, A1 = ±\b~m-2{m + 2J, A2 = -^a4b-2u>2A(m + 2J.
Tib \~ Zi
52. у»я = (А1Х~3 + A2xy-5)(y'xK.
Solution in the parametric form:
x = ar-^N1'2, у = Ът~2'\
where v=±, Ax= -^-a46w2A, A2 = ±f b3.
53. y'L = (Агх-^у-1'2 + A2y-3)(y'xK.
Solution in the parametric form:
x = ar-^N^N2, у = bT2'3N,
where v = i, A1= ±j-a3/2b~3/2, A2 = 2abu2A.
© 1995 by CRC Press, Inc.

54. V'ix = (A, + A2x-i/2y-2)(y'xK.
Solution in the parametric form:
x = o?, y
where v = ±, Ax= 2ab~2to2A, A2 = ±-|a3/2.
55. y'L = (AlX-2y + A2x-3)(y'xK.
Solution in the parametric form:
x = aT2'zN, у = bT
where v=±, Ax = ^fg-a3^3, A2 = —^a46"
56. y'L = (AlX-3 + A2x-2y-2)(y'xK.
Solution in the parametric form:
where v=\, Ax= --^a4&-2w2A, A2 = -^ a3.
> In the solutions of equations 57-72, the following notation is used:
-C2, т = / bL d
J ^±Ap\-2Pl-C2
and
C2, т= - Ci,
where functions p\ = pi (r) and p2 = p2 (т) are defined implicitly by the above elliptic
integrals. For the upper signs, they are the classical elliptic Weierstrass functions
pi = р(т + i, 2, C2) and p2 = р(т + ь -2, C2).
57. y'L
Solution in the parametric form:
x = apk, у = Ьт,
where Ax = ^ба"^, A2 = ab-2(-l)k+1; к = 1 and к = 2.
58. ух;х = (А1у-* + А2х2у
Solution in the parametric form:
х = ат~1рк, у = Ьт~1,
where AY = ab(-l)k+1, A2 = грба"^3; к = 1 and к = 2.
© 1995 by CRC Press, Inc.

59. y'L = (А1У-^ + A^y
Solution in the parametric form:
x = ar(T2pfe Tl), У = Ьт7,
where A1 =-^ab~bI7{-l)k+1, A2 = т^-a^77; к = 1 and к = 2.
so. y'L = (А1У-12/т + Mxiy
Solution in the parametric form:
х = ат-6(т2ркт1), у = Ьт~7,
where A1 = ^ab-2/7(-l)k+1, A2 = T^a^b6/7; к = 1 and к = 2.
Solution in the parametric form:
x = a[fk - (-l)feT], у = Ьт,
where Ax = ab-1A2(-l)k, A2 = -^отЧ'1 {^P) '^ к = 1 and к = 2.
62. yL
Solution in the parametric form:
х = ат, у = b[fk - (-l)feT],
where Л = 2a6-1f± — I 2, A2 = а-1ЬА1(-1)к; к = 1 and к = 2.
V a /
63. yl = (AlV-3/* + А2ху-^)(у'хM/2.
Solution in the parametric form:
where A1 = ab-1'2A2{-I)k, A2 = -la^^^172. fc = i and )fc = 2
64. yl = (А1Ж-5/4у + A2x-3/*)(y'xI/2.
Solution in the parametric form:
х = ат\ у = Ь[2т/к-2рк + (-1)к+1т2},
where A1 = -a1'ib-1(±—I/2, A2 = a-^bA^-lf- к = 1 and к = 2.
4 V a /
© 1995 by CRC Press, Inc.

65. V'L = (А1У-13/8 + А2ху-15/*)(у'хM/2.
Solution in the parametric form:
x = ат[2т3Л + 6r2pfe T 2 + (-l)feT4], у = Ьт~\
where А1 = аЬ~1^А2(-1)к, A2 = -a-V/8(=F-^-I/2; к = 1 and к = 2.
66. y'L = (AlX-15/sy + А2х-13/*)(у'хI/2.
Solution in the parametric form:
x = ат~8, у = 6т[2т3Л + 6r2pfe T 2 + (-l)fer4],
where Л = -^-а7/86(т-^-I 2. A2 = а-1/АЬА1(-1)к; к = 1 and к = 2.
8 V 2я /
67. yL = (А1У
Solution in the parametric form:
x = ar[5r3/fe - 20т2pfe ± 30 - (-l)fer4], у = Ът13,
here A1 = ab-5/l3A2(-l)\ A2 = -^а-1Ь711Н±^-I'2; к = 1 and к = 2
DO \ Lo J
68. y'L = (AlX-2°/13y + A2x-15/I3)(y'xI/2.
Solution in the parametric form:
x = ат13, у = Ьт[5т3/к - 20r2pfe ± 30 - {-1)кт%
^^1/2, А2 = а-ь/13ЬАг{-1)к- к = 1 and к = 2.
69. yL = (Ai+A2x)xf
Solution in the parametric form:
x = ар3, у = b[fk - (-1L
where А1 = ТтаЬ~2' Ai = ¦j2ab/3b-2{-l)k+1; к = 1 and к = 2.
70- y'L = (А1Ж-2/зу-т/з + A2y-3){y'xf.
Solution in the parametric form:
x = ap3[fk - (-1)кт}-\ у = b[fk - (-1)*т]-\
where Ax = ¦^1аъ'3Ь1'3{-1)к+1, A2 = т\аЬ; к = 1 and к = 2.
© 1995 by CRC Press, Inc.

71. у»я = (AlV-i/* +
Solution in the parametric form:
if к =
where Ax = ab~3/2A2, A2= { 1/2
±a-411/4(±j) if к = 2.
72. V'ix = (AlX-7/*y + A2x
Solution in the parametric form:
x = a[fk - (-l)fer]4/3, у = b{fk - г2 т 4p3fe),
((J if fc = l,
where A1={ ? 1/2 A2 = а
(() iffc 2
> In the solutions of equations 73-92, the following notation is used: functions Pi
and Pi are the general solutions of four modifications of the first Painleve equation
P{' = ±6P? + т, P% = ±6P| - т
(in the case of the upper sign, the equation for Pi is the canonical form of the first
Painleve equation; see Subsection 2.8.2). In addition,
Qi = ±6P2 + т, Q2 = ±6P22 - т,
Ri = 1P[ - t2, R2 = 2P'2 + t2,
Si = 3tP1'-3P1-t3, S'2 = 3tP2-3P2+t3,
Ti = t2Pi T 1, T2 = т2Р2 T 1,
Ui = {P[f - 2PiQi ± 8P3, U2 = (P^J - 2P2Q2 ± 8P3,
Vi = PlQ[ + P[ - Ql V2 = P'2Q'2 -P'2- Ql
Wi = T3P[ + 3t2P! Tl + т5, W2 = tzP'2 + 3t2P2 tI-t5,
Zi = 6(t3P[ - 4t2Pi ± 6) - t5, Z2 = 6(t3P2 - 4т2Р2 ± 6) + т5,
where primes stand for differentiation with respect to т.
73- у':х 23
Solution in the parametric form:
x = aPk, у = Ьт,
where Ai = ab-3(-l)k, A2 = =р6а6-2; к = 1 and к = 2.
© 1995 by CRC Press, Inc.

74. y'L
Solution in the parametric form:
x = aRk, у = Ьт,
where Аг = ab~2A2(-l)k+1, A2 = -2^4^ (±-^У/2; к = 1 and к = 2.
75- y'L
Solution in the parametric form:
x = ат, у = bRk,
where Ai = 2a-16-1(±—) , A2 = a-26^i(-l)fe+1; к = 1 and к = 2.
Solution in the parametric form:
x = ат~хРк, у = Ьт'1,
where Ax = аЬ2(-1)к, А2 = ^ба^3; к = 1 and к = 2.
77. y?a = (Агу-1'2 + A2xy-5/4)(y'xM/2.
Solution in the parametric form:
x = aSk, у = ЬтА,
where Ax = ab-3^A2(-l)k+1, A2 = -—a^b^^i^-^A ; к = 1 and к = 2.
Solution in the parametric form:
x = атА, у =
where A1 = ^-a-1/Ab-1(±^-)/, A2 = a-3/46^i(-l)fe+1; к = 1 and к = 2.
Z \ ZQ
79- y'L = (^iy-8/7 + A2x2y-15/7)(y'xK.
Solution in the parametric form:
x = атТк, у = Ьт7,
where Аг = -^ab-&'7{-l)k, A2 = т-^а-гЬг'7; к = 1 and к = 2.
© 1995 by CRC Press, Inc.

80. V'L = (AlV-V* + А2ху-™/*)(у'хM/2.
Solution in the parametric form:
x = ar~6Wk, у = Ьт~8,
where A1 = ab-1'SA2{-I)k, A2 =-а^/Чт^т) ; к = 1 and к = 2.
8 V b J
81. y»m = (А1Х
Solution in the parametric form:
x = от"8, у = bT~6Wk,
where А1 = -^а7/8Ь-1(т — I/2, A2 = a/8^i(-l)fe; к = 1 and к = 2.
8 V A J
82. y»m = (А1У
Solution in the parametric form:
x = aT~6Tk, у = Ът~7,
where Ax = -^ab-1'7{-l)k, A2 = ^^-a"^6/7; к = 1 and к = 2.
83. y'lx = (А1У-14/13 + A2xy-20/13)(y'xM/2.
Solution in the parametric form:
x = arZk, у = Ьт13,
where A1 = ab-6/13A2(-l)k+1, A2 = - — a"V/13(±—У7'; к = 1 and к = 2.
lo V Lou J
84. !&, = (AlX-2°/13y + A2x-^/13)(y'xI/2.
Solution in the parametric form:
x = ат13, у = brZk,
where A1 = ^-a7/13b-1(±-^-I/2, A2 = a-6/13bAi(-l)fc+1; к = 1 and к = 2.
85. у':х = (А1У + А2х
Solution in the parametric form:
x = aiPtf, у = ЬРк,
where Аг = т24а6, А2 = 2a3/26(-l)fe; /c = 1 and/c = 2.
© 1995 by CRC Press, Inc.

86. V'L = (Агх-^у-*/* + A2y-*)(y'xK.
Solution in the parametric form:
x = aP-1{P'kf, y = bP~\
where A1 = 2a3/2b1/2(-l)k, A2 = т24а62; к = 1 and к = 2.
87. y'L = {AlX-^y + A2xV3){y'xK.
Solution in the parametric form:
x = aP^/2, у = bllk,
where Аг = -^as/3b~3, A2 = =F8a6^i; к = 1 and к = 2.
88. y'L = (А1Ж-5/3у-7/з + A^xV3y
Solution in the parametric form:
where Ax = -^а&'Чг'3, A2 = T&a^bA^, к = 1 and к = 2.
89. y'L = (Агх-V* + A2y-3/*)(y'xK.
Solution in the parametric form:
x = aQ2k, у = b(P'kf,
where Ax = ^ба^Ц-2, A2 = ^ab-^2(-l)k+1; к = 1 and к = 2.
90. !&, = (А1У-3/2 + A2x-1/2y-5/2)(yxf.
Solution in the parametric form:
x = a(P^)-2Q2, y = b(P'ky2,
where Л = la&/2(-l)fe+1, A2 = T6a3/2b^2; к = 1 and к = 2.
91- V'L = (AiX-5/3y + A2x-*/3)(y'xf.
Solution in the parametric form:
x = a(P^K, у = bVk,
where A1 = --^-as/3b-3, A2 = а-1/3ЬА1(-1)к; к = 1 and к = 2.
92. y'L = (AlX-*/3y-5/3 + A2x-5/3y-7/3)(yxf.
Solution in the parametric form:
x = a{P'kKV-\ y = bV~\
where A1 = ^a7'3b-1l3{-l)k+\ A2 =-^a^3b^3; к = 1 and к = 2.
© 1995 by CRC Press, Inc.

> In the solutions of equations 93-96, the following notation is used:
1T2 Wl -fci+i 2Д2 _к?+1 _
г = < 1 2
+ -r2 + -;—i-Tfe+1 + 2B2ln|T| г/ k = ki^-l, k2 = -l;
i + Tr2 + !—Vrfe
4 /с + 1
Solution in the parametric form:
x = tF1'2, y = F,
where /ci = — 2mi — 3, k2 = —2m2 — 3, Ai = —B\, .
m—1 ч m+5
94- V'L = (Aixym + A2y 2
Solution in the parametric form:
x = aF~1/2G, y = bF m+i ,
171-1-3 t
where k\ = к = -, k2 =0, Ax = ——-a LbK+ \- n i л^\ , A2 =
Tib ~~\~ i- rC ~\ J-
95- l/i» = ( IL y-^W
Solution in the parametric form:
l
y = F~1/2G,
Jl±l
where k1 = k = ———, k2 = 0, Аг = --——a k+i b~ |—77-7-77-I . A2 =
1
-4a
96. y^
Solution in the parametric form:
'A
© 1995 by CRC Press, Inc.

97- V'L = (Aixmiymi + A2xm*ym*)(y'xK/2.
Solution in the parametric form:
x = CiT1/2expl - —
fdr
У =
fdr
where
/ =
T-l/2
T-l/2
x
2(mi
2(m2
if mi 7^ —1, m.2 7^—1,
if mi =m/ —1, m.2 = —1.
98- ухх = (Aixnyrni + A2x 7
Solution in the parametric form:
dr
x = G'i exp
) —Trai+тг m1-\-2n-\-3
+1 ут2)Ю ™i+n+2
И + 1
техр
n + 1
tz J - " ¦ \ mi +:
where z = z(t) is the solution of the algebraic equation
n+1
/ mi+n+2\/ n+1 \ mi+n4
V mi+l /V mi+l/
= T mi+n+2
mi+n+2
+ 1)
-rm2+1 if m2 ^ -1,
mi + n + 2
¦In It
if mo = —1.
> /n the solutions of equations 99-108, the following notation is used: functions P\
and P2 are the general solutions of four modifications of the second Painleve equation
(with parameter a = 0)
P" = тРг ± 2Pf, Pll = -tP2±2PI.
In the case of the upper sign, the equation for Pi is the canonical form of the second
Painleve equation (with parameter a = 0; see Subsection 2.8.2);
± Pt -
Q2 = rPl ± P24 - (P^J,
5i = 2P[QX - Pi т PiQl S2 = 2P^Q
where primes stand for differentiation with respect to т.
± P2Q2,
99.
Solution in the parametric form:
x = aPk, у = Ьт,
where Аг = Ъ3(-1)к, А2 = т2а&-2; к = 1 and к = 2.
© 1995 by CRC Press, Inc.

V'L = (Aixy-5 + A2x3y-6)(y'xf.
Solution in the parametric form:
x = ат^Рк, у = 6т,
where Аг = Ь3(-1)к, А2 = т2а64; к = 1 and к = 2.
101. iC = (А, + A2x-i/2y
Solutions in the parametric form:
= bP2k,
where Ax = т2аЬ~2, A2 = 2-a3/2&-3/2(-l)fe; к = 1 and к = 2.
102. jC = (А1Х-^у-2+А2у3
Solutions in the parametric form:
= bP~2,
here Ax = \az'2{-l)k, A2 = T^ab; к = 1 and к = 2.
103. y':x = (A1x-2y + A2)(yxJ.
Solutions in the parametric form:
x = aP2, у = Ъ[тР2к±Р?-{Р'кJ}, Р'к = {Рк)'т,
where Л = 2a36-3(-l)fe, A2 = ±2ab~2(-l)k; к = 1 and к = 2.
104. jC = (AlX-2y-2 + A2y-*)(yxf.
Solutions in the parametric form:
x = aP2k[rP2k±Pt-{P'kJY\ y = b[rP2k±Pt-{P'kJY\
where Ax = -2a3, A2 = =p2ab; к = 1 and к = 2.
105. y»m = (A! + A2xy-^2)(y'xf^.
Solutions in the parametric form:
x = aP-1^, у = bQ2k,
if к = 1,
where Ai = ^ab~1/2A2(—l)fe, A2 = .
© 1995 by CRC Press, Inc.

106. у»я = (AlX-^y +
Solutions in the parametric form:
x = aQ2k, у =
where Ai =
I An1/4~2(-
2a J
Solutions in the parametric form:
x = aSk, у = bQk,
where Ax = Tab~2A2(-l)k, A2 =
-^)/ а к = 2.
b J
108. у^ = (А1Ж-2У + А2)«K/2.
Solutions in the parametric form:
x = aQk, у = bSk,
where M = « A2 = Ta-4M-l)K
-a2b-2( ) if к = 2,
V a J
109. !&, = (А1Ж-7/5у-8/5 + ^a.-
Solution in the parametric form:
) F = {S'Tf-2SS';t, A2 = ^a12/
110. jC = (A^»-1 + А2Ж-2у-2)(у4K.
Solution in the parametric form:
A}
x =
(rZ'T + \Zf ± r2Z2] - A},
у = {ъс1Т2'* [(tzt + \zf ± r2z2] - A
where
у = {ъс1Т-2'* [(tz't + \zf ± r2z2] - Ay\
Z = { ClJl/z(T) + С2^1/з(т) for the upper sign,
| Ci/!/3(t) + C2K1/3(t) for the lower sign,
Ji/з and l^/з are Bessel functions, 1г/3 and ЛГ1/3 are modified Bessel functions;
A2 = |a36.
© 1995 by CRC Press, Inc.

111.
Solution in the parametric form:
4
A2
\
У =
where R= л/±Dт3 - 1), F = 2т f tR'1 dr+ С2тт *
2 г g4 a
'^-JT2) '
!, С = 4т^2Тт-2(Д^-1J,
> In the solutions of equations 112-113, the following notation is used:
E= f(l± г4)/2 dr + C2, k2 = ±1;
function E can be expressed in terms of the elliptic integrals or the lemniscate func-
functions.
112.
Solutions in the parametric form:
x = аС\Е~^{тЕ -к), у
where Аг = т-^а^Ъ^Ч, А2 = T-2-a~2b^5.
113. y»m = (A^y-11/5 + A2x3y-12/5)(y'xK.
Solutions in the parametric form:
x = аСгЕ{тЕ -к), у
where Аг = т-^а^Ь^к, А2 = T-2-a-2b2^5.
> In the solutions of equations 1Ц-115, the following notation is used:
Д = Cf - 2Cb R = C6A + МВт - 2т3I/2, z = 3 / t^R'1 dr,
W(z) =
I Л
/О
tan(±v/ZAz) + -?- if A < 0;
Gi
tanh(TVA z) + Q- if A > 0;
Gi
Cl2 V |C;
if A = 0, C2 < 0;
г/ А = 0, C2 > 0.
© 1995 by CRC Press, Inc.

114. y'lx = (А1Х~5/3 + А2х-5/3у~
Solutions in the parametric form:
x = aT"9/4(CiW2 - 2C2W + 2K/4FCiW - 6C2
у = bT-3/2{CxW2 - 2C2W + 2K/2,
where Аг = -2^1Ч~2СЪ A2 = 36a8/36/3B.
115. y»m = (AlX-5/3y-2/3 + A2x-
Solutions in the parametric form:
x = aT3/4(CiW2 - 2C2W + 2)/4FCiW - 6C2
у = br^iCiW2 - 2C2W + 2)/2,
where Аг = 36a8/3&-4/3B, A2 = -24a8/36-2/3d.
V'L =
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.4
for function x = x(y):
2(n + l) .
ж А
-7)у-2(у'K
. V'L = (-?* + Ах-7)у-2(у'х)
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.35
for function x = x(y):
-*)y-2(y'K
US- V'L = (-6* + Ax-*)y-2(y'x)
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.31
for function x = x(y):
119- V'L = (-12* + Ax-^)y-2(y'xK.
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.64
for function x = x(y):
-2)у-2(у'K
120. y'L = (-2x + Ах-2)у-2(у'х)
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.6
for function x = x(y):
© 1995 by CRC Press, Inc.

y'L = (те
Assuming y as the independent variable, we obtain an equation of the form 2.4.2.26
for function x = x(y):
122. y'L =
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.10
for function x = x(y):
123. y'L = (-ix + AxJ
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.12
for function x = x(y):
124. y'L = (-^-x х
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.66
for function x = x(y):
125- y'L = (lkx + Ax
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.29
for function x = x(y):
г" ~1Г2( -5-r Ат~7/ьЛ
126. y'L = Цх + Ax-^)y-\y'xf.
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.14
for function x = x(y):
127. y'L = (-?x + Ах
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.8
for function x = x(y):
2^
128. y'L = (-20* +
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.33
for function x = x(y):
/
129. y'L = (жж + Ax^)y-\y'xK.
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.37
for function x = x(y):
Xyy = У (, 4gX Ax J.
© 1995 by CRC Press, Inc.

13°- v'L = (Ax2 + -§ь-
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.60
for function x = x(y):
T" —ir2(-A-r2 ё-чЛ
131. y'ix = (Ax2 - JL-x)y-2(y'xf.
Assuming у as the independent variable, we obtain an equation of the form 2.4.2.62
for function x = x(y):
132. yl = [ (^++3J ХУ~2 + Axmy-m-^ (y'xf, тф-3,тф -1.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.5
for function x = x(y):
rprr -_ 1 l_o/~^/y> Ail~ -T
У У / i oN О У У **-
yy ' (m + 3)^
133. y»m = (Ax-2y - 2xy-2)(y'xK.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.7
for function x = x(y):
134. y'ix = (Ax-^y-m + ^rxy-*)(yxf.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.9
for function x = x(y):
135. y':x = (Ax-^y^ + -J^xy-*)(yxf.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.11
for function x = x(y):
136. y'lx = (Ax-^y^ - ^Xy-2)(y'xf.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.13
for function x = x(y):
137.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.15
for function x = x(y):
© 1995 by CRC Press, Inc.

138. V'ix =
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.27
for function x = x(y):
139.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.30
for function x = x(y):
140. y'L = (Ax-*y3 - 6xy-2)(y'xK.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.32
for function x = x(y):
tfy = Fy~2x - Afx-%
141. у»я = (Ax-^y-1'2 - 20xy-2)(y'xf.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.34
for function x = x(y):
x'^ = B0y-2x - Ay-^x-1'2).
142.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.36
for function x = x(y):
143. y'ix = (Ax^y-V* + %Xy-2)(yxK.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.38
for function x = x(y):
144. yU (^y + y)(yxf
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.61
for function x = x(y):
145. yL = (—kxy-2
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.63
for function x = x(y):
© 1995 by CRC Press, Inc.

146. !&, = (Аж-5/2у3/2 - 12xy-2)(y'xf.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.65
for function x = x(y):
147.
Assuming у as the independent variables, we obtain an equation of the form 2.4.2.67
for function x = x(y):
2.6.3. Equations of the Form
ylx = *Axnym(y'J J
Table 2.12 represents all solvable equations whose solutions are outlined in Subsection
2.6.3. The two-parameter families (in the space of parameters n, m, and /), one-parameter
families, and isolated points are represented in a consecutive fashion. Equations are ar-
arranged in accordance with the growth of /. The number of the equation sought is indicated
in the last column in this table.
!• V'L = Ax-m-2ymy'x - Ах-т-3ут+г, т ф -1.
Solution in the parametric form:
where A = =p(m + l)am+1b~m.
Vxx — AX У Ух АХ
Solution in the parametric form:
г ехр(тт2) dr + C2] ,
У = -y exp(Tr2) \j T-1 exp(Tr2) dr + C2]
3. y'L = Ax-™-2y™(yxf -AX-™-*y™+\yxf, тпф-2.
Solution in the parametric form:
where A = ±(m + 2)am+36-m.
© 1995 by CRC Press, Inc.

TABLE 2.12
Solvable cases of the equation y'^x = crAxnym(y'xI + Axn-1ym+1(yx)l~1
I
arbitrary
{1ф2)
arbitrary
arbitrary
arbitrary
(^1)
m + 3
m + 2
3n + 2
n + 1
1
1
1
3
~2
2
2
2
2
2
2
5
~2
3
3
3
m
arbitrary
(m ^ -1)
1-/
-2
0
arbitrary
(m^-1,-2)
0
arbitrary
(m ^ -1)
0
1
0
arbitrary
(m ф -1)
arbitrary
(m ф -1)
arbitrary
(m ^ -1)
arbitrary
(m ^ -1)
arbitrary
(m ^ -1)
-1
arbitrary
(m^-1,-2)
arbitrary
(m ^ -2)
arbitrary
(m ^ -2)
arbitrary
(m^-1,-3)
n
—m — 1
/-2
1
-1
1
arbitrary
(n^0,-l)
-m-2
arbitrary
(n ^ -1)
arbitrary
(n ^ 0, -2)
arbitrary
(n^0,-l)
arbitrary
(n^O)
m+1
-2m-2
m+1
2
0
arbitrary
(n^O)
1
-m-2
1
2
(T
-1
-1
-1
-1
m + 1
1
n
-1
j_
n
_2_
n
J_
n
-1
-1
-1
1
arbitrary
arbitrary
m + 1
-1
m + 1
m + 1
2
Equation
2.6.3.75
2.6.3.76
2.6.3.79
2.6.3.80
2.6.3.74
2.6.3.73
2.6.3.1
2.6.3.23
2.6.3.37
2.6.3.41
2.6.3.85
2.6.3.82
2.6.3.83
2.6.3.84
2.6.3.87
2.6.3.86
2.6.3.42
2.6.3.3
2.6.3.24
2.6.3.38
© 1995 by CRC Press, Inc.

TABLE 2.12 Continued
Solvable cases of the equation y^x = crAxnym(yxI + Axn-1ym+1(yx)l~1
/
0
0
0
0
0
0
0
0
1—1
1
1
1—1
1—1
1—1
1
1
1
1—1
1—1
1
1
1
1—1
1—1
1
1
1
h^
h^
h^
m
-3
-3
-3
0
0
0
0
0
-3
-2
-2
-2
-2
-2
-2
-2
-2
-2
-1
l
2
1
2
1
2
1
2
0
0
0
1
1
1
1
n
-1
г
2
2
-2
-1
-1
2
2
1
-2
-1
-1
-1
l
2
1
2
1
1
1
-1
-2
-1
l
2
1
-1
1
1
-4
-1
-2
1
a
2
-4
-1
-1
-2
-1
5
2
1
2
-1
1
2
arbitrary
-1
1
2
-1
arbitrary
-2
-1
-1
l
4
-1
-1
1
2
-1
arbitrary
-1
l
2
-2
-1
1
Equation
2.6.3.65
2.6.3.61
2.6.3.35
2.6.3.48
2.6.3.50
2.6.3.33
2.6.3.59
2.6.3.63
2.6.3.15
2.6.3.51
2.6.3.71
2.6.3.7
2.6.3.5
2.6.3.55
2.6.3.13
2.6.3.69
2.6.3.11
2.6.3.29
2.6.3.2
2.6.3.53
2.6.3.45
2.6.3.31
2.6.3.57
2.6.3.77
2.6.3.67
2.6.3.9
2.6.3.21
2.6.3.39
2.6.3.25
2.6.3.17
© 1995 by CRC Press, Inc.

TABLE 2.12 Continued
Solvable cases of the equation y'J,x = crAxnym(y'xI + Axn-1ym+1(yx)l~1
I
3
2
3
2
3
2
2
5
2
5
2
5
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
m
0
0
0
-1
-2
3
2
0
-5
-3
-3
-3
-2
-2
-2
-2
-2
-2
-2
3
2
3
2
1
2
0
0
0
0
0
0
0
0
-3
n
-1
1
2
1
0
1
1
1
2
-1
l
2
2
-1
-1
-1
0
l
2
1
2
-1
l
2
-1
-2
-1
-1
-1
l
2
1
1
2
1
a
-1
-2
1
arbitrary
-1
l
2
1
-2
2
-4
-1
arbitrary
-1
1
-1
-1
-1
l
2
1
2
-1
-1
-1
arbitrary
-1
l
2
2
arbitrary
-1
1
-1
Equation
2.6.3.27
2.6.3.43
2.6.3.19
2.6.3.81
2.6.3.28
2.6.3.44
2.6.3.20
2.6.3.22
2.6.3.52
2.6.3.54
2.6.3.26
2.6.3.72
2.6.3.8
2.6.3.6
2.6.3.4
2.6.3.46
2.6.3.78
2.6.3.40
2.6.3.56
2.6.3.32
2.6.3.14
2.6.3.16
2.6.3.70
2.6.3.30
2.6.3.12
2.6.3.58
2.6.3.68
2.6.3.10
2.6.3.18
2.6.3.47
© 1995 by CRC Press, Inc.

TABLE 2.12 Continued
Solvable cases of the equation y'? = crAxnym(y'xI + Axn-1ym+1(y'x)l~1
I
4
4
4
4
4
4
4
m
-2
-2
-2
l
2
2
5
1
1
n
-2
1
1
-2
1
-2
1
l
2
-1
1
2
1
4
2
~T
-1
2
Equation
2.6.3.66
2.6.3.34
2.6.3.49
2.6.3.62
2.6.3.60
2.6.3.36
2.6.3.64
4- v'L =
Solution in the parametric form:
A
x = - y exp(=FT2) | / т 1 exp(=FT2) <ir
-l
у = d \J г ехр(тт^) dr + C2
> /n the solutions of equations 5-12, the following notation is used:
f = exp(=FT2)dT + C2.
Solution in the parametric form:
x = C1 ехр(тт2)/-1, у = Ь[2т ± ех
where A = ±2b2.
V'L =
Solution in the parametric form:
x = а[2т ± ехр(тт2)/-1], у = d ехр(тт2)/-1,
where A = =F2a2.
V'L =
Solution in the parametric form:
x = d [2r/ ± exp(TT2)] ~\ y = bf [2тf ± ехр(Тт2)
where A = ±±-b2.
© 1995 by CRC Press, Inc.

8. vZx ^Ю y(yx)
Solution in the parametric form:
x = o/[2t/ ± exp(TT2)]-\ y = C1 [2т/ ± ехр(тт2)]~\
where A = ±-i-a2.
9- v'L = Axv'x - Av-
Solution in the parametric form:
х = ат, y = Ci[2t/ ± ехр(тт2)],
where A = =F2a~2.
Ю- y'L = Ax(y'xf - Ay(y'xf.
Solution in the parametric form:
ж = С1[2т/±ехр(тт2)]) у = Ьт,
where A = =f26~2.
П- V'L = 2Axy-2y'x - Ay-1.
Solution in the parametric form:
x = аС^т2! ± rexp(Tr2) ± /], у = bC^rf ± ехр(тт2)],
where A = ^\a~2b2.
12. y'x-x = Ax-1(yxK
Solution in the parametric form:
x = aCx[2Tf ± ехр(тт2)], у = bd[2T2f ± техр(тт2) ±
where A = ^\a2b~2.
> In the solutions of equations 13-22, the following notation is used:
E = ^t(t + 1) - ln(v/r + v/tTI) + C2,
13. !&, =
Solution in the parametric form:
x = aC\F~2, у
where A = —a~zl2b2.
© 1995 by CRC Press, Inc.

14. V'ix =
Solution in the parametric form
x =
у = bC^F~2,
where A = -a2b~zl2.
3y'x - Ay~2.
Solution in the parametric form:
x = aClF'1*
16.
where A = —2a 2b3.
Solution in the parametric form:
= bC{F
¦Зр-l./T + l
where A= -2a3
Solution in the parametric form:
x = aC-W^E
where A = a~2b~1.
t2F - E2),
18. y':x
Solution in the parametric form:
x = aC2F~\ у
where A = —а-16~2.
19. уг:х = Ах(УхK/2+Ау(у'хI/2.
Solution in the parametric form:
E-^rF2 + t2F - E2),
У = bCf.
?-i IT + 1
where A = 2 a
-2
("I)
1/2
© 1995 by CRC Press, Inc.

20.
Solution in the parametric form:
x =
у =
21. у?а = Ax-4yy'x - 2Ax~5y2.
Solution in the parametric form:
x = aClTE(TF2 + t2F - E2)'1, у = bC^TEF-^rF2 + t2F - E2)'1,
where A = -2a3b~1.
22. J&, = 2Ах2у-5(у'хK - Axy~\y'xf.
Solution in the parametric form:
x = aClrEF-^rF2 + t2F - E2)~\ у = bClTE(TF2 + t2F - E2)~\
where A = —2a~1b3.
23. y'^ = Axny'x + nAxn~1y, n ф -1.
Solution in the parametric form:
1
where A = (n + l)a-™-1/3.
24.
(yxf + y
Tib ~\~ A-
Solution in the parametric form:
тпф-2.
X = '
m + l
m+l
у = Ът ™+2
> In the solutions of equations 25-36, the following notation is used:
{tv + C2T~V for the upper sign,
вш(гЛпт) + Сг сов(гЛпт) for the lower sign,
1пт + С2 forv = 0,
{A + v)tv + A — v)Cit~v for the upper sign,
A — VC2) sin(jy In т) + (С2 + vC\) cos(v In t) for the lower sign,
© 1995 by CRC Press, Inc.

25- V'L = Ax-2yy'x - Ax~3y2.
Solution in the parametric form:
х = ат~2, y = br~2R~1Q,
where v = C\, A = ab~1; the solution is valid for all three cases of functions R and Q
given above.
26. y'ix = Ax*y-\y'xf - Axy-2(y'xJ.
Solution in the parametric form:
= Ьт~2,
where v = C\, A = a~1b; the solution is valid for all three cases of functions R and Q
given above.
27. y'L = Ax-^y'J3^ - Ax1/2
Solution in the parametric form:
where v = Сг, А= (——)
V b J
28. y'lx \ь/2
Solution in the parametric form:
/ 2b \ I/2
where v = C\, A = —
V a /
29. y'L = Axy-2y'x - Ay-1.
Solution in the parametric form:
x = aC\rR, у ¦¦
where A = a~2b2(l =F v2).
30. y'L =
Solution in the parametric form:
x = aCirQ, у =
where A = a2b~2(l =F v2).
© 1995 by CRC Press, Inc.

31. y'lx y^ y
Solution in the parametric form:
x = aT2R2, у = br2Q2,
where v = Cb A = a~1l2b1l2.
32. y'lx
Solution in the parametric form:
x = ar2Q2, у = bT2R2,
where v = Съ А = аг'2Ь-г'2.
33. у»я = Ax-1 - Ax-2y(y'x)-\
Solution in the parametric form:
x = ar2R2, y = bT2[Q2 + (lT
where v = d, A = 2а-16.
34. у':х
Solution in the parametric form:
x = aT2[Q2 + (lTv2)R% y = br2R2,
where v = C\, A = 2ab~1.
35. y»m = Ах2у~3 - Axy-2(yx)-\
Solution in the parametric form:
x = aClTR, у = bClT [Q2 + A т v2)Щ1/2,
where A = 4A т v2)a~AbA.
36. y'lx = Ax-2y{y'xf - Ax-3y2(y'xf.
Solution in the parametric form:
[ + (lTv2)R2]1/2, y = bClTR,
where A = 4A т v2)aAb~A.
© 1995 by CRC Press, Inc.

> In the solutions of equations 37-50, the following notation is used:
C\Jv{t) + C<2Xv{t) for the upper sign,
Z =
C\Iv(t) + C^Kyij) for the lower sign,
where Jv and Yv are Bessel functions, Iv and Kv are modified Bessel functions.
37- V'L = 2АхпУУ'х + nAaj"»2, n ф О, п ф -2.
Solution in the parametric form:
x =
у = 1>C?+1t-*'Z-1(tZ't + vZ),
where v = , A= a n lb .
n + 2 2
38- V'L = (m + l)Ax2ym(y'xK + 2Axym+1(y'xJ, тф-1, таф -З.
Solution in the parametric form:
x = aCim-2T-'*vZ-1{TZ'T + uZ), у = bdr2'2"',
m + 2 m + 3 i, m о
where v = , A = a~Yb~m~.
m + 3 ' 2
39- y'L = 1Ах~^уу'х - Ax~2y2.
Solution in the parametric form:
x = ClT2, у = bTZ~xZ'T,
where v = 0, A = -\b~x.
40. y'ix = Ax2y-2(y'xf - 2Axy-\y'xf.
Solution in the parametric form:
x = arZ-1^, у = ClT2,
w
here v = 0, A = -\a~x.
41. J&, = Axn(y'xK/2 + пАх^уШ1'2, пф0,пф-1.
Solution in the parametric form:
x = aC~\^-2\ у = ЪС1п+1т-2» \z~\tZ't + vZ) ±
where v =
(n + l)b
© 1995 by CRC Press, Inc.

42- V'L = (m + l)Axym(y'xM/2 + Aym+1(y'xK/2, m ф -1, m ф -2.
Solution in the parametric form:
x = aC*m-3T-*v \z~\tZ't + vZ) ± ^^2], у = bClT^,
where u=
m + 2 v 7 [ (m + 2)a
43. y^ =
Solution in the parametric form:
x = ClT\ у = b{rZ-xZ'T ± |r2),
where v = 0, A =-^-(-б)/2.
44. J&,
Solution in the parametric form:
x = a(TZ~xZ'T ± }т2), у = ClT\
where v = 0, A =-i(-a)-1/2.
45. !&, »^ ^
Solution in the parametric form:
where v = 0, A = -61/2.
46. y^
Solution in the parametric form:
x = ar2Z-\Z'Tf, y = C1Z~2,
where v = 0, A = —a1/2.
47. yl
Solution in the parametric form:
x = aZ-1BTZ'T±T2Z),
where v = 0, A = 4a.
48. у»я = Ax-2 -
Solution in the parametric form:
x = C1Z~1, y = bZ-1BTZ'T±T2Z),
where v = 0, A = 4b.
© 1995 by CRC Press, Inc.

49. y'^
Solution in the parametric form:
x = oCi[t2(Z;J + 2tZZ't ± t2Z2}, у = bdZ2,
where v = 0, A=^ab~1.
50. y':x = 2Ax'1 - Ах-*у(у'х)-\
Solution in the parametric form:
x = adZ2, у = ЪСг[т2{г'тJ + 2tZZ't ±t2Z2},
where v = 0, A = ^a~1b.
> In the solutions of equations 51-66, the following notation is used:
Г Ci J1/3(t) + C2Yi/3(t) for the upper sign,
I C\h/z{T) + C2K1/3(t) for the lower sign,
f/i = tZ't + \Z, U2 = Vl ± t2Z2, U3 = ±^t2Z3 - 2f/if/2)
where Ji/3 andY1/3 are Bessel functions, Ii/3 andK1/3 are modified Bessel functions.
51. у»я = Ax-2y-2y'x + 2Ах~3у-1.
Solution in the parametric form:
x = aC2TA/3Z2U-\ у =
where A = 2ab2.
52. y'ix = уЦух) + y\yx)
Solution in the parametric form:
x = aC-W-^Z-'U-'Us, у = bC2T^Z2U-\
where A = —2a2b.
53. y»m =
Solution in the parametric form:
x = aC-XT-^Z-2U2, у = bC2T-^3Z-2U~2Ul,
where A = ^p-jab1/2.
54. y'ix =
Solution in the parametric form:
x = аС-2т-А'3г-2Щ2и1 у = bClT-4/3Z-2U2,
where A = ^-g-
© 1995 by CRC Press, Inc.

55. y»m = 2Ах-1'2у-2у'я>
Solution in the parametric form:
x = aC$T-V3Z-2U?, у = bClT-^3Z-2U2,
where A = ±±a~1/2b2.
56. y'^
Solution in the parametric form
x = аС1Т-^г-2и2, у =
where A = т^аЧ'1/2.
57. !&, = Axy-1/^ + 2АУ1/2.
Solution in the parametric form:
x = adrWz-1!!!, у =
where A = 2a~41/2.
58. y'ix =
Solution in the parametric form:
x = aC?Ts/3Z-4Ul у =
where A = -2a1/26.
59. y'lx
Solution in the parametric form:
x = aClT-b'3Z-b'2Ul'2, у = bC?T-8'3Z-\Ui ± |-t2Z3[/i),
where A=^ra
60. y?m
Solution in the parametric form:
x = aC*T-s/3Z-\Ul ± ±t2Z3U{), у =
where А=^аЪ-*'ъ.
61. y?m
Solution in the parametric form:
x = aCfT-A/3Z-2Ul у = ЬС%т-*'3г-2(и$ ± f
where A = ±^a~5/44.
© 1995 by CRC Press, Inc.

62. у»я
Solution in the parametric form:
x
= aC%T-*'3Z-2(Ui ± fr2^I72, у =
where A = ±\aAb~bl2.
63. yxx -
Solution in the parametric form:
x = aClT2^ZU-1/2, у = bCtT-A'zZ-2U~2{Ul - 4[/f),
where A = -^-a~4b.
64. у':х
Solution in the parametric form:
т — nC^T~^ 67' Z71' 2(TT2 ATT6') ii —hC
where A = — ¦^-ab~4:.
65. yxx = 2Ах~гу~3 + Ax~2y~2{y'x)~1.
Solution in the parametric form:
.-„лМ/^^г1 i, — hr -r-2/37-1n- 1(tt2 att3\1^2
X — UL>1T Zj U2 , у — OOiT Zj U2 \U3—4:U2) ,
where A = —^ra~1bA.
66. уxX = Ax 2y 2{y'x) -
Solution in the parametric form:
x = aCtr-^Z-1^1 (U2 - 4f/|I/2) у =
where A=%-aAb~1.
> In the solutions of equations 67-72, the following notation is used:
M = Ci*(A, -i-, ±т) + С2Ф(А, |, ±т),
where Ф and Ф are linearly-independent solutions of the degenerate hypergeornetric
equation
tm';t + {\± t)m't - am = o.
The function Ф = Ф(А, -j> ±T) can be expressed in terms of a degenerate hypergeomet-
ric series (see Equation 2.1.2.65).
© 1995 by CRC Press, Inc.

67- V'L = А1ХУ'Х + А2У-
Solution in the parametric form:
x = ат1/2, у = M,
where AY = ±2a~2, A2 = ±4aA.
68. yL
Solution in the parametric form:
x = M, y = Ът1/2,
where Ax = т46А, A2 = T26.
69- y'L =
Solution in the parametric form:
x = M, y = ±Ът1/2М'т,
where Ax = тЬ2Х, A2 = ±b2(X + \).
70- y'L = Mx-\y'xf + A2x-2y(y'xJ-
Solution in the parametric form:
x = ±ат1/2М'т, у = М,
where Ax = =F«2(A + \), A2 = ±a2X.
71. у»я = AiX-iy-iy^ + А2х~2у-1.
Solution in the parametric form:
x = M~\ у = ±Ът1/2М-1М'т,
where Аг = ±Ъ2\, А2 = ±\Ъ2.
72. y'L = А1Х-*у-\ухK + A2x-2y-Hy'xJ-
Solution in the parametric form:
x = ±ат1/2М-1М'т, у = М~\
where Ax = т\а2, А2 = =Fa2A.
Зтг+2 2тг+1
73. y'L = АхпЮ n+1 + nAxn-1y{y'x) гь+i , n ф 0, п ф -1.
Solution in the parametric form:
, , n+l n+1 i_n,_2/ nb(i
where к = , A=—ттт-а о
n nzp V a
© 1995 by CRC Press, Inc.

тгг+3 1
74. y^ = А(т + 1)хут(у'х)™+2 + Аут+* (у'х) ™+* , т ф -1, т ф -2.
Solution in the parametric form:
m + 2 m + 2 2, Г a(m + Ш1 "»+2
where /c = -, A = —-. ттт7«~ Ь\
m + 1 (m + lJ/3 [
75. у^ = АЖ-"г-1у"г(у4)г
Solution in the parametric form:
1-2 f dr\ , _ -J52 / 1-2 f dr
where
76. y^ = Aa:'-2»1-'^I - Aa:'-»»2"'^I-1, * 7^ 2.
Solution in the parametric form:
(т+ /
\ J
= B - /) [/3 + е(г)т] ^ - 1, A = B -
( -?- ), у = С2ехр(т+ / ^
where
1°. Solution with Аф\:
у = clXA +
A +
2°. Solution with A = 1:
у = a?(Ci + Сг
78. y'ix =
1°. Solution with i^ 1:
2°. Solution with A = 1:
© 1995 by CRC Press, Inc.
C1
= ciyA + yt

> In the solutions of equations 79-80, the following notation is used:
1п|т| + С2 if/3 = 0.
79. y'L
Solution in the parametric form:
X = i
ехрГ- Jт^1 f~x dr\ y = bC1exp(- J т^Г1 dr\
where /3=|—1, Л = -а'-3Ь3-'.
80. y'L =
Solution in the parametric form
X = i
J>-* Г
where /3 = -^—j, A = -a1'1^
si. у4/я, = а1»-1(»4J + а2Я!-1»;.
1°. Solution with Ai ^ 1, A2 7^ -1:
y=(C7i
2°. Solution with ^i ^ 1, A2 = -1:
y = (C
3°. Solution with ^i = 1, A2 ф -1:
y = C
4°. Solution with Ax = 1, A2 = -1:
> In the solutions of equations 82-84, the following notation is used:
r.k+1
-T
© 1995 by CRC Press, Inc.

82- V'L = Axm+1ym(y'xJ - Axmym+1y'x, m ф -1.
Solution in the parametric form:
к = т.
83. у»я = Ах-2т-2ут(у'хJ - Ax-2m-zym+1y'x, m ф -1.
Solution in the parametric form:
x = ClT-1/2U1/2, у = C2U, k = m.
ггг+1 ггг+3
84. yxx = Ax 2 ym(y'x) —Ax 2 ут+1у'х, т ф—1.
Solution in the parametric form:
m + 3
x
= C2XU, y = C1T~1'2U1'2, k = —
85. yxx = Ахпут(ух) — Axn ym y'x, тп ф —1, n ф 0.
Solution in the parametric form:
_, f f dr \ i. f, f dr \ n
x = Ci exp I / —— I, у = C^t exp I /c / —— I, к = —-
m+1 '
where i71 = -F(t) is the solution of the transcendental equation
exp
(ir+ jfc-l)fc-i ^' -\m + l
Solution:
у = Ci exp [ /expf—xn) (F + С2)~Чх I,
where F= / A - ^irrn)exp( —xn ) dx.
87. y" ^ Aij/rTl(y/) + А2Ж 1yrn^1yf 1 тп ф —1.
Solution:
x = C1e,P[Je,P(-^Ty^)(F + C2)-4y],
where F = /A + A2ym+1) expf Ц-2/т+1) rf2/-
J Vm + 1/
© 1995 by CRC Press, Inc.

TABLE 2.13
Solvable cases of the equation y'? = A^x™1 ymi (y'x) x +
А2хП2ут2(у'хI2,11ф12
h
arbitrary
arbitrary
arbitrary
arbitrary
1
1
1
2
2
2
2
2
2
5
2
3
3
3
3
3
h
arbitrary
arbitrary
arbitrary
3-Zi
0
0
0
0
1
1
1
1
1
l
2
0
1
2
2
2
arbitrary
0
1-Zi
1-Zi
1
2
1
2
1
2
-1
arbitrary
arbitrary
(mi + -1)
arbitrary
(mi + -1)
-1
-1
arbitrary
arbitrary*
(mi ^ -2)
arbitrary
0
0
1
TO2
mi
0
1-/2
Zi-2
l
2
1
2
0
1
arbitrary
0
0
0
0
mi+ 2
mi + 3
arbitrary
0
0
0
0
arbitrary
Zi-2
Zi-2
0
0
0
arbitrary
0
0
0
arbitrary
0
mi +2
mi3
1
l
2
0
1
2
П2
0
ГЫ
h-2
1-Zi
0
1
0
arbitrary
-1
arbitrary
(n2 + -1)
-1
arbitrary
arbitrary
(n2 + -1)
mi
mi
-1
l
2
1
2
1
2
Equation
2.6.4.3
2.6.4.4
2.6.4.9
2.6.4.8
2.6.4.1
2.6.4.14
2.6.4.16
2.6.4.18
2.6.4.13
2.6.4.5
2.6.4.7
2.6.4.12
2.6.4.6
2.6.4.11
2.6.4.10
2.6.4.19
2.6.4.2
2.6.4.17
2.6.4.15
* For mi = -2, see Equation 2.6.4.8 with I = 3
2.6.4. Other Equations (Zi Ф l2)
Table 2.13 represents all solvable equations whose solutions are outlined in Subsection
2.6.4. Equations are arranged in accordance with the growth of l\. The number of the
equation sought is indicated in the last column in this table.
Solution in the parametric form:
x = Ci
У =
A2
2Ai
© 1995 by CRC Press, Inc.

2. V'ix = AlX
Solution in the parametric form:
x = Uci exp(-A2r) + ^Lrl , y = C1 exp(-A2r) - -^r2 + C2
3. У^ = AlV™(yxt + A2y™(y'J\
1°. Solution in the parametric form with m ф —1:
/
where / = d + (m + 1) f t(AiT11 + A2T12)'1 dr.
2°. Solution in the parametric form with m = — 1:
x = C2+ f(A1Th+A2Th)~1efdT, у = ef,
where / = Ci f t(AiT11 + Ajt'2) dr.
4. J&, = А1Ж"(у4)г1 + А2Ж"(У;)г2.
1°. Solution in the parametric form with n 7^ —1:
ж = /"^П", У = С2+ I'т(А1Т11 + А2Т12)'1 f~~^ dr,
where / = d + (n + 1) /(Лт'1 + ^т'2) dr.
2°. Solution in the parametric form with n = — 1:
ж = е/, у = С2+ /' T(AlTh +A2Th)~1efdr,
/ = d I (AlTh + ^т'2) dr.
where
5- У! = А1УтШ2 + А2хпу'х, пф-1, тф-\.
Solution:
dx
6- V'L = Aiy-HvLf + А2хпу'х, п ф -1.
1°. Solution with Ax ф -1:
У=
ы
п + 1
2°. Solution with Ax = -1:
у = C2 exp I Ci / exp( —^-жп+1) da; I.
© 1995 by CRC Press, Inc.

7-
V'L =
УтШ + A2x-Iy'x,
1°. Solution with A2 Ф -1:
ф -1.
X =
8.
2°. Solution with A2 = -1:
у = C2 exp
Solution in the parametric form:
x = С2 exp
у = Ci exp
J exp
-A2-l
[т(А1т1+А2т3-1-т2+тУ14т].
9.
It ^= Л i T * 71 *-llt 1 —1—
Solution in the parametric form:
x = C<i exp
у = C\ exp
,-i
f T(AlTh + A2rh - t2 + t)

10. y'lx = Axm+3ym(yxf - Axmym+3, m ф -2.
Solution in the parametric form:
rr = CiT1/2expl - —
here V = т~ '
VdT
exp
ЗА
2m + 4
-4
т""ехр
m + 2
11. yl = A^+2y-(^M/2 + Ax™y™+2(y'xI/2.
Solution in the parametric form:
4
-1/2
x = CiT1/2expl - —
Vdr
VdT
where function V = V(r) is defined in the parametric form т = т(м), V^ = V(u) as
follows:
1) For тф-l, тф -3/2,
© 1995 by CRC Press, Inc.

where
7 _ / C2Jv(u) + Yv(u) for the upper sign,
\ C2IV (u) + Kv (u) for the lower sign,
Jv and Yv are Bessel functions, Iv and Kv are modified Bessel functions,
TO +
±
Bm+ 3)
8(m + lJ
2 g
2) For m = -1,
where Z = C2 Jo(w) + СзУо(^), Jo and ^o are Bessel functions;
3) For m = -3/2,
, V =
1A
2A
k + u~k
C2uk + u
2A
1 (C2 -/c)sin(/clnu) + (l + /cC2)cos(/clnu) 2 j
2A C2sin(A;lnu)+ cos(A;lnu) 8>
where /c =
12.
1°. Solution with n2 7^ —1:
у = Ciexp / udx ,
where
и = exp
xpf ^2 xn'+1) \c2 + f(l - Aia;ni)exp
2°. Solution with n2 = -1, A2 ^ -1, A2 ф -щ - 1:
у = d exp / жА2 (С2 + -т-^-у
3°. Solution with n2 = -1, A2 = -1:
4°. Solution with n2 = —1, A2 = —n\ — 1:
у = Ci exp
2 - -^-x~ni - Ax In ж
© 1995 by CRC Press, Inc.

V'L = AiVmi(y'xJ + A2x-1ym2y'x.
1°. Solution with mi ф —1:
x = Сi exp ( I udy ),
where
и = exp —
2°. Solution with mi = -1, Ax ф 1, ^i 7^ m2 + 1:
3°. Solution with mi = -1, Аг = 1:
ж = d exp \J у-1 (c2 + In у + -^У^У dy
4°. Solution with mi = —1, Ax = m2 + 1:
> Jn ifte solutions of equations Ц-15, the following notation is used:
{Схткх + С2тк2 + С3ткз if B2(8Bf + 27B2) < 0,
ClTekT + C2eCTT if &Bf + 27B2 = 0,
CiekT + C2epT cos lot if В2{Щ + 21B2) > 0,
fel + C2/c2Tfe2 + C3k3rks if B2(8Bf + 27B2) < 0,
Q = ) Ci(l + кт)тект + С2аеат if &Bf + 27B2 = 0,
{ dkekT + C2epT(pcos lot-tosin lot) if B2(8Bf + 27B2) > 0,
where k\, k2, and k3 (real numbers) or к and p ± iu> (one real and two complex
numbers) are the roots of the cubic equation
A3 - BiA2 - \B2 = 0.
In the special case 8??3 = —27B2, we have к = y-Bi (multiple root) and a = —-jBi
(simple root).
Remark. In the expressions for R and Q, constant C3 may be set to any nonzero
number (for example, one may set C3 =
14. y»m = Aiy-my>x + А2ХУ-1/2.
Solution in the parametric form:
x = R, y = Q2; B1 = A1, B2=A2.
Solution in the parametric form:
x = Q2, y = R; Вг = -А2, В2 = -Аг
© 1995 by CRC Press, Inc.

> In the solutions of equations 16-17, the following notation is used:
fe + С2т~к) + C3 if B\ + 2B2 > 0,
R = I CiTexp(-l-BiT) + C2 if B\ + 2B2 = 0,
{ exp(-l-BiT) cos(wt) + C2 if B\ + 2B2 < 0,
B1 + 2к)тк + C2(B1 - 2к)т~к] if B\ + 2B2 > 0,
Q = { Ci(Bit + 2) exp(-l-BiT) + C2 if B\ + 2B2 = 0,
exp(-l-BiT)[Bi cos(wt) - 2w вт(шт)] if B\ + 2B2 < 0,
, to =-±^-(B2+ 2B2).
Solution in the parametric form:
x = R, y=iQ2; Вг = Аг, В2=А2.
Solution in the parametric form:
x = \Q2, y = R; Вг = -А2, В2 = -А1.
I8- V'L = Aix^y-^y'J2 + A2xn*y.
Solution:
\ f wldx
у = d exp /
\ f
- /
L J
where w = w(x) is the general solution of the second order linear equation
(u,rni — \\w" — АлГ)лтПх~Х11)' 4- АттП2(АттП1 — Л2«> — П
19. y'lx = A
Solution:
v' dy
x = G'i exp I '
where w = w(y) is the general solution of the second order linear equation
{A2ym- + l)w'^ - A2m2yn*-Xw'y - Aiym^{A2ym- + lJw = 0.
2.7. Equations of the Form у"х = f(x)g(y)h(yx)
See Section 2.3 for the case
/(ж) = const ж™, g(y) = const ym, h(w) = const.
See Section 2.5 for the case
f(x) = const xn, g(y) = const ym, h(w) = const wl.
© 1995 by CRC Press, Inc.

2.7.1. Equations of the Form y?x = f(x)g(y)
See equation 2.4.2.4.
See equation 2.4.2.35.
3- V'L = «-2Fy + Ay-4).
See equation 2.4.2.31.
4. у^х = х
See equation 2.4.2.64.
5- V'L = ж
See equation 2.4.2.6.
6- у1 = Ж-2(-^гу +
See equation 2.4.2.26.
7- У1 = ж-2(-^у
See equation 2.4.2.10.
8- yl = *-2(fy + A
See equation 2.4.2.12.
9- У1 = ж-2(^у + А
See equation 2.4.2.66.
Ю- У1 = ж-2(-^гУ +
See equation 2.4.2.29.
11- У^ = ж-2(-|-у +
See equation 2.4.2.14.
12. у! = ж-2(-^у +
See equation 2.4.2.8.
13. у»я = х~2B0у + A
See equation 2.4.2.33.
© 1995 by CRC Press, Inc.

See equation 2.4.2.37.
15- V'L = x~2(Ay2 - JLy).
See equation 2.4.2.60.
16- V'L = x~2(Ay2 + -&V).
See equation 2.4.2.62.
17. у^ = Ж
See equation 2.4.2.40.
18- tC = (Ax4 + Bx3)y~7.
See equation 2.4.2.39.
19- V'L = (A*2 + B)y~5.
See equation 2.4.2.16.
20. y'lx = (Ax-1 + Bx~2)y-2.
See equation 2.4.2.28.
21. y?m = (Ax-7/3 + Bx-10/3)y~5/3.
See equation 2.4.2.48.
22. V'L = (Ax'4/3 + Bx-10/3)y~5/3.
See equation 2.4.2.49.
23. y^ = (Ax'4/3 + Вх-7'3)у-ъ13.
See equation 2.4.2.24.
24. y^ = (Ax'2/3 + Bx-4l3)y~bl3.
See equation 2.4.2.90.
25. y^ = (A + Bx~2/3)y-5/3.
See equation 2.4.2.89.
26. y^ = (Ax2 + B)y~5/3.
See equation 2.4.2.47.
27. !&, = (Ax2 + Bx)y~5/3.
See equation 2.4.2.46.
© 1995 by CRC Press, Inc.

28. y'lx = A(ax~2/3 + Ьх~5/3Jу-5/3.
This is a special case of equation 2.7.1.37 with с = 1, d = 0.
29. y'lx = (Ах~8/5 +
See equation 2.4.2.25.
30. y'lx = (Ах~5/2 +
See equation 2.4.2.23.
31. y'lx = A(ax5 + Ьж4)-1/^-1/2.
This is a special case of equation 2.7.1.38 with с = 1, d = 0.
32. y'lx = A(ax15/8
This is a special case of equation 2.7.1.39 with с = 1, d = 0.
33. y'lx = A(ax7/3 -
This is a special case of equation 2.7.1.40 with с = 1, d = 0.
34. y'lx = (ax2 + bx + c)y~5/3.
The transformation x = x(t), у = (x't) leads to a third order equation:
Differentiating the latter equation with respect to t and dividing it by x't, we obtain
a constant-coefficient fourth-order linear equation: 3rr"t"t = Aax + 2b.
35. y'lx = (ах~10/3 + bx~7'3 + сх-4/3)у~5/3.
The transformation x = 1/t, у = w/t leads to an equation of the form 2.7.1.34:
w"t = (at2 + bt + c)w~5/3.
36. y'lx = (ax2 +bx + c)ny-2n~3.
This is a special case of equation 2.9.1.9 with /(?) = ?~2n.
37. y'lx = A(ax + bJ(cx + d)-10/3y-5/3.
CLX I h 1]
The transformation ? = , w = leads to the Emden—Fowler equation
ex + d cx + d
of the form 2.3.1.9: w'^ = AA~2?2w~5/3, where A = ad - be.
38. y'lx = A(ax + b)-xl2(cx + d)-^-1/2.
CLX I h 1]
The transformation ? = , w = leads to the Emden—Fowler equation
ex + d ex + d
of the form 2.3.1.25: w'^ = M"^/2^^, where A = ad - be.
© 1995 by CRC Press, Inc.

39. !&, = A(ax + Ь)
CLX ~\~ Ь XI
The transformation ? = , w = leads to the Emden—Fowler equation
of the form 2.3.1.17: w'^ = AA^^^w'1^2, where A = ad - be.
40. !&, = A(ax + b)/ /
О IT I n 1]
The transformation ? = -, w = leads to the Emden—Fowler equation
ex ~\~ d ex ~\~ d
of the form 2.3.1.20: w'^ = AA~2^-15^w2, where A = ad - be.
- y'xx = ^4.ехр(ож2 + bx) exp(fey).
The sbubstitution kw = ky + ax2 + bx leads to an equation of the form 2.9.1.1: w'^x =
k 2ak-1.
2.7.2. Equations Containing Power Functions (h / Kmst)
See equation 2.6.2.116.
2- V'L = (-?*
See equation 2.6.2.117.
3- у':х = (-6х + Ах-*)
See equation 2.6.2.118.
4- V'L = (-12* +
See equation 2.6.2.119.
5- y':x = (-2x + Ax-2)
See equation 2.6.2.120.
6- V'L = (-k*
See equation 2.6.2.121.
See equation 2.6.2.122.
8- V'L = (-Tx
See equation 2.6.2.123.
9- V'L = (-^-x + Ax-
See equation 2.6.2.124.
© 1995 by CRC Press, Inc.

y'L = (ж
See equation 2.6.2.125.
See equation 2.6.2.126.
See equation 2.6.2.127.
- y'L = (-20*
See equation 2.6.2.128.
See equation 2.6.2.129.
See equation 2.6.2.130.
See equation 2.6.2.131.
y'L =
See equation 2.6.2.15.
See equation 2.6.2.111.
19- y'L = x
See equation 2.6.2.96.
20. y'L = x-2(Ay-i + By-2)(y'xK.
See equation 2.6.2.110.
21. y'L = ж
See equation 2.6.2.34.
22. y'L = ж-5/3(Ау~4/3 + Ву-10/3)(у'хK.
See equation 2.6.2.36.
23. y'L = х-ь'*(Ау-Ы* + By-V*)(y'xf.
See equation 2.6.2.14.
© 1995 by CRC Press, Inc.

24. V'L = x-5/3(Ay-2/3 + By-^)(y'xf.
See equation 2.6.2.115.
y»m = x-
25. y»m =
See equation 2.6.2.114.
26. у':х = х
See equation 2.6.2.35.
27.
See equation 2.6.2.33.
» = Ax-5/3(ay-2/3 + by~b^f (y'f
28. y»m = Ax-5/3(ay-2/3 + by~b^f (y'xf.
This is a special case of equation 2.7.2.37 with с = 1, d = 0.
29. !&, = x
See equation 2.6.2.109.
30. y^ = ж
See equation 2.6.2.13.
31. y^ = Ax-i/2(ay* + Ьу*)-1/2(у'хK.
This is a special case of equation 2.7.2.38 with с = 1, d = 0.
32. y^ =
This is a special case of equation 2.7.2.39 with с = 1, d = 0.
33.
This is a special case of equation 2.7.2.40 with с = 1, d = 0.
34. yL =
Assuming у as the independent variable, we obtain an equation of the form 2.7.1.34
x'yy ^
for function x = x(y): x'yy = —(ay2 + by +
35. y»m =
»m = ж-
Assuming у as the independent variable, we obtain an equation of the form 2.7.1.35
for function x = x(y): x'^y = -(ay-10/3 + Ьу~4/3 + c^'
36. у»я = x-2n-3(ay2 + by + c)n(y'xK.
Assuming у as the independent variable, we obtain an equation of the form 2.9.1.9
with /(?) = -?n for function x = x(y): x'^y = -(ay2 + by + c)nx~2n-3.
© 1995 by CRC Press, Inc.

37. y'lx = Ax-*'a(ay + ЬJ(су + d)
Assuming у as the independent variable, we obtain an equation of the form 2.7.1.37
for function x = x(y): а#„ = -A(ay + bJ(cy + //
)-2(y'f
38. y'ix = Ax-i/2(ay + b)-^(cy + d)-2(y'xf.
Assuming у as the independent variable, we obtain an equation of the form 2.7.1.38
for function x = x(y): а#„ = -A(ay + b)-^2(cy + d)-2x-^2.
39. y»m =
Assuming у as the independent variable, we obtain an equation of the form 2.7.1.39
for function x = x(y): а#„ = -A(ay + b)-^3(cy + d^^x-1/2.
40. y»m = Ax2 (ay + Ь)-15/Цсу + d)-20^(y'xf.
Assuming у as the independent variable, we obtain an equation of the form 2.7.1.40
for function x = x(y): з%у = -A(ay + Ь)-1Ъ'7{су + d)-20'7x2.
41. y'ix = Ax-i/2y~2 [(y'xJ + B2]
Solution in the parametric form:
x = a(u2 - 1)~\tu ± RJ, у = Ьт-^и2 - 1)/2,
where R = аД2 - 2т + d, и = Ttanh(c2 + f R^drj, A = -\a~xl2b2,
42. y'ix = Ax-V2y-2 [(y'xJ - B2]
Solution in the parametric form:
x = a(u2 + 1)~\tu ± RJ, у = Ът-^и2 + 1)/2,
where R = ^-т2 - 2т + Сь и = ±tan(c2 + I R'1 dr\ A = --l-
43. y'ix = Ax-i/2y~2 [B2 - (y'xf\
Solution in the parametric form:
x = a(l- u2)~\tu t RJ, У = br-^l - u2)~1/2,
where R = a/t2 - 2т + Сь и = ±tanh(c2 + f R^drj, A = -\a~xl2b2,
© 1995 by CRC Press, Inc.

44. y»m = Ах-*у[
Solution in the parametric form:
x = ат-^и2 - 1)/2, у = b(u2 - 1)~\ти ± RJ,
where R = л/т2 - 2т + d, и = Ttanh(c2 + j RT1 dr), A = -\azb~z/2,
В = 2a~1b.
45. y'ix = Ax-2y
Solution in the parametric form:
= ar-^l - u2)~1/2, y = b(l и2)~\
= b(l- и2)\ти
where R = ^т2-2т~1 + С1, и = ±tanh(c2 + /R'1 dr\ A = -±a3b~3/2,
В = 2а-гЬ.
46. y'ix = Ax-2y-i/2(y'xJ[B2 - (y'xJ}1/2.
Solution in the parametric form:
x = ar-^l + u2)~1/2, y = b(l + и2)~\ти ± RJ,
where R = s/-t2 - 2т + Сь и = ±tan(c2 + f Д dr), A = -^a^/2,
В = 2а-гЬ.
2.7.3. Equations Containing Exponential Functions (h / smst)
Preliminary Comments.
1. With / ф 1 — m, the equation
y>:x = Ae*ym(y>J A)
has the particular solution
у = BeXx, where A = -, В = (А\г)л.
1 — m — I
2. With m^O and / ф 1, equation A) can be reduced, with the aid of the transformation
to the generalized Emden—Fowler equation with respect to w = w(t):
1 2m+l
w"t = Bti-iw-1(w't) m , B)
l
where В = —m\A{l — /)] m . Equations of the form B) are outlined in Section 2.5.
© 1995 by CRC Press, Inc.

When obtained the general solution w = w(t) of the Emden—Fowler equation B), the
solution of the original equation A) can be written in the parametric form with the formulae
i_
x = lnw, у = k(w't) m,
i_
where к = [A(l - /)] m .
3. With / ^ n + 2, the equation
у>:х = АхпеУ(у>хI C)
has the particular solution
y = Xln(Bx), where A = / - n - 2, B=[
V
V A
4. Taking у as the independent variable and x as the dependent one, we obtain from
equation C) an equation of the form A) for x = x(y):
xyy
5. With пф—\ and 1ф1, equation C) can be reduced, with the aid of the transformation
t = {y'xI-\ u = xn+\
to the generalized Emden—Fowler equation for и = u(t):
< = ^—t^Tu-^T(u'tf. D)
It ~\ i-
Equations of this form are outlined in Section 2.5.
When obtained the general solution и = u(t) of the Emden—Fowler equation D), the
solution of the original equation C) can be written in the parametric form with the formulae
n+1
X = U n+1 , У = — ln(l4) +
1- V'L = Aex(v'J-
1°. Solution in the parametric form with / ф 1:
A{1-1)
y=Cl I ^A ± t) llt d
¦t + U2.
2°. Solution in the parametric form with / = 1:
A /
x = ln(±-^-V у = d — ехр(±т) dr + C2.
\ A / J т
© 1995 by CRC Press, Inc.

2.
V'L = Ae*ym(y'xJ.
1°. Solution in the parametric form with m
—1:
dr
where function / = /(т) is defined implicitly by the relation
т А
(L L_
V т m + 1
^m+l
.+ 1/ (m+l)/-r m + 1
2°. Solution with m = — 1:
da;
у = C2 exp
v'L = AeXv-
1°. Solution with A > 0:
where Jo and Kq are modified Bessel functions.
2°. Solution with A < 0:
у = С^оB^^Ае*/2) + C2Y0B^Ae*/2),
where Jo and lo are Bessel functions.
V'L = Ae*y-i/2(y'xK/2.
Solution in the parametric form:
x = t2 - ln(Af), y = C1 [2r/ - exp(r2)]2,
where / = /ехр(т2) dr + C<i-
V'L = Ae*y(y'xK/2.
Solution in the parametric form:
T+l
where / = ln(v/f
l) +C2.
V'L = АеУ(у'хI.
1°. Solution in the parametric form with / ф 2:
ж =
2°. Solution in the parametric form with 1 = 2:
dr + C2, y = ln(±^-
V A
© 1995 by CRC Press, Inc.

1°. Solution in the parametric form with n ф — 1:
1
x = т exp
„+iUT+aJ- »=Ут
where function / = /(т) is defined implicitly by the relation
fi__L
п+1
Vt n + 1/ (п + 1)/-т n+1
2°. Solution with n = -1:
x = C2 exp
-lnr
у - Ae» +
Solution in the parametric form:
ж = С1[2т/-ехр(т2)]2) у = т2
where / = / ехр(т2) dr + C2.
Solution in the parametric form:
r — 9Г72 I 1 f
X — ZOj ( 1 — J
where f =
т
+ C2.
1°. Solution in the parametric form with A > 0:
ж = CiJ0Bt) + С2УоBт), у = \h(t/VA),
where Jo and lo are Bessel functions.
2°. Solution in the parametric form with A < 0:
where Jo and Ко are modified Bessel functions.
11. Ух-х = Ае*еУ(ухI.
Solution in the parametric form:
ж =
where / =
2-/ 1-/
ln|r
lnlrl+d
+C
7i if / Ф l,:
if / = 1;
if / = 2.
© 1995 by CRC Press, Inc.

2 + Ьу)(у'K
12. у»я = Aexp(kx) exp(oy2 + Ьу)(у'хK.
Taking у as the independent variable, we obtain an equation of the form 2.7.1.41 for
function x = x(y): x'yy = —Аещ>(ау2 + by) ехр(/сж).
13. У1 = Ae*y-i/2(y'xK/2^y'x - IB.
Solution in the parametric form:
x = In[ат(coshи)~г], y = Bcosh и (т tanhи ± R) ,
where a =-A-1 B-1'2, R = л/21пт + т2 + Q, u = C2TfR~1dT.
14. y'lx = Aexy
Solution in the parametric form:
x = lnfai^cosu)], у = В cos2 и (т tan и ± RJ,
here a =-А-1 В-1/2, R= л/21пт - т2 + Q, u = C2TJR~1dT
15. y'lx = Ах-^еУух^ух - В.
Solution in the parametric form:
x = ——соъ2и{тЬа,пи± RJ, у =
2B
= A~1V2, R= а/21пт - т2 + Сь и = С2 Т J R^dr.
- у'х
Solution in the parametric form:
x = —— cosh и (rtanhu ± ДJ, у = In[Ьт(coshu) 1],
2B
where b = А'1 л/2, Д= а/21пт + т2 + Сь и = С2 Т / R^dr.
2.7.4. Equations Containing Hyperbolic Functions (h / S)nst)
Solution in the parametric form:
x = a cosh и (т tanh и ± Д), y = u/u>,
where A = а, Д= а/21пт + т2 + Съ u = C2TfR~1dT.
2. y^ = Ax[sinh(u>y)]-2yx-
Solution in the parametric form:
x = a sinh и (т coth и ± R), y = u/u>,
where A = a~2, R= а/21пт + t2 + d, u = C2 T / Д^.
© 1995 by CRC Press, Inc.

3- y'L
Solution in the parametric form:
) 1/2(tu±R), y = u>~1ln(u+\/u2
here A = 2a~2y/aU, R= ^С1-т2-2т~1, и = ±tan(C2 + / R'1 dr).
Solution in the parametric form:
x = a(u2 - 1)~1/2(tu ±R), y = ±u>~1 ln(u + \/v? - 1),
where A = ±2a-2v/otJ, R= ^/С1+т2-2т~1, и = Ttanh(C2 + /R'1 dr).
Solution in the parametric form:
x = и'1 ln(u + \Ju2 + 1), у = b(u2 + 1)~1/2(ти ± R),
hereA=-2b-2Vbu>, R= ^С1-т2-2т~1, и = ±tan(C2 + / R'1 dr).
6- yL
Solution in the parametric form:
x = ±u>~1 ln(u + \Ju2 - 1), у = b(u2 - 1)~1/2(ти ± R),
where A = T2b-2Vbu>, R= ^С1+т2-2т~1, и = Ttanh(C2 + / Д"
7- y'L = A[cosh(u>x)}-2y(y'xJ-
Solution in the parametric form:
= u/u>, у = 6 cosh u (rtanhu ± R),
wh
here A = -b~2, R= л/21пт + т2 + Сь и = С2 Т / R'1 dr.
8. y'ix = A[sinh(u,x
Solution in the parametric form:
= u/u>, у = b sinh и (т coth и ± R),
whereA=-b~2, R= а/21пт + т2 + Съ u = C2
© 1995 by CRC Press, Inc.

2.7.5. Equations Containing Trigonometric Functions (h / smst)
1.
2.
> In the solutions of equations 1-4, the following notation is used:
R = а/21пт-т2 + С1, и = C2 ± f R'1 dr.
yxx = Ax[cos{<jjy)]-2y'x.
Solution in the parametric form:
x = a cos и (rtanu ± R), y = u/ui, where A = a~ .
V'L = Ax[sin(u>y)}-2y'x.
Solution in the parametric form:
x = a sin и (rootи =F R), у = u/u>, where A = a .
v'L = а[со5(шх)]~2у(у'хJ-
Solution in the parametric form:
x = u>~1u, у = bcosu (rtanu ± R), where A = — b2.
4. y'lx = A[S\n(u:x)]-2y(yxy.
Solution in the parametric form:
x = to~1u, у = bsinu (rootи =F R), where A = — b2
> In the solutions of equations 5-8, the following notation is used:
R = vV2 - 2т + d , u = ±tanh|C2-
5.
3.
6.
7.
8.
Solution in the parametric form:
x = a(l — u2) (tutR), У = u~x arccosu, where A = 2a~2(—au>I^2.
Solution in the parametric form:
x = a(l — u2) (tu^f R), у = lo~1 arccosu, where A = 2a~2(atoI'2.
Solution in the parametric form:
x = lo~1 arccosu, у = b(l — u2) (
Solution in the parametric form:
x = lo~1 arccosu, у = b(l — u2
where A = — 2b~2(—btoI'2.
where A = -26FшI/2.
© 1995 by CRC Press, Inc.

2.7.6. Some Transformations
For the sake of visualization, we also use the symbolic notation {/, g, h} to denote the
equation
y'^ = f1(x)9l(y)h1(y'x). A)
1. Taking у as the independent variable and x as the dependent one, we obtain an
equation of the similar form for x = x(y):
XyV = gi{y)h{x)hl{x'y), where h^w) = -w3hi(l/w).
Denote this transformation by T¦
2. The Backlund transformation
x =
j h ,wy У= fi(x)dx, where w = y'x B)
leads to an equation of the similar form for function у = y(x):
У'к = h{x)g2{y)h2(y'x),
where the functions f2, g2, and h2 are defined in terms of the original functions /1, gi, and
hi parametrically by the formulae
* i-\ - f dw
J2(x) =w, x= ,
J hi(w)
1 _ r
92{y) = 7-7-7, У= / h{x)dx,
dgi _ 1
—г-, w= —-
Denote transformation B) by Q. For equations of the form A) wherein /1, gi, and hi are
power functions of their arguments, transformation Q (to a precision of constant factors)
is considered in Subsection 2.5.3. For equations A) with exponential functions /1 and gi,
transformation Q is discussed in Subsection 2.7.3.
When found the solution у = у(х) of the transformed equation, the formulae
make it possible to obtain the solution of the original equation A) in the parametric form
x = x(x), у = y(x).
3. The twofold application of transformation Q to the original equation yields an equa-
equation of the similar form:
У XX J О\ /УО\У} О\Уx)'
where the functions /3, g$, and ft.3 are defined in terms of the original functions /1, gi,
and hi parametrically by the formulae
1
1 - f wdw
9з(У) = —, У =
/wi
M
w ' J hi(w) '
dh = , ,_ч
© 1995 by CRC Press, Inc.

{si, /ь K}^/ / \ Х^{д2, Л, К)
{/ь 91, /»i}'
FIGURE 4
The treefold transformation Q yields the original equation.
Different compositions of transformations T and Q generate six different equations of
the analogous form which are shown in Figure 4.
1 и
4. In the special case g(y) = ym, h = 1, the transformation x = —, у = — leads to an
т т
equation of the similar form:
Denote this transformation by Ti.
For g(y) = ym and h = 1, different compositions of transformations JF, Q, and Ti generate
twelve different equations of the form A).
2.8. Some Nonlinear Equations with Arbitrary Parameters
2.8.1. Equations Containing Power Functions
bx + c)n.
This is a special case of equation 2.9.1.2 with /(?) = ?n.
2- V'L = (ay + Ьх2Г + с
The substitution aw = ay + bx1 leads to an equation of the form 2.9.1.1:
to" =anwn + c+—.
xx a
3. y'lx = Хх~2п-3(ху + a)n.
This is a special case of equation 2.9.1.7 with /(?) = A?n, 6 = с = 0.
4- У^ = (oa;2 + Ьж + С)У~5-
This is a special case of equation 2.9.1.9 with /(?) = ?~2.
© 1995 by CRC Press, Inc.

5. y'lx = A(ax + Ь)п(сх + d)-n-m~3ym.
CLX ~\~ Ь XI
The transformation ? = , w = leads to the Emden—Fowler equation
ex + a cx + a
(see Section 2.3): w'^ = AA~2?nwm, where A = ad - be.
6- V'L = cxmy-nk-m-3(ayn + bxn)k.
L
= c?-"fe-m-3(a?n + b)k
This is a special case of equation 2.9.1.5 with /(?) = c?-"fe-m-3(a?n + b)
7. y'lx = cxmy-2nk-2m-3(ay2n + bxnf.
This is a special case of equation 2.9.1.6 with /(?) = c?-2nfc-2m-3(a?2n +
8. у = (ay -\- bxy -\- ex -\- ay -\- /Зх + -у) , а ф 0.
The substitution 2au = 2ау + bx + a leads to an equation of the form 2.9.1.9:
u" = u~3f
XX v
VAx2 + Bx -
._. ло, л9 Ч-3/2 . Aac — b2 „ 2ав — ba „ 4a7 — a2
where /(?) = ? (a? +1) , .A = , В = ^
4a 2a 4a
9. y'lx = Xy~1/3 + (ax2 + bx + c)y~5/3.
The transformation x = x(t), у = (x't) leads to a third order equation:
9r'r'" _ (™"\2 — ± \(^'\2 _|_ JL (п'г2 хАт4-И
Differentiating the latter equation with respect to t and dividing it by x't, we arrive
at a constant-coefficient fourth-order linear equation: 3rr"t"t = 2Аж + 4аж + 2b.
10. y?m = Лж-^Зу-1/3 + (ах'10/3 + bx-7/3 + сЖ-4/3)у-5/3.
The transformation x = 1/t, у = w/t leads to an equation of the form 2.8.1.9:
w't't = Aw;/3 + (at2 + Ы + c)w~b'3.
+ У3 + ахпу = 0.
This is a special case of equation 2.9.2.1 with f(x) = axn.
12- y'L + B°У + bxn)y'x + bnx^y = 0.
This is a special case of equation 2.9.2.13 with f(x) = bxn.
13- y'L = axn(xy'x - yJ + bxm.
This is a special case of equation 2.9.3.4 with f(x) = bxm, g(x) = 0, h(x) = axn.
14. y'L = ax
This is a special case of equation 2.9.4.30 with f(x) = axn,
© 1995 by CRC Press, Inc.

15. !&, = ax-n-3yn(xy'x - у)т.
This is a special case of equation 2.9.4.31 with /(?) = a?m.
16. y^ = ax-^yny'x(xy'x - y)"\
This is a special case of equation 2.9.4.33 with f(y) = ay71, g(?)
This is a special case of equation 2.9.4.32 with /(?) = a?fe.
18. y^ = fca^y^A
The Legendre transformation x = w't, у = tw't — w, where w = w(t), leads to the
generalized Emden—Fowler equation
W>>t = li^T^)-.
A good deal of solvable equations of this type are outlined in Section 2.3 and Sec-
Section 2.5.
2n-{-m—nk
19- V'L = axn-1ym-1(y'x) »+"» (xy'x - y)k.
This is a special case of equation 2.9.4.36 with /(?) = a!;.
20- V'L = axn(xy'x -y) + bxm(xy'x - y)k.
This is a special case of equation 2.9.4.39 with f(x) = axn, g(x) = bxm.
21- XV'L = пУ'х + ах2п+г + bx2n+1ym.
This is a special case of equation 2.9.2.4 with f(y) = a + bym.
22- *y'L = ~(n + 1)|? + ож"-1 + bxnm+n~1ym.
This is a special case of equation 2.9.2.5 with /(?) = a + b?m.
23. yy^ - \ (y'xJ = ax2 + bx + с
The substitution у = у;4/3 leads to a special case of the equation 2.8.1.9 with A = 0:
4w'xX = 3(arr2 + bx + c)w-5/3.
24. Zyyxx - 2(y'xf = ax2 + bx + с
The substitution у = w3 leads to the equation 2.8.1.4: 9wxx = (ax2 + bx + c)w~5.
25. 1yy'lx = (у'хУ + bxny2 - a.
This is a special case of equation 2.9.3.7 with f(x) = —bxn.
© 1995 by CRC Press, Inc.

26- УУях = п(УхJ - ау4п~2 + Ъхту2.
This is a special case of equation 2.9.3.8 with f(x) = —bxm.
This is a special case of equation 2.9.3.9 with f(x) = —axk, g(x) = —bxm.
28. (n + 2)yy'x-x - (n + l)(y'xJ = (ax2 + bx + c)"
The substitution у = wn+2 leads to an equation of the form 2.7.1.36:
W*x = (n + 2J (^ + bx + cTw~2n~3-
29. yyxx = (y'xJ + axnyy'x + bxmy2.
This is a special case of equation 2.9.3.6 with f(x) = —axn, g(x) = —bxm.
30. ayyxx + b(y'xJ + (xn + X)myy'x = 0.
Solution: y^T = d f exp[-— f (xn + A)mdx\ dx + C2.
31. (y + ax)y'x-x = bxn(xy'x - yJ.
The substitution у = —ax + xz leads to the equation
/iTn+3('r/"i2 — П
— OX (Z~.) — U.
Having set w = z'x/z, we obtain the Beroulli equation
xw'x + 2w + x(l - bxn+2)w2 = 0.
32- x2y'x'x = n(n + l)y + ax3n+2 + bxnm+3n+2ym.
This is a special case of equation 2.9.1.11 with /(?) = a +
33- x<2y'L = fe(fe + !)У + ахкт+зк+2(Ьх2к+1 + c)nym.
The transformation ? = bx2k+1 + c, w = yxk leads to the Emden—Fowler equation
(see Section 2.3)
34- x2y'L + ху'х = avn + ь-
This is a special case of equation 2.9.2.8 with f(y) = ayn + b.
35- x^y'L = -(n + m+ l)xy'x - nmy + axnk+n-2myk.
This is a special case of equation 2.9.2.9 with /(?) = а?к.
© 1995 by CRC Press, Inc.

36. x yxx + axyx + by = cxnym.
1 1 — a 1
The transformation x = ?а, у = fm, where a = ±—=, /3 = ±
D = A — аJ — 46, leads to the Emden—Fowler equation
whose solvable cases are outlined in Section 2.3.
37- *2yL = «(«
This is a special case of equation 2.9.4.37 with f(x) = axn~2.
38. (ax2 + b)y'^x + axy'x + cyn = 0.
This is a special case of equation 2.9.2.11 with f(y) = cyn.
39. (ay + Ьх2)Ух-х = 1.
This is a special case of equation 2.8.1.2 with n = —1, с = 0.
40. xyyxx = x(y'xJ — yy'x + axkys.
This is a special case of equation 2.9.4.14 with /(?) = a?, g(?) = 1, k = n — 1, s = m + 2.
Having set — = г4(ж), we obtain
У
~u'xuxxx + 3«J2 = (ax + b)(u'xf-
Taking и as the independent variable, we obtain a constant coefficient linear equation
for x = x(u): х'Ции = ax + b.
42.
The transformation ? = In
w'lc — w'c = a~2w~2.
x + a
x
, w = — leads to an equation of the form 2.2.1.7:
x
43. (y2 + ax2 + 2bx + cfy'^x + sy = 0.
Dividing by the coefficient of y'J.x and multiplying by ax(xy'x —y) + bBxy'x — y) + cy'x,
we arrive at a total differential equation. Integrating the latter, we obtain
2
(ax2 + 2bx + c)(y'xf - 2(ax + b)yy'x + ay2 + y2 + J2V+2bx + c = С
© 1995 by CRC Press, Inc.

44. (ax + bJ(cx + dJyxx = sy + A(ax + b)k(cx + dI m kym.
The transformation
ax + b \ ( ax + b\ m_i у
1 w = ' '
cx + a J \cx + a J ex + a
leads to an autonomous equation:
«4'? - Bn + 1)«4 + (n2+n- sA~2)w = AA-2wm,
к
where n = , A = ad — be.
m — 1
45. (o2 - *2)(b2 - y2)y':x + (о2 - x2)y(y'xJ = х(Ь2 - y2)y'x.
Solution: arcsin — = C\ + C2 arcsin —.
b b
46. (ayn + bxn)y'lx + cxn-3 = 0.
This is a special case of equation 2.9.1.5 with /(?) = —c(a?n + b)~1.
47. (ayn + Ъх^у'^ + суп~3 = 0.
This is a special case of equation 2.9.1.5 with /(?) = — c?n~3(a?n + b)~ .
48. (оу2гг + Ъхп)у'1х + cy2n~3 = 0.
This is a special case of equation 2.9.1.6 with /(?) = -c?2n~3«2n + 6)~\
49. (oy" + bxn)y'^x + cxmyn-m-3 = 0.
This is a special case of equation 2.9.1.5 with /(?) = -c?n~m~3«n + 6).
50. (оУ2гг + Ьхп)у'^х + cx™y2n-2m-3 _ 0
This is a special case of equation 2.9.1.6 with /(?) = -c?2n~2m~3«2n + 6).
51- (V'LT = ос(ху'х - y) + Cy'x + 7.
Differentiating the equation with respect to ж, we obtain
Equating the second factor to zero and integrating, we find
^ ^ 2 С0. B)
The integration constants Cj and the parameters a, C, and 7 are related by the
constraint 4Cf = /?Ci—aCo+7 which is obtained by substituting the above solution B)
into the original equation.
In addition, there is the solution that corresponds to the first factor in A):
C0, where /3Ci - aC0 + 7 = 0.
© 1995 by CRC Press, Inc.

2.8.2. Painleve Equations
Preliminary comments.
Painleve equations are met with in different fields of contemporary physics. They re-
resulted from solving the problem of extracting equation classes of the form
d2y „/ dy\
\ =R[x, y, -f-\,
V ax J
where R is a rational function of у and dy/dx with analytic coefficients whose integrals have
no movable critical points. This problem was solved in works by P. Painleve and B. Gambier.
A total of 50 equations were extracted, of which 44 equations have the general solutions
that are expressed in terms of elementary functions, solutions of some linear equations, or
solutions of the other six equations.
The six equations not integrated by Painleve and Gambier are referred to as Painleve
transcendents (or irreducible Painleve equations), and their solutions are called transcen-
transcendental Painleve functions. The absence of movable critical points makes it possible to use
solutions of Painleve transcendents as basic functions for representing solutions of other
nonlinear differential equations, along with quadratures and solutions of variable-coefficient
linear differential equations. The canonical forms of Painleve transcendents are specified
below.
1. First Painleve transcendent:
y'L = 6у2 + *• (!)
The solution of the first Painleve equation is a unique function of x. It can be presented,
in the vicinity of movable pole xq, in terms of the series
3=7
where xq and С are arbitrary constants; coefficients uj (j > 7) are uniquely defined in terms
of Xo and C.
For large values of \z\, the following asymptotic formulae holds:
y(aO ~ a^Vf-s5/4 - a, 12, b),
where the elliptic Weierstrass function p(?; 12, b) is defined implicitly by the integral
c =
л/4р3 -I2p-b
a and b are some constants.
The first Painleve transcendent A) is invariant with respect to stretching variables
x = Ax, у = A3y, where A5 = 1, i.e., it admits discrete symmetry of the fifth order.
2. Second Painleve transcendent:
v'L = 2У3 + xy + a. B)
© 1995 by CRC Press, Inc.

The solution of the second Painleve equation is a unique function of x. Denote the
solution by y(x, a) with fixed parameter a. Then, the following relation holds:
y(x,-a) = -y(x,a),
while solutions y(x, a) and y(x, a — 1) are related by the Backlund transformations
r ^ r ^ 2a-1
{1) {)+
Therefore, in order to study the general solution of equation B) with arbitrary a it is
sufficient to construct the solution for all a out of the band 0 < Re a < -j.
Three solution corresponding to a and a ± 1 are related by the rational formulae
(fi - I n I \ (Л.ч i I ?УгУ11 I ^J ex I 1 i I 10/^v 1 i?/
2(ya_i + ya)Byl + x) + 2a - 1
where ya stands for y(x,a).
Solutions y(x, a) and y(x, —a — 1) are related by the Backlunds transformations
y(x,a) = y(x, -a - 1) - ^
-a -
The second Painleve transcendent B) for all a = n + 4- (n is an integer) possesses
the one-parameter family of solutions which are generated by the general solution of the
Fuchsian equation
3=1
and are expressed in terms of Airy functions and their derivatives.
There exist rational solutions of the form у = R1/R2, where Д1 and Д2 are polynomials
in x, only when a = 0, ±1, ±2, ±3, ... A rational solution can be written as
1 dPa 1 dQa
У а =
Pa dx Qa dx '
where Pa and Qa are polynomials, a is an integer.
In the special case a = 1, we have
Pi = 1, Qi = x hence y\ =
x
In the special case a = 2, we have
о 1 Зж2
P2 = x, Q2 = x + 4 hence y2 = 5
x x6 + 4
The following recurrence formulae take place:
Pa+l = Qa, Qa+1 = [xQ*a + {Q'af - 4QeQ^]/Pe,
© 1995 by CRC Press, Inc.

where prime denotes differentiation with respect to x.
If we stretch and shift the variables in accordance with the rule x = s2x — 6e~10,
у = ey + e~5 and let a = 4e~15, equation B), in the limit e —>• 0, transforms to equation A).
3. Third Painleve transcendent:
V'L = ^~ - — + —("У2 + /3) + 7У3 + —• C)
у у x у
The solution of the third Painleve equation is a unique function of x.
Any solution of the Riccati equation
y'x = ky2 + kx у + с, D)
where к2 = 7, с2 = —6, к/3 + с(а — 2/с) = 0, is a solution of equation C). Substituting
и' 1
x = At, у = —-г2*-, where A2 = -—, into D), we obtain a linear equation:
ku kc
whose general solution is expressed in terms of Bessel functions:
U = T^k [C7i J_a_ (t) + CaFoL (tI .
L 2fe 2k
In some special cases, equation C) can be integrated by quadrature. Rewrite equa-
equation C) in the form of integro-differential relations by two ways
and
/ /. г с о "I
— / I o~ ~г ]У I ~г I ~г ^У I ^ CLZ. X — о . 10 1
у J \Уу j vy ^J
It is obvious from E) that for a = C = 'j = 6 = 0, the general solution has the form
у = CiX°2. Adding F) multiplied by 2 to E), we obtain
У J У v У " J V2/ 'J J У
Subtracting F) times 2 from E), we find
— ) —2—- + (—z—7у2)е2г+2( ay)ez = — 4 / (/ye2zy2 + aezy)dz. (8)
у J у \ у ' ^ у ' J
Substituting 6 = C = 0 into equation G) and 7 = a = 0 into equation (8), we arrive at
У J У
У J У Г У
© 1995 by CRC Press, Inc.

Equations (9) and A0) are integrable by elementary functions. Substituting у = е z/v
into (9), we obtain
(v'zf = 2av + j+(l + C1)v2. A1)
As a result find
2a
z(a2 hi x + 2aC In x + C2 - 7)
1
У= <
I Cx2m + KlXm + K2
if Ci + 1^0, C = 6 = 0,
sy sy^ ryj 1 I \]-t )
where C/0, Kx = — — -, K2 = , „ , m2 = 1 + Ci. Accordingly,
Oi -|- 1 4O A -|- Oi)
equation A0) is reduced to equation A1) with the substitution у = vez.
If C = —a and 6 = —7, the transformation у = e~%w brings equation C) to the equivalent
form
1 2a
w" -\ w' = sin w + 2j sin 2w.
x x
If we perform the transformation x = l+s2x, y=l+2ey, a = -ye, C= y?~6+2/3e~3,
7 = ^-e~6, 6 = —^-e~6, then equation C), in the limit e —>• 0, transforms to equation B).
4. Fourth Painleve transcendent:
y" = — 1 y3 -\- 4xy2 + 2(ж2 — <x)y -\ . A2)
1y 1 У
If we pass on to the new independent variable x = ez, the solutions are unique functions
of z.
The Laurent-series expansion of the solution of equation A2) in the vicinity of any pole
x = Xq has the form
00
m m . 2 \ nf \2 V* i \i
ж — Жо 3 *r^L
3—^
where m = ±1, С is an arbitrary number, and a,j (j > 3) are uniquely defined in terms of
a, C, xq, and C.
If the condition C + 2A + amJ = 0 , where m = ±1, is satisfied, then every solution of
the Riccati equation
y'x = my + 2mxy — 2A + am)
is simultaneously a solution of the fourth Painleve transcendent A2).
Equation A2) is invariant with respect to the transformation у = Xy, x = Xx, a = aX ,
f3 = J3, where A4 = 1. Two solutions of equation A2) corresponding to different values of
parameters a and C are related to each other by the Backlund transformations
1
. e,,2^ r,2 — _9/3
т{yxq
zsy
{' 2 sy2), P2 = -2J3,
^r{yxP
zsy
as-l- \pf, 4a = -2s - 2a - 3sp,
© 1995 by CRC Press, Inc.

where у = у(х,а,0), у = y(x,a,C), s is an arbitrary parameter.
If we perform the transformation x = 2~2/3ex — e~3, у = 22/3ey + e~3, a = —a— у
j3 = — y?~12, equation A2), in the limit e —>• 0, transforms to equation C).
5. Fifth Painleve transcendent:
1 2 [ay-\ 1+7 1
x1 V у J x
2y(y - 1) ж ж2 V у/ ж у-1
If we pass on to the new independent variable x = ez, the solutions are unique functions
of z.
Solutions of the fifth Painleve transcendent A3) corresponding to different values of the
parameters are related by two equalities
y(x, a, C, 7,6) = y(-x, a, C, -7,$),
y(x,a,C,j,6) = _ _a _—?-.
Having set x = e* in A3), we obtain
У
If 7 = 6 = 0, equation A4) is reduced, by means of integration, to a first order autonomous
equation:
which is readily integrable by quadrature.
If the condition
7 = v/z26(l+ ч/
is satisfied, any solution of the Riccati equation
xy'x = V2ay2 + (V-26x-V2a- \/-2C)y+ V-2C A5)
is simultaneously a solution of the fifth Painleve transcendent A3). equation A5) can be
reduced to the degenerate hypergeometric equation 2.1.2.65.
If we perform the transformation у = 1 + ey, C = —e~2C, a = e~2C + e~1a, 7 = ?7,
6 = eS, equation A3), in the limit e —>• 0, transforms to equation C). In a similar manner,
as a result of the transformation у = V2ey, x = 1 + л/2 ex, a = \e~A, 7 = —?~4,
6 = — y?~4 — e~26, equation A3), in the limit e —>• 0, transforms to equation A2).
6. Sixth Painleve transcendent:
y" = —( 1 1 ^(y'J — ( 1 1 W
zy У — -1 У — x ж ж — i у — ж^
у(у-1)(у-ж) \_ , ах , ж-1 , сж(ж-1)
ж2(ж - IJ L У (У - 1) (У - ж)
In equation A6), the points x = 0, x = 1, and ж = оо are critical. Painleve found
two integrable case of the equation. First, ifa = /3 = 7 = <5 = 0, the general solution of
equation A6) has the form
У = EidU! + C2LO2, X),
© 1995 by CRC Press, Inc.

where E(u, x) is the elliptic function, defined by the integral
и =
dy
о y?y(y-l)(y-x) '
A7)
with periods 2u>i and 2u>i which are functions of x. Second, ifa = /3 = 7 = 0, 6 = -j, the
general solution of equation A6) has the form
у = E(w + C\ui\ + C2OJ2, x),
where w ф 0 is any particular solution of the linear equation
2x~1 . j , 1
x(x -
4x(x -
¦w = 0,
E is the elliptic function defined by formula A7).
Solutions of the sixth Painleve transcendent A6) corresponding to different values of
the parameters are related by three equalities
y(x, -13, -a, 7, 6) = —j -,
y( — , a, f3, 7, S)
V x J
1
y(x, -C, -7, a, 6) = 1 -
/ X X \ x
y[x, -13, -a, -6 + —, -7+— = — —.
V 2 2 У у(ж, а, 13, 7, 6)
The consecutive use of these equalities yields 24 equations of the form A6) with different
values of the parameters related by the known transformations.
All the solutions of the Riccati equation
Vx ~
x(x -l)V + x(x -l)V + x-1
are simultaneously solutions of equation A6) if \2a — a/—2/3 ф 1 and the condition
A8)
2-sfby, C/3 - a + 7 - 6) + 2a/-2/3 Ca - /3 - 7 + 6) + ^-a/
+ (a + C + 7 + 6f + 2 (a - /3 - 7 - 4a/3 - 2aj - 2C6) = 0
is satisfied (one should take such a value of a/—a/3 which coincides with i/a\J—C). In
equation A8),
A =
If we perform the transformation z = 1 + ex, 6 = e 26,
in the limit e —>• 0, transforms to equation A3).
= e lry — e 26, equation A6),
© 1995 by CRC Press, Inc.

2.8.3. Equation Containing Exponential Functions
1. y^x = X2y + oexp[A(n + 3)x]yn.
This is a special case of equation 2.9.1.15 with /(?) = a?n.
2- V'L = х2У + ae»xym, А ф 0.
The transformation ? = e2Xx, w = yeXx leads to the Emden—Fowler equation (see
Section 2.3)
го'' = —rr?nw;m, where n =
¦« " 4A2 ч ' w c c " " 2A
3- y'L = х2У + aex^m+^x(be2Xx + c)nym, X ф О.
The transformation ? = be2Xx + c, w = yeXx leads to the Emden—Fowler equation
(see Section 2.3)
4. y'L = X2y + Аех(т+^х(ае2Хх + b)n(ce2Xx + d) n m 3yn.
The transformation
ae2Xx + b _ yeXx
ce2Xx + d ' ce2Xx + d
leads to the Emden—Fowler equation (see Section 2.3)
w'^ = ABAX)~2Cwm, where A = ad -be.
5. y'lx = ay'x + be2axyn.
This is a special case of equation 2.9.2.17 with f(y) = byn.
This is a special case of equation 2.9.2.18 with /(?) = б^™.
7. у" -\- ay' -\- by = ceXxyrn.
The substitution ? = ex leads to an equation of the form 2.8.1.36:
8- V'L = -(A* + »)У'Х
This is a special case of equation 2.9.2.19 with /(?) = a^n-1.
9- y'L = ХУ'Х + ЬхУ + ае2Хху~3.
This is a special case of equation 2.9.2.20 with f(x) = —bx.
Ю- y'L = ХУ'Х + Ье>ххУ + ae2Xxy~3.
This is a special case of equation 2.9.2.20 with f(x) = —be]xx.
© 1995 by CRC Press, Inc.

• V'L = ay'x + b ехрBож + cyn).
This is a special case of equation 2.9.2.17 with f(y) = bexp(cyn).
This is a special case of equation 2.9.2.1 with f(x) = aeXx.
13. yxx = axeyy'x + aey.
Solution: у = dx - ln(-a / xeClXdx + C2).
14. y" = 2aexyy' -\- aexy2.
Solution in the parametric form:
where Z = CiJi(r) + C2Fi(r) or Z = CJ^t) + C2.Ki(t), Л and Fi are Bessel
functions, I\ and ATi are modified Bessel functions.
15. y'lx = axnevy'x + anxn~1ev.
Г f 1
Solution: у = C\x — In C2 — a / ж™ ещ>(Сгх) dx .
L J J
16. у»я = ае'у-1/^ + y
Solution in the parametric form:
x = ln(±-^-/J т г2, у = C\ [2r ± exp(Tr2)/]'
-i-i
where / = I / exp(=Fr2) dr + C2
beAa;)y; + XbeXxy = 0.
Integrating the equation, we obtain the Riccati equation:
y'x + ay2 + beXxy = С
Solution:
To the limiting case C2 —>• —1 corresponds у = —x — ln(Ci — ax).
This is a special case of equation 2.9.3.16 with f(x) = —cxn.
© 1995 by CRC Press, Inc.

20. у^х = a(y'xJ - be4ay + ceXx.
This is a special case of equation 2.9.3.16 with f(x) = —ceXx.
21- V'L = аЮ2 + ЬхпеаУ + cxm.
This is a special case of equation 2.9.3.15 with f(x) = —bxn, g(x) = —cxm.
22- V'L = a(v'xJ + beay+cx + kxm.
This is a special case of equation 2.9.3.15 with f(x) = —becx, g(x) = —kxm.
23. V'L = аЮ2 + Ь
This is a special case of equation 2.9.3.15 with f(x) = —beXx, g(x) = —ce^x
24. y'L + ayn(y'xJ + Ье*У + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = ayn, g(y) = beXy + с
25. y'L = ayn(y'xJ + beXxy'x.
This is a special case of equation 2.9.3.2 with f(y) = ayn, g(x) = beXx.
26. у'х'х +
This is a special case of equation 2.9.3.1 with f(y) = aeXy, g(y) = byn + с
27. у'х'х +
This is a special case of equation 2.9.3.1 with f(y) = aeXy, g(y) = 6eM3/ + с
28. y'L =
This is a special case of equation 2.9.3.2 with f(y) = aeXy, g(x) = bxn.
29. y'ix = аехУ(у'хJ + Ье»ху'х.
This is a special case of equation 2.9.3.2 with f(y) = aeXy, g(x) = belxx.
30. y'lx = aeXx(xy'x - yJ +
This is a special case of equation 2.9.3.4 with f(x) = 6емж, g(x) = 0, h(x) = aeXx.
31- V'L + beaxym(y'xf + ay'x = 0.
This is a special case of equation 2.9.3.24 with f(y) = bym.
32. y'L + Ьеах+хУ(у'хK + ay'x = 0.
This is a special case of equation 2.9.3.24 with f(y) = beXy.
33. y'L = аех(у'хУ + аеху(у'хУ.
a I yeCiy dy +
© 1995 by CRC Press, Inc.

34. у»х (ух) + (ух)
Solution in the parametric form:
= \пт.
35. y'lx = ах2еУ(у'хK + 2ахеУ(у'х)
Solution in the parametric form:
where Z = C1J1(t) + C2Y1(t) or Z = Ci/i(t) + C2Ki(t), Л and Y-± are Bessel
functions, Ii and Ki are modified Bessel functions.
зб. y'L =
Solution in the parametric form:
Ж = С2[2т±ехр(Тт2)/]2,
-i-i
where / = / exp(=Fr2) di
37. y'L = anexyn-1(y'xf + aexyn(yxJ.
Solution: x = C1y-lnfa fyneClVdy + C2j.
38. у':х = ае
Solution:
x = -
+ л
1 — G2
To the limiting case C2 —>• 1 corresponds x = —y — ln(Ci + ay).
39. y'lx = aex(y'x) ' + at
Solution in the parametric form:
x = 1пт2, у = -2o-'1t-\Z-x(tZ't + 2Z) T }r2],
where
Г C\J2(t) + C2Y2(t) for the upper sign,
1_ С\12(т) + С2К2{т) for the lower sign,
J2 and Y2 are Bessel functions, I2 and K2 are modified Bessel functions.
© 1995 by CRC Press, Inc.

40. y'L = ахеУ(у'хM/2 + аеУ(у'хK/2.
Solution in the parametric form:
x = -2a~2T~i[Z-1(TZ'T + 2Z)T }т2], у = 1пт2,
where
f C\J2(t) + C2Y2(t) for the upper sign,
I C\I2(t) + C2-K2(j) for the lower sign,
J2 and У2 are Bessel functions, I2 and K2 are modified Bessel functions.
41. y'^ = -ay'x + beamxyk(y'x)m+2.
This is a special case of equation 2.9.4.49 with f(y) = —byk, n = m + 2.
42. yxx = — —y'x + beaxym~k+1(y'x)k.
This is a special case of equation 2.9.4.52 with /(?) = 6?.
43. y'lx = y'x + Aexp[(n + 2 - l)x]ym(y'J.
The substitution ? = ex leads to the generalized Emden—Fowler equation
which is outlined in Section 2.5.
44. y'ix = -{y'xf + Axn exp[(m + I - l)y] {y'J.
The substitution w = ev leads to the generalized Emden—Fowler equation
which is outlined in Section 2.5.
This is a special case of equation 2.9.4.51 with /(?) = b?.
46- у':х = аУпШ2
This is a special case of equation 2.9.4.10 with f(y) = ayn, g(y) = beXy.
47. yL =
This is a special case of equation 2.9.4.10 with f(y) = aeXy, g(y) = by11.
48. у»я = аехУ(у'хJ + Ье™(у'х)к.
This is a special case of equation 2.9.4.10 with f(y) = aeXy', g(y) = 6eM3/.
© 1995 by CRC Press, Inc.

49. y'lx = axn(xy'x - у) + ЬеХх(ху'х - y)k.
This is a special case of equation 2.9.4.39 with f(x) = axn, g(x) = beXx.
50. y'ix = aeXx(xy'x - y) + bxn(xy'x - y)k.
This is a special case of equation 2.9.4.39 with f(x) = aeXx, g(x) = bxn.
Solution in the parametric form:
= т - I — - C2,
у =
where F = [a{2 - l)eT + d] l~2 + 1.
52- V'L = aeXx(xy'x -
'x - y)k.
This is a special case of equation 2.9.4.39 with /(ж) = aeXx, g{x) = be1111.
53-
This equation is encountered in the combustion theory and hydrodynamics.
The transformation ? = In x, w = Xy + (n+1) In x leads to an autonomous equation
of the form 2.9.1.1: w'L = aXew. Having integrated the latter equation, we obtain
the solution of the original equation in the parametric form:
я = exp[C1±/(*)], y=i.-IL±l[C71±/(*)],
where
/(*) =
1 , VC2 + 2aAe* -
m 7.
2aAe*
- arctan
2aAe*
if C2 > 0,
if C2 = 0,
if C2 < 0.
This is a special case of equation 2.9.2.4 with f(y) = aeXy.
55. xy'ix = ny'x + ax2n+1 exp(Ay™).
This is a special case of equation 2.9.2.4 with f(y) = aexp(Aym).
56.
The transformation ? = жу^, w = xn~m+1 eXy after dividing by w leads to a first order
linear equation:
a(;mw'c = AC + n - m + 1.
© 1995 by CRC Press, Inc.

This is a special case of equation 2.9.4.11 with f(y) = aeXy.
58. xv" + v' = (axneXy + bxm~
The transformation ? = xy'x, w = xn~m+1eXy leads to a first order equation with
separation of variables:
Cm(aw + b)w'( = (AC + n - m + l)w.
59- 2vv'L = Ю2 + beXxv2 - a-
This is a special case of equation 2.9.3.7 with f(x) = —beXx.
60- VV'L = n(v'J2 ~ ay4n~2 + beXxy2.
This is a special case of equation 2.9.3.8 with f(x) = —beXx.
This is a special case of equation 2.9.3.9 with f(x) = —axm, g(x) = —beXx.
This is a special case of equation 2.9.3.9 with f(x) = —aeXx, g(x) = —bxm.
63. yy'L = n(v'xJ + o,eXxy2 + Ье>лхуп+1.
This is a special case of equation 2.9.3.9 with f(x) = —aeXx, g(x) = —be]xx.
64. yy'L = (v'xJ + axnyy'x + beXxy2.
This is a special case of equation 2.9.3.6 with f(x) = —axn, g(x) = —beXx.
65. yy'L = Ю2 + aeXxyy'x + bxny2.
This is a special case of equation 2.9.3.6 with f(x) = —aeXx, g(x) = —bxn.
66. yy'L = Ю2 + aeXxyy'x + be»xy2.
This is a special case of equation 2.9.3.6 with f(x) = —aeXx, g(x) = —belxx.
67. yy'L = Ю2 + beaxyn(y'x)k.
This is a special case of equation 2.9.4.53 with /(?) = b?, g(() = (k, n = m — к + 2.
68. yy'lx = (y'xJ + (aeXxyn + Ьу2~т)(ух)т.
The transformation ? = y'x/y, w = eXxyn+m~2 leads to a first order equation with
separation of variables:
© 1995 by CRC Press, Inc.

69- x2v'L = axn+2ey + n.
This is a special case of equation 2.9.1.17 with /(?) = a?.
70. x2y'x[x + xy'x = aeXy + b.
This is a special case of equation 2.9.2.8 with f(y) = aeXy + b.
71- x2iga + xy'x = кхпеаУ + b.
This is a special case of equation 2.9.2.23 with /(?) = k^ + b.
72. (ax2 + b)y'lx + axy'x + ceXy = 0.
This is a special case of equation 2.9.2.11 with f(y) = ceXy.
73. (ae2x + b)y'lx + ae2xy'x + cyn = 0.
This is a special case of equation 2.9.2.14 with g(x) = ae2x + b, f(y) = —cyn.
74. (ae2x + b)y'lx + ae2xy'x + ceXy = 0.
This is a special case of equation 2.9.2.14 with g(x) = ae2x + b, f(y) = —ceXy.
2.8.4. Equations Containing Hyperbolic Functions
1. y'lx = \2y + a(cosh\x)-n-3yn.
This is a special case of equation 2.9.1.21 with /(?) = a?n.
2- V'L = х2У + °(sin
This is a special case of equation 2.9.1.20 with /(?) = a?n.
3. yxx = X2y + asinhn(Xx)
у
The transformation ? = tanh(Arr), w = ;—r- leads to the Emden—Fowler equa-
cosh(Aa;)
tion
which is outlined in Section 2.3.
4. yxx = X2y + acoshn(Xx) sinh~"~r"~3(A«)yr".
у
The transformation ? = cothfArr), w = -,—r- leads to the Emden—Fowler equa-
sinh(Aa;)
tion
w'pp = <xA~ ^nwm
which is outlined in Section 2.3.
5. y'^x = bcosh(Aa;)y + ay~3.
This is a special case of equation 2.9.1.12 with f(x) = — 6cosh(Arr).
© 1995 by CRC Press, Inc.

6- v'L = bsinh(Aa;)y + ay~3.
This is a special case of equation 2.9.1.12 with f(x) = —6sinh(Aa;).
7- v'L + 3W'X + V3 + ° cosh(Aa;)y = 0.
This is a special case of equation 2.9.2.1 with f(x) = acosh(Arr).
8- V'L + 3УУ'Х + У3 + (а sinh x + b)y = 0.
This is a special case of equation 2.9.2.1 with f(x) = a sinh ж + b.
9- V'L + a.yn(y'xf + b cosh™ у + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = ayn, g(y) = 6coshm у + с.
Ю- V'L + аУпЮ2 + b tanh™ у + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = ayn, g(y) = 6tanhm у + с.
11- y'L = ayn(y'S + bsinhm(Xx)yx.
This is a special case of equation 2.9.3.2 with f(y) = ayn, g(x) = 6sinhm(Aa;).
12. y'L = ayn(y'xJ +btanhm(\x)y'x.
This is a special case of equation 2.9.3.2 with f(y) = ayn, g(x) = 6tanhm(Aa;).
13- y'L + a cosh" У Ю2 + bym + c = 0.
This is a special case of equation 2.9.3.1 with f(y) = a cosh™ y, g(y) = bym + с
14- y'L + о cosh" У Ю2 + b cosh™(Ay) + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = acosh™ y, g(y) = bcoshm(\y) +c.
15. y'L = a sinh" у (у'хJ + bxmy'x.
This is a special case of equation 2.9.3.2 with f(y) = a sinh™ y, g(x) = bxm.
16. y'L = a sinh" у (у'хJ +bsinhm(\x)y'x.
This is a special case of equation 2.9.3.2 with f(y) = asinh"y, g(x) = 6sinhm(Aa:).
!7- y'L + a tanh" У (У'хJ + bym + c = 0.
This is a special case of equation 2.9.3.1 with f(y) = atanh™ y, g(y) = bym + с
18- y'L + atanhny(yxJ + btanh™(Ay) + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = atanh™ y, g(y) = 6tanhm(Ay) +c.
19. y'L = o, tanh" у (у'хJ + bxmy'x.
This is a special case of equation 2.9.3.2 with f(y) = atanhny, g(x) = bxm.
© 1995 by CRC Press, Inc.

20. y'L = a tanh" у (у'хJ + b tanh™(Aa;)y^.
This is a special case of equation 2.9.3.2 with f(y) = atanhny, g(x) = 6tanhm(Aa;).
21. y'L = ayn(y'xJ + bcosh™ у (у'х)к.
This is a special case of equation 2.9.4.10 with f(y) = ayn, g(y) = 6coshm y.
22. y'L = ayn(y'x? + btanhm у (у'х)к.
This is a special case of equation 2.9.4.10 with f(y) = ayn, g(y) = 6tanhm y.
23. y'L = a cosh" у (y'xJ + by™(y'x)k.
This is a special case of equation 2.9.4.10 with f(y) = a cosh™ y, g(y) = bym.
24. y'L = otanh" у (у'хJ + bym(y'x)k.
This is a special case of equation 2.9.4.10 with f(y) = atanh™ y, g(y) = bym.
25. xy'L = ny'x + ax2n+1 coshm(Xy).
This is a special case of equation 2.9.2.4 with f(y) = acoshm(Xy).
zd. xyxx — пУх "г ах sinn \Лу).
This is a special case of equation 2.9.2.4 with f(y) = asinhm(Ay).
27. xy'L = ny'x + ax2n+1 tanhm(Ay).
This is a special case of equation 2.9.2.4 with f(y) = atanhm(Ay).
28. xy'L = ny'x + ax2n+1 cothm(Xy).
This is a special case of equation 2.9.2.4 with f(y) = acothm(Ay).
29. 2yy^ = (y'xJ + bcoshm(Xx)y2 - a.
This is a special case of equation 2.9.3.7 with f(x) = —bcoshm(Xx).
30- VV'L = п(УхJ - ay4n~2 + bcoshm(Xx)y2.
This is a special case of equation 2.9.3.8 with f(x) = —bcoshm(Xx).
31. yy'L = n(y'xJ + axmy2 + b coshk (Xx)yn+1.
This is a special case of equation 2.9.3.9 with f(x) = —axm, g(x) = —bcoshk(Xx).
32- yy'L = n(y'xJ + axmy2 + bsinhk(Xx)yn+1.
This is a special case of equation 2.9.3.9 with f(x) = —axm, g(x) = —6sinhfe(Aa;).
33. yy'L = n(y'xJ + axmy2 + btanhfc(Aa;)y"+1.
This is a special case of equation 2.9.3.9 with f(x) = —axm, g(x) = —6tanh (Xx).
© 1995 by CRC Press, Inc.

34. W'L = Ю2 + axnyy'x + b coshm(\x)y2.
This is a special case of equation 2.9.3.6 with f(x) = —axn, g(x) = —bcoshm(Xx)
35- W'L = Ю + a coshn (Xx)yy'x + bxmy2.
This is a special case of equation 2.9.3.6 with f(x) = —acoshn(Xx), g(x) = —bx"
36. x2y'x[x + xy'x = a cosh"(Ay) + b.
This is a special case of equation 2.9.2.8 with /(y) = a cosh™ (Ay) + b.
37'• x2v'L + xy'x = a sinh"(Ay) + b.
This is a special case of equation 2.9.2.8 with f(y) = asinhn(Ay) + b.
38- x2v'L + xy'x = a tanh"(Ay) + b.
This is a special case of equation 2.9.2.8 with f(y) = atanhn(Ay) + b.
39- *2V'L + xy'x = a coth"(Ay) + b.
This is a special case of equation 2.9.2.8 with f(y) = acothn(Ay) + b.
40. (ax2 + b)y^, + axy'x + cosh" (Ay) + с = 0.
This is a special case of equation 2.9.2.11 with f(y) = cosh™ (Ay) + c.
41. (ax2 + b)y'lx + axy'x + sinh"(Ay) + с = 0.
This is a special case of equation 2.9.2.11 with /(y) = sinhn(Ay) + c.
42. (ax2 + b)y'lx + axy'x + tanh"(Ay) + с = 0.
This is a special case of equation 2.9.2.11 with /(y) = tanhn(Ay) + c.
43. (ax2 + b)y'lx + axy'x + coth"(Ay) + с = 0.
This is a special case of equation 2.9.2.11 with /(y) = cothn(Ay) + c.
2.8.5. Equations Containing Logarithmic Functions
1. yxx = ax~3(lny — In ж).
This is a special case of equation 2.9.1.5 with /(?) = aln?.
2. у^ = ож
This is a special case of equation 2.9.1.6 with /(?) = 2aln?.
3- У1 + oy"(y^2 + Ыпт у + с = 0.
This is a special case of equation 2.9.3.1 with /(y) = ayn, g(y) = 61nm у + с
© 1995 by CRC Press, Inc.

4- V'L = аУпЮ2 + b lnm(Xx)y'x.
This is a special case of equation 2.9.3.2 with f(y) = ayn, g(x) = 61пт(Аж).
5- V'L + о In" у (y'xJ + bym + c = O.
This is a special case of equation 2.9.3.1 with f(y) = a In™ y, g(y) = bym + с
6. у':х УЮ + ух
This is a special case of equation 2.9.3.2 with f(y) = a In™ y, g(x) = bxm.
7. y':x = a In" у (y'xJ + Ь \ът{\х)у'х.
This is a special case of equation 2.9.3.2 with f(y) = a In™ у, д(х) = 61пт(Аж)
8- y'L = ax-2y-Hy'J4 ~ 2ож-3 lny (y'xf.
Solution in the parametric form:
[2)]1/2) y = C1F,
V f "I —^
where F = ехр(тт2) / ехр(тт2)dr + C2\ , A = (±-i-aC2I/4.
9- У^ уу(У;)
Solution in the parametric form:
1/2
x = CiF, y = X[(F±2TJ±4:ln(CiF)
r p -| —1
where F = ехр(тт2) / ехр(тт2)dr + C2\ , A = (±-i-aC2I/4.
This is a special case of equation 2.9.4.10 with f(y) = ay71, g(y) = 61nm y.
11. y'lx = a In" у (y^J + bym{y'x)h.
This is a special case of equation 2.9.4.10 with f(y) = a In™ y, g(y) = bym.
12- xy'L = пУ'х + ax2n+1 lnm(Xy).
This is a special case of equation 2.9.2.4 with f(y) = alnm(Ay).
13. xyxx = — (n + l)y^, + ож"—1Aпу + n 1пж).
This is a special case of equation 2.9.2.5 with /(?) = aln?.
This is a special case of equation 2.9.2.22 with /(?) =
© 1995 by CRC Press, Inc.

15- xv'L + xBay + In x + b)y'x + у = 0.
Integrating the equation, we obtain Riccati equation
y'x + ay2 + (In x + b)y = C.
16. yy'lx = n(y'xJ + axmy2 + Ыпк(Хх)уп+1.
This is a special case of equation 2.9.3.9 with f(x) = —axm, g(x) = —bIn (Аж).
17. yy'lx = (y'xJ + a lnn(Xx)yyx + bxmy2.
This is a special case of equation 2.9.3.6 with f(x) = —а1пп(Аж), g(x) = —bxm.
18. yy'lx = (y'xJ + axnyy'x + b lnm(Xx)y2.
This is a special case of equation 2.9.3.6 with f(x) = —axn, g(x) = —blnm(Xx).
This is a special case of equation 2.9.3.17 with /(?) = ln?.
20. x2y'x'x = x2 (y + a In x + b)n + a.
This is a special case of equation 2.9.1.22 with /(?) = ?n.
21. x2yxx = n(n + l)y + ax3n+2(lny + п1пж).
This is a special case of equation 2.9.1.11 with /(?) = aln^.
1 — m
22. x2y'lx + \y + Ax~2~ (a In x + b)nym = 0.
у
The transformation ? = alnx + b, w = —— leads to the Emden—Fowler equation
(see Section 2.3)
23. ж2у^ + xy'x = a In"(Ay) + b.
This is a special case of equation 2.9.2.8 with f(y) = a In™ (Ay) + b.
24. (ax2 + b)y'lx + axy'x + с In"(Ay) = 0.
This is a special case of equation 2.9.2.11 with /(y) = clnn(Ay).
2.8.6. Equations Containing Trigonometric Functions
1- V'ix = -А2У + o(cos Аж)"у~"~3.
This is a special case of equation 2.9.1.27 with /(?) = a?~n~3.
2. yxx = —X2y + o(sin Аж)"у~"~3.
This is a special case of equation 2.9.1.26 with /(?) = a?~n~3.
© 1995 by CRC Press, Inc.

3. y^x = - A2y + a cos"(Aa;) si
The transformat
(see Section 2.3)
The transformation ? = cot(Arr), w = -.—r- leads to the Emden—Fowler equation
sin(Aa;)
4. y'lx = -A2y + osin"(Aa;)[sin(Aa;) + bcos(Xx)]my-n-m-3.
The transformation ? = 1 + 6cot(Arr), w = leads to the Emden—Fowler
equation (see Section 2.3)
5. у»я = -X2y + Asinn(Xx + a) sin^CAa; +
The transformation
sin(Aa; + а)
w =
sin(Aa; + b) ' sin(Aa; + b)
leads to the Emden—Fowler equation (see Section 2.3)
w'^ = A[\sm(b - a)]-2Cw~n~m~3.
6. yxx = bcos(Xx)y + ау~3.
This is a special case of equation 2.9.1.12 with f(x) = —bcos(Xx).
7. yxx = bsin(Xx)y + ay~3.
This is a special case of equation 2.9.1.12 with f(x) = —6sin(Aa;).
8- y'L = 2(cos Х)~2У + °(cot a;)"+3y".
This is a special case of equation 2.9.1.29 with /(?) = a?n.
9. y'lx = 2(sinx)-2y + o(tana;)"+3y".
This is a special case of equation 2.9.1.28 with /(?) = a?n.
4 + ny +
This is a special case of equation 2.9.2.30 with /(?) = a(rm~1.
+ У3 + а sin(Xx)y = 0.
This is a special case of equation 2.9.2.1 with f(x) = asin(Aa;).
12- V'L + 3УУ'Х + У3 + (a cos x + b)y = 0.
This is a special case of equation 2.9.2.1 with f(x) = a cos ж + b.
13- V'L + B°У + b sin x)v'x + b(cos x)y = 0.
Integrating, we obtain the Riccati equation y'x + ay2 + b(sinx)y = C.
© 1995 by CRC Press, Inc.

14- V'L + аУпЮ2 + b sin™ у + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = ayn, g(y) = 6sinm у + с.
15- v'L + аУпЮ2 + b tan" у + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = ayn, g(y) = 6tanm у + с.
16. y'L = ayn(y'xf +bsinm(Xx)y'x.
This is a special case of equation 2.9.3.2 with f(y) = ayn, g(x) = 6sinm(Aa;).
17. y'L = ayn(y'xJ +btanm(Xx)y'x.
This is a special case of equation 2.9.3.2 with f(y) = ayn, g(x) = 6tanm(Aa;).
18- y'L + a sin" У Ю2 + bym + c = 0.
This is a special case of equation 2.9.3.1 with f(y) = a sin™ y, g(y) = bym + с
19- y'L + osin" у (y'xf + bsinm(Xy) + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = a sin™ y, g(y) = bsinm(Xy) + с
20. y'L = asin" у (ухJ + Ьхту'х.
This is a special case of equation 2.9.3.2 with f(y) = a sin™ y, g(x) = bxm.
This is a special case of equation 2.9.3.2 with f(y) = asin"y, g(x) = 6sinm(Aa:).
22. yxx + a tan" у (y'x) + by171 + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = atan™ y, g(y) = bym + с
23. y'L + a tan" у (y'xJ + b tan™ (Ay) + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = atan™ y, g(y) = 6tanm(Ay) + с
24. y'L = a tan" у (у'хJ + bxmy'x.
This is a special case of equation 2.9.3.2 with f(y) = atan™ y, g(x) = bxm.
25. y'L = о tan" у (у^J + btan™(Aa;)y;.
This is a special case of equation 2.9.3.2 with f(y) = atan"y, g(x) = 6tanm(Arr).
26. y'L = ayn(y'xJ + bsin™ у (y^)fc.
This is a special case of equation 2.9.4.10 with f(y) = ayn, g(y) = 6sinm y.
27. y'L = a.yn{y'xf + btan™ у (у'х)к.
This is a special case of equation 2.9.4.10 with f(y) = ayn, g(y) = 6tanm y.
© 1995 by CRC Press, Inc.

28. y'L = о sin" у (у'хJ + Ьу™(у'х)к.
This is a special case of equation 2.9.4.10 with f(y) = a sin™ y, g(y) = bym.
29. y'L = atari" у (y'xf + by™(y'x)k.
This is a special case of equation 2.9.4.10 with f(y) = a tan™ y, g(y) = bym.
30. xy'^x = ny'x + ax2n+1 cos™ (Ay).
This is a special case of equation 2.9.2.4 with f(y) = acosm(Xy).
31. xVx-x = ny'x + ах2п+г sinm(Xy).
This is a special case of equation 2.9.2.4 with f(y) = asmm(Xy).
32- xy'^x = ny'x + ax2n+1 tanm(Xy).
This is a special case of equation 2.9.2.4 with f(y) = atanm(Ay).
33. xy'ix = ny'x + ax2^1 cotm(Xy).
This is a special case of equation 2.9.2.4 with f(y) = acotm(Xy).
34. Zyy'xx = (y'x) + bsin(Xx)y2 — a.
This is a special case of equation 2.9.3.7 with f(x) = —6sin(Aa;).
35. yy'ix = n{y'xJ - ay4n~2 + bsin(Xx)y2.
This is a special case of equation 2.9.3.8 with f(x) = —bsin(Xx).
36. yy'lx = n(y'xJ + axmy2 + bcosk(Xx)yn+1.
This is a special case of equation 2.9.3.9 with f(x) = —axm, g(x) = —bcosk(Xx).
37. yy'lx = n(y'xJ + axmy2 + bsink(Xx)yn+1.
This is a special case of equation 2.9.3.9 with f(x) = —axm, g(x) = —6sinfe(Aa;).
38. yy'lx = n(y'xJ + axmy2 + b tanfc (Xx)yn+1.
This is a special case of equation 2.9.3.9 with f(x) = —axm, g(x) = —6tanfe(Aa;).
39. УУхх = n(y'xJ + asin(Xx)y2 + b sin(^x)yn+1.
This is a special case of equation 2.9.3.9 with f(x) = —asin(Aa;), g(x) = —bsin^/ix).
40. yy'lx = (y'xJ + axnyy'x + bsinm(Xx)y2.
This is a special case of equation 2.9.3.6 with f(x) = —axn, g(x) = —6sinm(Aa;).
41. yy'lx = (y'xJ + axnyy'x + btanm(Xx)y2.
This is a special case of equation 2.9.3.6 with f(x) = —axn, g(x) = —bta,nm(Xx).
© 1995 by CRC Press, Inc.

42- W'L = Ю2 + asinn(Xx)yyx + bxmy2.
This is a special case of equation 2.9.3.6 with /(ж) = —asinn(Aa;), g(x) = —bxm.
43. x2y'x[x + xy'x = a sin" (Ay) + b.
This is a special case of equation 2.9.2.8 with f(y) = asin"(Ay) + b.
44. x2y'x'x + xy'x = a tan" (Ay) + b.
This is a special case of equation 2.9.2.8 with f(y) = atan™(Ay) + b.
45. x2y'x'x + ax2 tan x y'x + n(ax tan x - n - l)y = bxnm+2(cosxJaym-3.
This is a special case of equation 2.9.2.33 with /(?) = b?m~3.
46. (ax2 + b)y'lx + axy'x + cos" (Ay) + с = 0.
This is a special case of equation 2.9.2.11 with f(y) = cos™ (Ay) + c.
47. (ож2 + b)y^ + ожу^ + sin" (Ay) + с = 0.
This is a special case of equation 2.9.2.11 with f(y) = sinn(Ay) + c.
48. (ax2 + b)y'lx + axy'x + tan" (Ay) + с = 0.
This is a special case of equation 2.9.2.11 with f(y) = tann(Ay) + c.
49. (ax2 + b)y'lx + axy'x + cot" (Ay) + с = 0.
This is a special case of equation 2.9.2.11 with f(y) = cot™ (Ay) + c.
50. sin ж y^ + -f- cos ж y'x = ayn + b.
This is a special case of equation 2.9.2.14 with д(ж) = sin ж, /(у) = ay™ + b.
51. sin ж yxx + \ cos ж y'x = a sin"(Ay) + b.
This is a special case of equation 2.9.2.14 with g(x) = sin ж, /(у) = a sin™ (Ay) + b.
52. sin ж y^ + -|- cos ж y^, = a cos" (Ay) + b.
This is a special case of equation 2.9.2.14 with д(ж) = sin ж, /(у) = acos™(Ay) + b.
53. sin ж y^ + -|- cos ж y^, = a tan" (Ay) + b.
This is a special case of equation 2.9.2.14 with g(x) = sin ж, /(у) = a tan™ (Ay) + b.
54. sin2 ж y^ = n(n + 1 - n sin2 ж)у + o(sin x-)™m+3n+2ym.
This is a special case of equation 2.9.1.30 with /(?) = a?m.
55. COS2 Ж y^ = П(П + 1 - П COS2 ж)у + O(COS ж)™™+3тг+2уттг
This is a special case of equation 2.9.1.31 with /(?) = a?m.
© 1995 by CRC Press, Inc.

2.8.7. Equations Containing the Combinations of Exponential,
Hyperbolic, Logarithmic, and Trigonometric Functions
1. у»я = X2y + аеЗХхAпу + Xx).
This is a special case of equation 2.9.1.15 with /(?) = aln?.
2- V'L = ~аУ'х + ЬеахAпу + ax).
This is a special case of equation 2.9.2.18 with /(?) =
3. y^x = ay'x + be2axln
This is a special case of equation 2.9.2.17 with f(y) = blnn(Xy).
4. у»я = ау'х + be2axsinn(Xy).
This is a special case of equation 2.9.2.17 with f(y) = bsmn(Xy).
5. у»я = ay'x + be2ax tan" (Ay).
This is a special case of equation 2.9.2.17 with f(y) = bt&nn(\y).
6- V'L + atanxy'x + b(atanx - b)y = cebmx(cosxJaym-3.
This is a special case of equation 2.9.2.32 with /(?) = c^m.
7- V'L = a(y'xf - Ье^У + csinh(Xx).
This is a special case of equation 2.9.3.16 with f(x) = —csinh(Aa;).
8. y'lx = a(y'xJ + bcoshn(Xx)eay + cxm.
This is a special case of equation 2.9.3.15 with f(x) = — 6coshn(Arr), g(x) = —cxm
9. у»я = a(y'xJ + Ь 1пп(Хх)еаУ + cxm.
This is a special case of equation 2.9.3.15 with f(x) = —61пп(Аж), д(х) = —cxm.
10. y'lx = a(y'xJ + Ь 1пп(Хх)еаУ + cevx.
This is a special case of equation 2.9.3.15 with f(x) = —61пп(Аж), д(х) = —cevx.
11- V'L = a(y'xf - Ье^У + csin(Xx).
This is a special case of equation 2.9.3.16 with f(x) = — csin(Aa?).
12- y'L = a(v'xJ + bsinn(Xx)eay + cxm.
This is a special case of equation 2.9.3.15 with f(x) = —6sinn(Aa;), g(x) = —cxm.
13- y'L = аЮ2 +bsinn(Xx)eay + ce»x.
This is a special case of equation 2.9.3.15 with f(x) = —6sinn(Aa;), g(x) = —cevx.
© 1995 by CRC Press, Inc.

V'L
This is a special case of equation 2.9.3.1 with f(y) = aeXy, g(y) = Ыпп у + с.
. y'L + аехУ(у'хJ + bsinn у + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = aeXy, g(y) = b sin™ у + с.
y'L =
This is a special case of equation 2.9.3.2 with f(y) = aeXy, g(x) = Ыпп(/лх).
17. V'L = аеХуШ2 +Ьв-тп((лх)ух.
This is a special case of equation 2.9.3.2 with f(y) = aeXy, g(x) = bsmn(/ix).
18- V'L = аехУ(у'хJ +btann(fix)yx.
This is a special case of equation 2.9.3.2 with f(y) = aeXy, g(x) = bt&nn(p,x).
19- y'L + о In" у (у'хJ + ЬехУ + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = a In™ y, g(y) = beXy + с
20. y'L = ob" у (у'хJ + be^y'x.
This is a special case of equation 2.9.3.2 with f(y) = a In™ y, g(x) = beXx.
21. y'L = a\nny(y'xJ + bsinm(Xx)y'x.
This is a special case of equation 2.9.3.2 with f(y) = a\nn y, g(x) = 6sinm(Aa;).
22- y'L + a sin" У Ю2 + beXy + с = 0.
This is a special case of equation 2.9.3.1 with f(y) = a sin™ y, g(y) = beXy + с
23. y'L = as™ny(yx) + yx
This is a special case of equation 2.9.3.2 with f(y) = a sin™ y, g(x) = beXx.
24. y'L = a sin" у (y'xJ + b \nm(Xx)y'x.
This is a special case of equation 2.9.3.2 with f(y) = asin"y, g(x) = 61пт(Аж).
25. y'L = a tan" у (у'хJ + beXxy'x.
This is a special case of equation 2.9.3.2 with f(y) = a tan™ y, g(x) = beXx.
26. y'L + beax cosh(Ay) (y'xf + ay'x = 0.
This is a special case of equation 2.9.3.24 with f(y) = bcosh(Xy).
27. y'L + beax sin(Xy)(y'xK + ay'x = 0.
This is a special case of equation 2.9.3.24 with f(y) = bsm(Xy).
© 1995 by CRC Press, Inc.

28. xyxx = ax In ж eyy'x + aey.
Solution: у = - In \eClX (C2--^f x^e'0^ dx) + -?- Inx\.
29. W'L = aex(y'xf + aexy In у (y'xJ.
Solution: x = - In \eClV (С2 + -?г[ 2/~1e~Cl3/ dy] - -?- In yj .
30- W'L = Ю2 + aeXxyy'x + b sin" (yx)y1.
This is a special case of equation 2.9.3.6 with /(ж) = — aeXx, g(x) = — bsmn(vx).
31. (ae2x + Ъ)ухх + ae2xy'x = cosh"(Ay) + с.
This is a special case of equation 2.9.2.14 with g(x) = ae2x + b, f(y) = cosh™ (Ay) +c.
32. (ae2x + Ъ)у'1х + ae2xy'x = tanh"(Ay) + с
This is a special case of equation 2.9.2.14 with g(x) = ae2x + b, f(y) = tanhn(Ay) + с
33. (ae2x + h)y'ix + ae2xy'x = In" (Ay) + с
This is a special case of equation 2.9.2.14 with g(x) = ae2x + b, f(y) = lnn(Ay) + с
34. (ae2x + b)y'lx + ae2xy'x = sin" (Ay) + с
This is a special case of equation 2.9.2.14 with g(x) = ae2x + b, f(y) = sin™(Ay) + с
35. (ae2x + b)y'lx + ae2xy'x = tan" (Ay) + с
This is a special case of equation 2.9.2.14 with g(x) = ae2x + b, /(y) = tann(Ay) + с
36. sin x yxx + \ cos x y'x = aeXy + b.
This is a special case of equation 2.9.2.14 with g(x) = sin ж, /(у) = аеХу + b.
37. sin x yxx + \ cos x y'x = a cosh"(Ay) + b.
This is a special case of equation 2.9.2.14 with g(x) = sin ж, /(у) = acoshn(Ay) + b.
38. sin x yxx + \ cos x y'x = a sinh"(Ay) + b.
This is a special case of equation 2.9.2.14 with g(x) = sin ж, /(у) = asinhn(Ay) + b.
39. sin x yxx + \ cos x y'x = a tanh"(Ay) + b.
This is a special case of equation 2.9.2.14 with g(x) = sin ж, /(у) = atanhn(Ay) + b.
40. sin ж yxx + \ cos ж y'x = a In" у + b.
This is a special case of equation 2.9.2.14 with g(x) = sin ж, /(у) = a In™ у + b.
© 1995 by CRC Press, Inc.

2.9. Equations Containing Arbitrary Functions
Notation: f, g, h, ip, and ф are arbitrary composite functions of their arguments indi-
indicated in parentheses just after the function name (the arguments may depend on x, y, y'x).
2.9.1. Equations of the Form F(x,y)y%x + G(x,y) = 0
i- v'L = f(v)-
This autonomous equation is met with in mechanics, the combustion theory, and the
theory of mass transfer with chemical reactions.
The substitution u(y) = y'x leads to a first order equation with separated variables:
uu'y = f{y).
ЛГ 1~1/2
Ci + 2 / f(y) dy\ dy = C2± x.
•J J
Particular solutions: у = А%, where A% are roots of the equation f(Ak) = 0.
2- V'L = /(°У + Ьх + с).
The substitution w = ay+bx+c leads to an equation of the form 2.9.1.1: w'J.x = af(w).
3- V'L 2
The substitution w = у + ax2 + bx + с leads to an equation of the form 2.9.1.1:
<x = /M + 2a-
4- v'L = f(v + ахП + Ьх2 + cx) - an(n -1)»"-2.
The substitution w = у + axn + bx2 + cx leads to an equation of the form 2.9.1.1:
<x = /И + 2b-
The transformation ? = 1/x, w = y/x leads to an equation of the form 2.9.1.1:
6- V'L =
'L = x-
Having set w = уж'2, we obtain
Integrating the latter equation, we arrive at an equation with separation of variables.
1-1/2
dw = C2±ln\x\.
ЛГ 
Cx + \w2 + 2 / f(w) dw
J J
xx x3 \ x x2 x
The transformation ? = 1/x, w = y/x leads to an equation of the form 2.9.1.3:
© 1995 by CRC Press, Inc.

8. !&, = x-^fiayx-1/2 + Ьх1'2).
The substitution w = ay + bx leads to an equation of the form 2.9.1.6: wxx =
9. " -¦-»*< y
л/ах2 + bx + с
Setting м(ж) = y(ax2 + bx + c)^1/2 and integrating the equation, we obtain a first
order equation with separation of variables:
(ax2 + bx + cf(u'xf = [\b2 - ac)u2 + 2 /w/H du + Cx.
10.
у/ах2 + bx + с
Setting w = ay + fix + 7 and denoting /(z) = —3~<p(z)> we obtain an equation of the
form 2.9.1.9:
wxx =
Vax2 + bx + с
This is a special case of equation 2.9.1.14 with ф = x~n.
12. y'L + f(x)y = ay-3.
Yermakov's equation.
Let w = w(x) be a nontrivial solution of the linear equation wxx + f(x)w = 0.
/ciir xi
——, z = — leads to an autonomous equation of the form
«r w
2.9.1.1: z'^ = az~3.
( f dx \ 2
Solution: dy2 = aw2 +w2(C2 + Ci / -^5- .
V J w2 J
13. (axn + b)y'^x = an(n - 1)хп~2у + y~2f
n-2.,±.,-2*l ^
axn + b
/ciir 11
— Г7Г, w = leads to an equation of the
(axn + bJ axn + b
form 2.9.1.1: w'^ = vj-2f(w).
/ciir 11
——, w = —— leads to an equation of the form 2.9.1.1:
ф2 ф
Ci + 2 / f(w)dw\ dw = C2± ¦
J J J
dx
ф2(х)
© 1995 by CRC Press, Inc.

15- V'L = х2У + e3Xxf(yeXx).
This is a special case of equation 2.9.1.14 with ф = e~Xx.
16. у»я = f(y + aeXx + b)- a\2eXx.
The substitution w = y+aeXx + b leads to an equation of the form 2.9.1.1: wxx = f(w).
!7- x2iga = x2f(xney) + n.
The substitution у = w — n In x leads to an equation of the form 2.9.1.1: wxx = f(ew).
18* Vxx = f(V "I" osinha; + b) — osinha;.
The substitution w = y+a sinh x+b yields an equation of the form 2.9.1.1: w'J.x = f(w).
19- y'xx = f(y + о cosh ж + b) — a cosh x.
The substitution w = y+a cosh x+b yields an equation of the form 2.9.1.1: wxx =f(w).
20. y'L = А2У + (sinhXx)-3f(-
sinh Лж
This is a special case of equation 2.9.1.14 with ф = sinh Xx.
21. «С = Аа
This is a special case of equation 2.9.1.14 with ф = cosh Xx.
22- x2v'L =2
The substitution w = y+a In x+b leads to an equation of the form 2.9.1.1: wxx = f(w).
23. ,:, = -
ж2 In ж Aпж)?
This is a special case of equation 2.9.1.14 with ф = In ж.
24. y'L = /(у + о, sin ж + b) + a sin ж.
The substitution w = y+asinx+b leads to an equation of the form 2.9.1.1: wxx = f(w)
25. y'L = /(y + a cos ж + b) + a cos ж.
The substitution w = y+a cos x+b leads to an equation of the form 2.9.1.1: wxx = f(w)
26. v" = -X2v -
sin Аж
This is a special case of equation 2.9.1.14 with ф = sin Аж.
27. y'L = -X2y + (cos Аж)-3/
cos А
This is a special case of equation 2.9.1.14 with ф = cos Аж.
© 1995 by CRC Press, Inc.

28- V'L = 2(sina;)-2y + (tan жK/(у tana;).
This is a special case of equation 2.9.1.14 with ф = cot x.
29. y'lx = 2(cosx)~2y + (cot жK/(у cot ж).
This is a special case of equation 2.9.1.14 with ф = tana;.
30. sin2 xy'^x = n(n + 1 - n sin2 x)y + sin3n+2 xf(y sin x).
The substitution ж = ? + -|- leads to an equation of the form 2.9.1.31:
cos2 ? y'^ = n(n + 1 - n cos2 i)y + cos3™+2
31. cos2 ж yxx = n(n + 1 — n cos2 ж)у + cos3"+2 ж/(у cos" ж).
r
The transformation ? = / cos2™ xdx, w = у cos™ x leads to an autonomous equation
of the form 2.9.1.1: w'^ = f(w).
32. y" =<p-3f(V-+il>) +-^HLy- " ¦ ¦
The transformation
/dx
—
leads to an autonomous equation of the form 2.9.1.1: w't't = f(w).
Solution:
dw = ± Г dx where = Г
) + d J <P2 J
2.9.2. Equations of the Form F(x,y)y%x + G(x,y)y'x + H(x,y) = 0
V'L + 3УУ* + У3 + f(x)v = 0-
The substitution у = w(Jwdx)^1 leads to the linear equation: w'^x + f(x)w = 0.
2- y'L + [2oy + f(x)]y'x + af(x)y2 = g(x).
Having set и = y'x + ay2, we obtain u'x + f(x)u = g(x). Thus, the original equation is
reduced to the first order linear equation and the Riccati equation.
3- V'L + [3y + f(x)]y'x + у3 + /(ж)у2 + g(x)y + h(x) = 0.
The substitution у = и'х/и leads to a third order linear equation:
<xx + f(xXx + 9(x)u'x + h{x)u = 0.
© 1995 by CRC Press, Inc.

Multiplying both sides by x 2n 1, we obtain an equation of the form 2.9.2.14.
1°. Solution with пф-\:
I -1/2 „+1
dy = ± T + C2.
n + 1
2°. Solution with n = -1:
Л Г 1 ~xl2
C\ + 2 / f(y) dy\ dy = ±In |ж| + C2-
The transformation ? = ж™, «; = уж™ leads to an autonomous equation of the form
2.9.1.1: n2w'^ = f(w).
6* xv'xx ~ nv'x + f(x)y = ax2n+1y~3.
The substitution w = yx~nl2 leads to Yermakov's equation 2.9.1.12:
к/' -i- г~2[г f (тЛ -n(n -I- 9ili» — mu~3
7- *y'L = f(y)y'x-
The substitution w(y) = xy'x/y leads to a first order linear equation: yw'y = —w +1 +
Substitutions ж = ±e* leads to an equation of the form 2.9.1.1: y"t = /(y).
9- x2v'L + (n + m + l)xy'x + nmy = xn-2mf(yxn).
1°. For n ф m, the transformation ? = xn~m, w = yxn leads to an autonomous
equation of the form 2.9.1.1: (n — mJw'^ = f(w).
2°. For n = то, the transformation ? = lnx, w = yxn leads to an autonomous equation
of the form 2.9.1.1: w'^ = f(w).
This is a special case of equation 2.9.3.23 with g(x) = h{x) = 0.
11. (ax2 + b)y'lx + axy'x + f(y) = 0.
/dx
—, leads to an autonomous equation of the form
Vax2 + b
2.9.1.1: y? + f(y) = 0.
12. x3y'x-x = f(y/x)(xyx-y).
This is a special case of equation 2.9.4.34 with g(z) = z.
© 1995 by CRC Press, Inc.

Integrating, we obtain the Riccati equation y'x + f(x)y + ay2 = C.
^ ), 9=9(x).
Intergating, we obtain an equation with separation of variables:
With g(x) > 0, the solution is
c1+2 J ^
f Г f
/ f(y) dy\ dy = C2 ± / ¦
J I J
v'L + Bfv + 9)y'x + fxy2 + 9xy = o, f = f(x), g = g(x).
Integrating, we obtain the Riccati equation y'x + fy2 + gy = C.
v'L-— vL
/m у
—-7г dx, w = —- leads to an autonomous equation of the
¦ф2 ф
form 2.9.1.1: w'^ = f(w).
Multiplying both sides by e 2ax, we obtain an equation of the form 2.9.2.14.
Solution:
r r1/2 i
7i + 2 f(y)dy\ dy = C2±-eax.
The transformation ? = eax, w = yeax leads to an equation of the form 2.9.1.1:
u& = a-2f(w).
1°. For ц ф v, the transformation ? = e*^"^, ад = уе]хх leads to an autonomous
equation of the form 2.9.1.1: (/л — vJw'^ = /(ад).
2°. For ц, = v, the substitution ад = yelxx leads to an autonomous equation of the form
2.9.1.1: w'^ = /(ад).
20. y'L ~ Wx + f(x)y = ae2Xxy-3.
The substitution ад = ye~Xxl2 leads to Yermakov's equation 2.9.1.12:
© 1995 by CRC Press, Inc.

22.
23.
ХУХХ ~ пУх ~ °(°ж + п)У = x2n+1e3axf(yeax).
This is a special case of equation 2.9.2.16 with ip = xn, ip = e~ax.
The transformation z = xneay, w = xy'x leads to a first order equation with separation
of variables: z(aw + n)w'z = [f(z) + l]w.
The transformation z = xneay, w = xy'x leads to a first order equation with separation
of variables: z(aw + n)w'z = f(z).
24. x2y'Jx - ax2y'x - n(ax + n + l)y = x3n+2e2ax f (yxn).
This is a special case of equation 2.9.2.16 with (p = eax, ip = x~n.
25. (ae2Xx + b)y'^x + aXe2Xxy'x + f(y) = 0.
This is a special case of equation 2.9.2.14 with g(x) = ae2Xx + b.
26. yL+9xyL + fy = ae~2gy~3' f = f(x), 9=9(x).
The substitution w = ye9^2 leads to Yermakov's equation 2.9.1.12:
27. y'L-— vL-4— + a)y = e3axv2f(yeax), <p = <p(x).
The transformation ? = f ipe2ax dx, w = yeax leads to an equation of the form 2.9.1.1:
2y'x'x
28- x2y'x
The substitution x = e* leads to an equation of the form 2.9.1.3: y't't = f(y + at + bt2).
29. y'ix + Xmtan(\x)yx + f(x)y = a[cos(Aa;)]2™y-3.
This is a special case of equation 2.9.2.26 with g = —mlncos(Aa;).
30. yxx — (n + 1) tan x y'x — ny = cos"~2 x f(y cos" ж).
This is a special case of equation 2.9.2.16 with (p = cos"™ x, ip = cos~n x.
х -n[(m+l)tan2x + l]y = cos2m+n x f (y cosn x).
This is a special case of equation 2.9.2.16 with (p = cosm~n x, ip = cos~n x.
32- y'L + atanxyx + b(atanx - b)y = cos2a xe3bxf(yebx).
This is a special case of equation 2.9.2.16 with ip = cos° x, ф = e~bx.
© 1995 by CRC Press, Inc.

33. x2yxx + ax2tanxy'x + п(ах tanx -n- l)y = x3n+2 cos2a x f(yxn).
This is a special case of equation 2.9.2.16 with ip = cos° x, ip = x~n.
34. x2yxx - ax2 cot x y'x - n(ax cot x + n + l)y = x3n+2 sin2a x f(yxn).
This is a special case of equation 2.9.2.16 with ip = sin° x, ip = x~n.
35. sin ж y^x + \cosxy'x = f(y).
This is a special case of equation 2.9.2.14 with g = sin ж.
36. cos x y'lx - \ sin x y'x = f(y).
This is a special case of equation 2.9.2.14 with g = cos ж.
2.9.3. Equations of the Form
м
+ Y, Gm(x,y)(y'x)m = O (M = 2, 3, 4)
m=0
2-
The substitution w(y) = (y'x) leads to a first order linear equation:
w'y + 2f(y)w + 2g(y) = 0.
Dividing by y'x, we obtain a total differential equation. Its solution is found from the
equation
\n\y'x\= J f{y)dy + J g{x)dx + C.
Solving the latter for y'x, we arrive at an equation with separation of variables. In
addition, у = C\ is the solution with arbitrary constant C\.
3- y'L= П У ' (xy'x-y)yx.
xy
XXI*
The transformation z = xnym, w = —— leads to an equation with separation of
У
variables: z(mw + n)w'z = [f(z) — l](w2 — w).
-y) + h(x)(xyfx - yJ.
The substitution w(x) = xy'x — у leads to the Riccati equation:
w'x = xf(x) + xg(x)w + xh(x)w2.
5. yy'L + Ш2 + f(x)yy'x + g(x) = o.
The substitution и = у2 leads to a linear equation: uxx + f(x)u'x + 2g(x) = 0.
© 1995 by CRC Press, Inc.

6- yy'L - Ю2 + f(x)yy'x + g(x)y2 = о.
The substitution и = y'x/y leads to a first order linear equation: u'x + f(x)u+g(x) = 0.
7- ^VV'L - Ш2 + f(*)y2 + o = 0, о > 0.
If и and v are two solutions of the linear equation 4yxx + f(x)y = 0, which satisfy the
condition (uv'x — u'xvJ = a, then у = uv is a solution of the original equation.
8- yy'L - МУ'ХJ + f(*)y2 + ay4™ = 0.
1°. With n = 1, this is an equation of the form 2.9.3.6.
2°. With пф1, the substitution w = y1~n leads to Yermakov's equation 2.9.1.12:
wxx + A - n)f(x)w + a(l - n)w~3 = 0.
9- yy'L - "Ю2 + f(*)y2 + 9(х)Уп+1 = 0.
The substitution w = yx~n leads to a nonhomogeneous linear equation:
<x + A - n)f(x)w + A - n)g(x) = 0.
Ю- WL + "Ю2 + f(x)v'x + 9(x)y2 = 0.
The substitution w = ya+1 leads to a linear equation: wxx + f(x)w'x + (a+l)g(x)w = 0.
11. w" — 2(i/ ) — (fv + 2a)y' ¦+- f'v2 + а' у = 0, f = f(x). a = a(x).
Integrating, we obtain the Riccati equation: y'x + Cy2 + fy + g = 0.
12. yy'L — (у'х) "I" (fy2 "Ь 9)у'х "b fxy3 — 9а;У ^ 0» / = f{x)i 9 = 9(x)-
Integrating, we obtain the Riccati equation: y^, + /y2 + Cy — g = 0, where С is an
arbitrary constant.
The substitution w(x) = xy'x/y leads to the Bernoulli equation 1.1.5:
xw'x = w + [f(x) — l]w2.
14- W'L + f(x)(v'xJ + 9(x)yy'x + Цх)у2 = 0.
The substitution и = y'x/y leads to the Riccati equation:
u'x + (l + f)u2+gu + h = 0.
15- y'L - *Ю2 + f(*)eay + 9(x) = 0.
The substitution w = e~ay leads to a nonhomogeneous linear equation:
W'L ~ ag{x)w = af(x).
© 1995 by CRC Press, Inc.

I6- y'L - a(y'xJ + be4ay + f(x) = °-
The substitution w = e~ay leads to Yermakov's equation 2.9.1.12:
wxx ~ af(x)w = cibw~ .
17- yy'L = f(eaxyn)(y'xJ-
The transformation z = eaxyn, w = y'x/y leads to a first order separable equation:
z(nw + a)w'z = \f(z) — l]w2.
18- y'L = xf(y)(y'xf.
Taking у as the independent variable, we obtain a linear equation for x = x(y):
xyy ~ ~J\y)x-
19- y'L + [xf(y) + g(y)](y'xf + Hy)(y'xJ = 0.
Taking у as the independent variable, we obtain a linear equation for x = x(y):
This is a special case of equation 2.9.4.35 with к = 2.
21- x3y^x + [x4f(y) + a}(y'xf = 0.
Taking у as the independent variable, we obtain an equation of the form 2.9.1.12 for
x = x(y): x'yy - f(y)x - ax~3 = 0.
22. y'L = x-1[f(y)+g(y)(xy'x -y) + h(y)(xy'x - yJ}y'x.
The substitution w(y) = xy'x — у leads to the Riccati equation:
w'y = f(v) + 9{y)w + h(y)w2.
23. yL = *
The transformation z = y/x, w = xy'x/y leads to the Riccati equation:
zw'z = f(z) + [zg(z) - l]w + z2h(z)w2.
24. y'L + e3
The substitution ? = e ax leads to an equation of the form 2.9.4.2 with g(z) = a3z3:
\3 n
25. y'lx = x
This is a special case of equation 2.9.4.8 with n = —3/2, m = 1, f(z) = zip(z).
© 1995 by CRC Press, Inc.

26. y'L + f(y)(y'xL + д(у)Ю2 + Ну) = о.
The substitution w(y) = (y'x) leads to the Riccati equation:
w'y + 2f(y)w2 + 2g(y)w + 2h(y) = 0.
27. xyL + *2m+14
This is a special case of equation 2.9.4.46 with n = 4, ip = x~m.
28. (y + ax)y'^ = fB
The substitution у = —ax + xz leads to the equation
xzz'x'x + 2zzx-x3f(x)(zxf = 0.
Setting w = z'x/z, we obtain the Bernoulli equation:
xw'x + 2w + [x - x3f(x)]w2 = 0.
2.9.4. Equations of the Form F(x,y,yfx)y%x + G(x,y,yfx) = 0
1- v'L = f(x)g(y'a).
The substitution u(x) = y'x leads to a first order equation with separation of variables:
< = f{x)g{u).
2- y'L = fivMv'J-
The substitution u(y) = y'x leads to a first order equation with separation of variables:
uu'v = f{y)g{u).
In addition, there may exist solutions of the form у = Ax + C, where A are roots
of the equation g(A) = 0, С is an arbitrary number, or у = В, where В are roots of
the equation f(B) = 0.
3- yL = f(ax + by +
4-
With b = 0, we have an equation of the form 2.9.4.1. With b ф 0, the substitution
u(x) =y+(ax + c)/b leads to an equation of the form 2.9.4.2: uxx = f(bu)g(u'x — — ).
The substitution w(x) = xny'x leads to a first order equation with separation of vari-
variables: xw'x = f(w) + nw.
,.» ,. —2n—1 f („,п„.Г \
Ухх — У Т\У Ух)'
The substitution w(y) = yny'x leads to a first order equation with separation of vari-
variables: yww'y = f(w) + nw2.
6.
у
The substitution w(x) = xy'x/y leads to a first order equation with separation of
variables: xw'x = f(w) + w — w2.
© 1995 by CRC Press, Inc.

7
л/хУ
Setting и = y'x and passing on to new variables
du
f
= /
J
-тГТ,
we have у = (w't) . Differentiating the latter with respect to x, we obtain a second
order linear equation w"t = g(i)w, where function g(t) is defined parametrically:
i f du
g= fu, t =
Л~тг,,.ггг\ 2п-\-т
•by), i ч .
The transformation z = xnym, w = xy'x/y yields
1 2n+m
z(mw + n)w'z = z m+i f(z)w n+m + w — w .
2n+m
Divide both sides of this equation by w n+m sxid introduce a new variable
? = w n+m — uj n+m
As a result we obtain a first order linear equation:
9- t?, + f(x)y'x + g(x)(y'x)k = 0.
The substitution u(x) = y'x leads to the Bernoulli equation u'x + f(x)u + g{x)uk = 0.
The substitution w(y) = y'x leads to the Bernoulli equation w'y = f(y)w + g(y)
xyxx + xnm-2m+1f(y)(y'x)n + my'x = 0.
This is a special case of equation 2.9.4.46 with ip = x~
n n — kn — km —1 f fe—1 _k n
xx km x
Passing on to new variables z = xnym, w = xy'x/y, we arrive at
z(mw + n)w, = —Ц -w — w+ f(z)wK+ .
km
Multiplying the latter by w~k~1, we obtain, with the aid of the substitution
m 1_k n _k
a first order linear equation:
w
,fe-i
© 1995 by CRC Press, Inc.

is.
Passing on to new variables z = xnym, w = xy'x/y, we arrive at
/ \ / 777-1 -L ~r /C) о n/\ &>_i_o
г(яио + n)u^ = w H ^ -ur + f(z)wk+2.
Multiplying the latter by w~k~'2, we obtain, with the aid of the substitution
m h n h -i
к к + 1
a first order linear equation:
У
The transformation z = xnym, w = xy'x/y leads to a first order separable equation:
z(mw + n)w'z = f(z)g(w).
+ b
Setting и = у'х, we rewrite the equation as follows:
Differentiating both sides with respect to x and passing on to new variables
1 f du r-
1 f
2 J
Л\ > " — V •*">
U)
we obtain an equation of the form 2.9.1.12:
z"t = au(t)z - bz~3.
„ fivL)
Ухх i ; ; ^ '
у ay + bx*
Setting и = y'x, we rewrite the equation as follows:
[u'x/f(u)]-2 = ay + bx2.
Differentiating both sides with respect to x and introducing a new independent vari-
variable t = —-—, we obtain a second order linear equation for x = x(t), integrable
J f(u)
by quadrature:
2x"t = 2bx + au(t),
where function и = u(t) is defined imlicitly: t = I .
© 1995 by CRC Press, Inc.

V'L=
vax + by2
Taking у as the independent variable, we obtain an equation of the form 2.9.4.16 for
x = x(y):
V'L = (ax2 + bxV + СУ2 + ax+f3y + iy1
The transformation x = At + Bu + C, у = Dt + Pu + Q, where и = u(t), reduces this
equation, by selecting appropriate constants A, B, C, D, P, and Q, to an equation
of the form 2.9.4.15, 2.9.4.16, or 2.9.4.17.
19. y»=
\/axy + bx3/2 + ex
Setting и = y'x, we rewrite the equation as follows:
[Vx~u'Jf(u)] =ay-
Differentiating both sides with respect to x and changing to new variables
1 f du r
t= ^Г / TT> z = v1'
2 J f(u)
we obtain a second order linear equation:
2z"t = 2au(t)z + b.
1 f du
1 f
where function и = u(t) is defined in the inplicit form: t = — /
\/axy + by3!2 + cy
Taking у as the independent variable, we obtain an equation of the form 2.9.4.19 for
x = x(y):
cy)-1/2f(l/x')(x'f.
y'L=
22.
\/axy + bx2 + еж3/2 + dx
The substitution aw = ay + bx leads to an equation of the form 2.9.4.19:
w,, /K - b/a)
yaxw + еж3/2 + dx
fW
\Jaxy + by2 + cy3l2 + dy
Taking у as the independent variable, we obtain an equation of the form 2.9.4.21 for
x = x(y):
x»y = -(axy + by2 + суг'2 + dyf1/2f(l/x'y)(x'yf.
© 1995 by CRC Press, Inc.

23. y'ix = \xf(y'x) + yg{y'x) + h{y'x)} .
The Legendre transformation x = w't, у = tw't — w (y'x = t, yxx = l/w"t) leads to a
second order linear equation:
tg(t)]w't - g(t)w + h{t).
24. y'L = a + f(x)^/(y'xJ - lay.
Setting и = y'x, we rewrite the equation as follows:
Differentiating both sides with respect to x and dividing by (u'x — a), we obtain a
linear equation:
f<x=f'xu'x+fu-afx.
25.
The Legendre transformation x = w't, у = tw't — w leads to an equation of the form
2.9.4.7: < = — г
The transformation x = tw't — w, у = —w't, where w = w(t), leads to an equation of
the form 2.9.4.7: w"t =
26.
f(-w't)Vtw
27. y'L = х~2(ху'х - y)f(y'x).
The Legendre transformation x = w't, у = tw't — w (y'x = t, yxx = l/w't't) leads to an
equation of the form 2.9.3.13: w'L = V
28.
The Legendre transformation x = w't, у = tw't — w leads to an equation of the form
2.9.3.18: w't't = t[f(w)]-l(w'tK.
29.
x
The substitution w = y'x — — leads to an equation with separation of variables:
X
xw'x = —w + f(w).
30. yxx = f(x)g(xy'x - y).
The substitution w = xy'x — у leads to an equation with separation of variables:
w'x =xf(x)g(w).
31- y'L = x-n-3ynf(xy'x - y).
The transformation x = 1/t, у = w/t leads to an autonomous equation of the form
2.9.4.2: w't't = wnf(-w't).
© 1995 by CRC Press, Inc.

32.
xy
This is a special case of equation 2.9.4.36 with к =
n
33. y'ix = x-1f(y)g(xy'x - y)y'x.
The substitution w(y) = xy'x — у leads to an equation with separation of variables:
w'v = f(y)g(w).
34. y'ix = x-3f(y/x)g(xyfx - y).
The transformation x = 1/t, у = w/t leads to an equation of the form 2.9.4.2:
w't't = f(w)g(-w't)-
35. y'^ =
The transformation z = y/x, w = xy'x/y leads to the Bernoulli equation
zw'z = -w + zkf(z)wk.
There are particular solutions у = x and у = Ax, where A are roots of the equation
f(A) = 0; with к > 0 we also have у = C, where С is an arbitrary number.
Л„п„,тгг\ 2гг+тгг—пк
У ) / / \ ~T / / \ к
xx xy x x
The transformation z = xnym, w = xy'x/y leads to the equation
fc-l 2n+m-nk
z(mw + n)w'z = zn+m f(z)w n+m (w — l)k + w — w2.
2n+m
Multiplying both sides by w n+m and passing on to a new variable
? = w n+m — w n+m ,
we arrive at the Bernoulli equation
fc-i
(n + m)zCz = -( + z n+m f(z)C.
37. y'L = n(n - l)x~2y + f(x)(xy'x - ny)m.
The substitution w = xny'x — nxn~1y leads to a first order equation with separation
of variables: w'x = xn+m-nmf(x)wm.
38. y'L = n(n - l)x~2y + f(x)g(xny'x - nx^y).
The substitution w = xny'x — nxn~1y leads to a first order equation with separation
of variables: w'x = xnf(x)g(w).
39. y'L = f(x)(xy'x -y)+ g(x)(xy'x - yf.
The substitution w(x) = xy'x — у leads to the Bernoulli equation
w'x = xf(x)w + xg(x)wk.
© 1995 by CRC Press, Inc.

40. y'L = x-1[f(y)(xy'x - y)+g(y)(xy'x - y)k]y'x.
The substitution w(y) = xy'x — y leads to the Bernoulli equation w'y = f(y)w + g(y)wk.
y'L =
(xy'x-v)\f(-)v'a + д(-)Ш
|_ *¦ X ' *¦ X '
The transformation z = y/x, w = xy'x/y leads to the Bernoulli equation: zw'z =
[zf(z)-l]w + zkg(z)wk.
42. yxx = x {xyx — у) JI — ) + x
The transformation x = —1/t, у = —w/t leads to an equation of the form 2.9.4.10:
w't't = f(w)(w'tJ+g(w)(w't)k.
4<i- vxx = (yx) + (хУх-у) н ух(хУх-у)-
xy xy
The transformation z = xnym, w = xy'x/y followed by the substitution
С = W n+m — yj n+m
leads to the Bernoulli equation
fe-i
(n + m)zC = [g(z) -l]C + z n+m /B)^.
2тг+тгг+3 тг+1
44- y'L = ХПут(у'х) п+ггг+2 F(C), C = (xy> _ y)(yx) n+m+2
The Legendre transformation x = w't, у = tw't — w leads to an equation of the form
2.9.4.36:
2a+l-ab ,
where a = - n^+2 , b = -m.
45. y'L = ПХ V } Ш^+^ + 9{ У } y'Jxy'x - y)
xy xy
+ ( v
+ Ш
xy
The transformation z = xnym, w = xy'x/y followed by the substitution
?B) = w n+m — w n+m
leads to the Riccati equation
(n + m)z('z = z n+m h(z)C + [g(z) - 1}( + z
© 1995 by CRC Press, Inc.

46- у1 + У2-
The substitution ? = f <p(x) dx leads to an equation of the form 2.9.4.2:
The substitution w(y) = y'x^[g leads to an equation with separation of variables:
ww'y = f(y)h(w).
48. fyL + ifLy'x = f9(yWJ2 + fn4yWxJn, f = f(x).
f dx
The substitution f = / —, leads to an autonomous equation of the form 2.9.4.10:
49- yL + ea(n-2
This is a special case of equation 2.9.4.46 with ip = e~ax.
50. y'L = -x-Wx + x-2f(xneay)g(xyx).
The transformation z = xneay, w = xy'x leads to an equation with separation of
variables: z(aw + n)w'z = f(z)g(w).
51. yl = ^\zJ^'J b*k
Passing on to new variables z = хпеауу w = xyfXJ we have
/ \/ Oj L /Co p / \ U
z(aw + njti;^ = zT^ + w + J\z)w ¦
Multiplying both sides by w~k and introducing a new variable
Oj
v =
Oh ^ 1 h
+ w
we obtain a first order linear equation:
52.
Passing on to new variables z = eaxym, w = y'x/y, we have
2 Oj Z /C
/ \ f 2 Oj
zimw + a)wz = — w
m
w + j{z)w
m 1 — к
Multiplying both sides by w~ and introducing a new variable
m 9 h a -i h
+ u'
v =
we obtain a first order linear equation:
mzv'z = (к — 2)v + mf(z).
© 1995 by CRC Press, Inc.

53. y'L = v-4vLJ + vf(eaxym)g(y'Jv)-
The transformation z = eaxym, w = y'x/y leads to an equation with separation of
variables: z(mw + a)w'z = f(z)g(w).
54. y'L = x-2f(xneay) ехр(--%У;).
The transformation z = xneay, w = xy'x leads to the equation
z(aw + n)w'z = w + f(z) exp( w)
which can be reduced, with the aid of the substitution С = wexpl—w), to a first
V n /
order linear equation: nzC,z = С + f{z)-
55. y'L = -—VL **(—) + f(eaxym)v'x-
тп \ У /
The transformation z = eaxym, w = y'x/y leads to an equation
z(mw + a)w' = wlnw — w + f(z)w.
m
Dividing both sides by w and passing on to a new variable v = raw + a In w, we obtain
a first order linear equation: mzv'z = —v + mf(z).
56. y^ = (yx)
The transformation z = xneay, w = xy'x leads to the equation
z(aw + n)w' = w w2lnw + w2f(z).
n
Dividing both sides by w and passing on to a new variable v = alnw — nw~1, we
obtain a first order linear equation: nzv'z = —v + nf(z).
2.9.5. Equations of the Form F(x, y, y'x, y"x) = 0
i- у = f(y'D-
The substitution w(y) = \{y'x) leads to an equation of the form 1.9.1.2: у = f(w'y).
2-
The transformation t = y2, w = (y'x) leads to an equation of the form 1.9.1.7: w =
tf2(w't).
3. у = xf(x3y'L)-
The transformation x = 1/t, у = w/t leads to an equation of the form 2.9.5.1: w =
© 1995 by CRC Press, Inc.

4. y = ax2+bx + c + f(y'x-x).
The substitution w = у — ax2 — bx — с leads to an equation of the form 2.9.5.1:
ХУх — У — J\X Ухх )¦
This is a special case of equation 2.9.5.10 with (p = xn.
The Legendre transformation x = w't, у = tw't — w (y'x = t, yxx = l/w't't) leads to an
equation of the form 2.9.5.4: w = at2 + bt + с + f{l/w'{t).
7- fiv'L) + *y'L = v'x-
Solution: 2y = Cxx2 + 2xf{C1) + C2-
Differentiating with respect to x, we obtain
From the equation y'xxx + y'x = 0, we find
у = Asin(x + d) + C2, where A2 = /(C2).
Equating the expression in the square brackets to zero, we arrive at another solution
in the parametric form:
x =
9- f{*){y'L ~ оJ = Ш2
Differentiating with respect to x, we obtain
(y':x - a)BfyZx + fxy'L ~ 2У'Х ~af'x) = 0. A)
/dx
——, w = y'x,
v/
we arrive at a second order linear equation of the form 2.1.9.1:
// 1 ?l
w^-w = -2-afx,
whose right-hand side is to be expressed in terms of ?. Substituting the solution of
the latter equation into the original one, we obtain a relation connecting integration
constants.
Equating the first factor in A) to zero, we find the second solution у = ^а{х+СJ.
10. xy'x-y = f(<pyxx), <p = <p(x).
f x
The transformation ? = / — dx, w = xy'x — у leads to an equation of the form 1.9.1.2:
J 4>
w =
© 1995 by CRC Press, Inc.

п.
Solutions can be found from the equality
(y - Cxf = 2C2{x - A) + Cl, where F(d, A) = 0.
The question of whether there are other solutions calls for further investigation.
12. F(y'^, xy'lx - y'x, x2Vx-x - 2xy'x + 1y) = 0.
It is known that all the functions of the form
where A = A(C1,C2) is determined from the equation F(A,C1,C2) = 0, are the
solutions of the original equation.
v v
It is known that all the functions of the form
where A = A(C\,C2) is determined from the equation F(C\,C2,A) = 0, are the
solutions of the original equation.
14. xyx-y = f(e^y'x-x).
This is a special case of equation 2.9.5.10 with (p = eXx.
2.9.6. General Equations Admitting the Order Reduction
1- Vxx = ПЪ Ух)'
The substitution w(x) = y'x leads to a first order equation: w'x = F(x,w).
2. y'^ =
The substitution w(y) = y'x leads to a first order equation: ww'y = F(y,w).
3- y'xx = F(ax + byi y'x)-
The substitution bw = ax + by leads to the equation of the form 2.9.6.2:
' x b
4. y'lx = xk-2F(x~ky, x1~ky'x).
Homogeneous equation in the extended sense.
The transformation t = In ж, w = x~ky leads to an equation of the form 2.9.6.2:
w"t + Bk - l)w't + k(k - l)w = F(w, w't + kw).
© 1995 by CRC Press, Inc.

Homogeneous equation in the extended sense.
The transformation z = xnym, w = xy'x/y leads to a first order equation:
z(mw + n)w'z = F(z, w) + w — w .
6- y'L = F(x, *y'x - v)-
The substitution w(x) = xy'x — у leads to a first order equation: w'x = xF(x, w).
7. y'L =
The substitution w(x) =y'x leads to a first order equation: xw'x = —w + xF(x, w).
8- y'L = x~2F(y, xy'x - y).
The substitution w(y) = xy'x — у leads to a first order equation: (y + w)w'y = F(y, w).
9- y'L + [/(«) + g(y)]y'x + ГА*)у = o.
r
Integrating yields a first order equation: y'x + f(x)y + / g(y) dy = C.
ю- hy'xy'L + hyy'L + Mv'J2 + Uyy'x + hy1 = o, fk = h(x).
The substitution w(x) = y'x/y leads to the Abel equation
(fiw + f2)w'x + fcw3 + (/2 + h)w2 + Uw + h = 0.
ii. f(y'x)y'L + a(y)y'x + h(x) = o.
Integrating, we obtain
f f f
/ f(u) du+ g(y) dy+ h(x) dx = C, where и = y'x.
J J J
12. Py'L + ffxy'x = *(»» /l/i)» / = /(^)-
The substitution w(y) = fy'x leads to a first order equation: ww'y = Ф(у,1и).
13. x(x - aJy'L = f(y/x).
/ x — a \ xi
The transformation ? = In I 1, w = — leads to an equation of the form 2.9.6.2:
V x J x
w'Je — w't = a~2f(w). For some specific functions /, the solutions of this equation are
given in Subsection 2.2.1.
„ (ex -\- d)n~1 ( (ax -\- Ь)пг
I4- VL =  Л 4., / T Z3T^
""" (ax + b)n+2 \(cx + d)n+
The transformation
ax + b \ (ax + b)ny
cx + dJ' (cx + d)n+l
leads to an equation of the form 2.9.6.2:
w'L - Bn + l)w'? + n(n + l)w = A~2f(w), where A = ad -be.
© 1995 by CRC Press, Inc.

1б- V'L =
This equation can be derived by eliminating subsidiary function z from the simulta-
simultaneous first-order equations
z'x=F(x,z), A)
Ух - ay2 = z. B)
If one succeed in finding the general solution z = z(x, C\) of equation A), the original
equation is reducible to the Riccati equation B) with a known right-hand side.
16. y'L = e--F(e-t/, e™y'x).
The substitution w = eaxy leads to an equation of the form 2.9.6.2:
wxx — 2aw'x + a2w = F(w, w'x — aw).
17. y'L = yF(e™y™, y'Jy).
Exponential homogeneous equation.
The transformation z = eaxym, w = y'x/y leads to a first order equation:
z(mw + a)w'z = F(z, w) — w2.
18- y'L = x~2F(xneay, xy'x).
Exponential homogeneous equation.
The transformation z = xneay, w = xy'x leads to a first order equation:
z(aw + n)w'z = F(z, w) + w.
19. y'lx = e2ayF(xeay, e-ayy'x).
The transformation z = xeay, w = e~ayy'x leads to a first order equation:
(azw + l)w'z = F(z, w) — aw.
20. y'L = аеУу'х + F(x, y'x - oe»).
This equation can be derived by eliminating subsidiary function z from the simulta-
simultaneous first-order equations
z'x = F(x,z), A)
y'x - aey = z. B)
If one succeed in finding the general solution z = z(x, C\) of equation A), the original
equation is equation B) of the form 1.7.2.5 which is readily integrable.
21- y'L = x-2F(ay + b\nx, xy'x).
The transformation z = ay + binx, w = xy'x leads to a first order equation:
(aw + b)w'z = F(z, w) + w.
© 1995 by CRC Press, Inc.

22- V'L = yF(ax + b\ny, y'Jy).
The transformation z = ax + blny, w = y'x/y leads to a first order equation:
(bw + a)w'z = F(z, w) — w2.
23.
Let F ф <p(x)y + ф{х), i.e., the equation is a nonlinear one. Then, its order can be
lowered by one if the right-hand side of the equation has the following form:
F(x,y) = r3/2E^(u) + J[^ffZx(u + V) + f^g'^E-1] dx}, A)
where
E = exp (k f Г1 dx\ V = j f^^gE-1 dx, и = f'^E^y - V;
ф = Ф(и), f = f(x), g = g(x) are arbitrary functions, к is an arbitrary constant.
The integral in A) may always be expressed in terms of E and V. The following
cases are possible:
Case 1. For f'J'xx ф О,
Case 2. For / = ax2 + bx + c, fx ф -2k, fx ф -|/с,
F(x, у) = Г3/2ЕФ(и) + -^ [2fg'x - fxg + 2kg] + (k2 + 1
where A = 4ac — b2.
Case 3. For / = C - 2kx,
F(x,y) = Г2[Ф(у + W) + fg'x + 2kg], W = J f^gdx.
Case 4. For /= J-fcr + /3,
F(x,y) = ф(г2у -и) + r2(fg'x + jk) + jk2u, и = J r2gdx.
In all these cases, the transformation
= JГЧх, u = f-1'2E-1y-V
t
leads to an autonomous equation:
u"t + 2ku't + k2u = Ф(и),
© 1995 by CRC Press, Inc.

which is reducible, with the aid of the substitution z(u) = u't, to the Abel equation:
zz'u + 2kz + k2u = Ф(и)
(see Subsection 1.3.1).
With к = 0, the solution of the original equation for the first and second cases is
as follows:
for case 1,
/ —, = ± / —%- + C2, where Ф(» = / ФШ dw.
J a/2*(u) + Ci J J J
for case 2,
r=± / —2—I bC2, where Ф(и;) = / Ф(у;)сгу;.
Remark. The original equation can be reduced, with the aid of point transfor-
transformations, to an autonomous form only for function F of the form A).
2.9.7. Some Transformations
The transformation ? = 1/x, w = y/x leads to the equation w'L + F(?, w) = 0.
2. y'^x = n(n + l)x~2y + x3nF(x2n+1, xny) = 0.
The transformation ? = x2n+1, w = xny leads to the equation Bn+ lJw'/^ = F(?, w).
3-
ax + b ax + b
О1Г I fj 1]
The transformation ? = —, w = — leads to the equation
ax + b ax + b
w'^ + A-F(C, w) = 0, where A = ad - be.
4- x2v'L + axv'x + ьу + F(x> v) =
The transformation ж = ?", у = ^w, where parameters v and ц, are found from the
simultaneous algebraic equations
2n + l + (a-l)v = 0, /л2 + (a - 1)цр + bv2 = 0,
leads to an equation of the form
© 1995 by CRC Press, Inc.

5. y^x = n(n + l)x~2y + x3nF(ax2n+1 + b, xny).
The transformation ? = ax2n+1 +b, w = xny leads to the equation
6. y^x = X2y + e3XxF(ae2Xx + b, eXxy).
The transformation ? = ae2Xx + b, w = eXxy leads to the equation
_3Aa; / nf>2Xx i и pXxii
7. W = Aw + — —— F\
(ce2Xx + dK V ce2Xx + d ce2Xx + d
The transformation
ae2Xx + b eXxy
«- Ce2Xx+d' W~ ce2Xx+d
leads to the equation
w'^ = BAX)-2F(?, w), where A = ad - be.
соШ(Лж), -
sinh(
Ь(Лж)
V
The transformation ? = coth(Aa?), w = —т-—г leads to the equation
smh(Aa;)
wcc =
9. y'x'x = \2y + cosh-3(\x)F(tanh(\x), V
V
The transformation ? = tanh(Aa;), w = —7-—r- leads to the equation
cosh(Aa;)
Wcc =
- x2vL + TV + VxFfalnx + Ь, -^—\ = 0.
V v x J
у
The transformation ? = alnrr + 6, w) = —— leads to the equation
\Jx
11. \x2 - 1|3/V = F(in
( , r
V x + 1 ^\x2 -1\
ax — а у
The transformation ? = In , w = —, leads to the equation
x + 1 V|2l|
^ = F(?, w) + w.
© 1995 by CRC Press, Inc.

V'L + Х2У + sin-3(\x)F(cot(\x), . V ) = 0.
V sin(Aa:) /
v
The transformation f = cot(Arr), w = -.—r- leads to the equation
sm(Aa;)
V'L + Х2У + cos-3(Xx)F(tan(Xx), ^—) = 0.
V соэ(Лж) /
V
The transformation f = tan(Arr), w = ;—— leads to the equation
сов(Аж)
14. V'L + А2У + -п-3(АЖ + b)Fl . ? I. ' ^-TV-Пл = °-
хх V sin(Aa; + b) sin(Aa; + 6) /
The transformation ? = — —-, ад = т- тт- leads to the equation
вт(Аж + b) вт(Аж + b)
w'^ + [Asin(> - a)]-2F(?,w) = 0.
, 2 x3/2 „ ( У \
15. (ж + 1) yxx + i71! arctanж + o, — I = 0.
The transformation ? = arctan x + 6, ад = —т^^^^ leads to the equation
л/ж2 +1
u;^ + ад + F(?, ад) = О.
16. (ж2 + lf/2yxx + Ffarccot ж + 6, V \ = 0.
V л/ж2 + 1 /
2/
The transformation ^ = arccot ж + 6, ад = —r^=^= leads to the equation
, w) = 0.
The transformation ж = <р{?), у = w^/aip^ leads to the equation
3 ff'^V] _2 , 3/2 /
4 V ip'p I \ * V ' V
The sign of parameter a must coincide with that of derivative ip'*.
Taking у as the independent variable, we obtain the following equation for x = x(y):
x'yy - g{x, y)x'y - f(x, y) = 0.
© 1995 by CRC Press, Inc.

19. F(x, у, y'x, yl) = 0.
The Legendre transformation x = w't, у = tw't — w, where w = w(t), in view of the
relations y'x = t and y'J.x = l/w"t, leads to the equation
F[w't, tw't-w, t, -\
\ wtt
Given the solution of the original equation, the solution of the transformed equation
is written in the parametric form:
t = y'x> w = xy'x-y, where y = y(x).
© 1995 by CRC Press, Inc.

Chapter 3
Third Order
Differential Equations
3.1. Linear Equations
3.1.1. Preliminary Comments
1. A linear homogeneous equation of the third order has the form
h(x)y':xx + h(x)y':x + h(x)y'x + fo(x)y = 0. A)
Let yo = yo (x) be a nontrivial particular solution of this equation. The substitution
y = yo(x) I z(x)dx
J
yields a linear equation of the second order:
hVoz" + Cf3y'o + /2У0У + (З/з^' + 2f2y'o + fiyo)z = 0, B)
where prime denotes differentiation with respect to x.
2. Let г/i = y\(x) and y2 = 2/2[x) be two nontrivial linearly-independent particular
solutions of equation A). Then, the general solution of this equation can be written in the
form
2/ = Ci2/i + C22/2 + C3 B/2 / yiipdx-yx l у2фdx\, C)
where
г/> = ехр(- / -?-dx) (yiy'2 - y[y2) ¦
v J h '
For specific equations described below in 3.1.2—3.1.8, often only particular solutions will
be given, while the generalized solution can be obtained with formula C).
3. A linear nonhomogeneous equation of the third order has the form:
h(x)y':xx + h(x)y':x + h(x)y'x + fo(x)y = g(x). D)
Let 2/1 = 2/1 (x) and 2/2 = 2/2 (x) be two linearly-independent particular solutions of the
corresponding homogeneous equation A). Then, the general solution of equation D) is
defined by formula C) with
+ ^r f-?-AeFdx),
^ J h '
f-?-AeFdx), where F=[^dx, A = (y[y2 - yiy'2).
J h ' J h
© 1995 by CRC Press, Inc.

4. The substitution
y = zexp
(- — / ~^-
reduces equation A) to the form wherein the second derivative is absent:
(fil + (fii)z + (^Va ^4>\4>i + 2У^2 + <Po)z = 0.
where yk = fk/h (k = 0, 1, 2).
3.1.2. Equations Containing Power Functions
1- tC + Ay = 0.
Solution:
+ C2x + C3x2 if A = 0,
{
„ _kx kx/2 (n kxV3 . kxV3 \
Cie Kx + eKx/ I C2 cos — \- C3 sin —-— I if А ф 0,
\ 2 2 J
where к is the real root of the equation A = k3.
2- VZX + ХУ = ax2 +bx + c.
Solution: у = w + — (ax2 + bx + c), where w is the general solution of the equation
3.1.2.1: w'?xx + \w = 0.
3- v'vL = axv + b-
This is a special case of equation 5.1.2.4.
4- vZx + (ax + b)y = °-
For a = 0, this is an equation of the form 3.1.2.1.
For а ф 0, the substitution a? = ax + b leads to an equation of the form 3.1.2.3:
VtK + <y = 0.
5- v'JL* + ax3v = bx-
The substitution ^ = x2 leads to an equation of the form 3.1.2.90: 2^yl'L + 3yl', +
6- tC + (Зо2ж - о3ж3)у = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
2
yx + (ax - а)у = Cex
(see 2.1.2.28 for the solution of the corresponding homogeneous equation).
© 1995 by CRC Press, Inc.

7- tC = axny.
1°. Forn=-9, -7, -6, -9/2, -3, -3/2,1, and 3, see equations 3.1.2.183,3.1.2.180,
3.1.2.175, 3.1.2.185, 3.1.2.150, 3.1.2.184, 3.1.2.3, and 3.1.2.5, respectively.
2°. The transformation x = t~1, у = wt~2 leads to an equation of the similar form:
' 6
w'xxx = -at~n~6w.
3°. For n Ф —3, the transformation ? = ж("+3)/3) u = xn^y leads to an equation of
the form 3.1.2.151:
izu%? + A - v2)?u'? + (v2 - 1 - avziz)u = 0, where гу = —^—.
555 ** n-\- i
8- tC + [о3ж3гг - Зо2ткв2"-1 + on(n - l)xn-2]y = 0.
/ ctxn \
Particular solution: wq = exp ( — ).
V n+ 1 /
/ axn+1 \ f
The substitution у = expl ) / z(x) dx leads to a second order equation
V n+1/J
of the form 2.1.2.44: z%x - 3axnz'x + Ca2x2n - Sanx^z = 0.
9- y*L + abv'x + °2жC - b - ax2)v = o-
By integrating, we obtain a nonhomogeneous second-order linear equation:
yxx + axyx + (a2x2 +ab-a)y = C exp
ax2
(see 2.1.2.28 for the solution of the corresponding homogeneous equation).
vZL + axvL + anv = o. n = l, 2, з, ...
Solution: у = wx , where w is the solution of the second order equation wxx +
axw = C.
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.2.2:
wxx + axw = 0.
12- tC + ожу^ + Ь(ах + Ь2)у = 0.
Particular solution: yo = e~ .
The substitution w = y'x + by leads to an equation of the form 2.1.2.12: wxx —
bw'x + (ax + b2)w = 0.
13- tC + axy'x + (abx + a + Ь3)у = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
Ухх - by'x + (ax + b2)y = Ce~bx
(see 2.1.2.103 for the solution of the corresponding homogeneous equation).
© 1995 by CRC Press, Inc.

. tC + (ax + b)y'x + ay = O.
By integrating, we obtain a nonhomogeneous second-order linear equation:
y'lx + {ax + b)y = C
(see 2.1.2.7 for the solution of the corresponding homogeneous equation).
- tC + (ax + b)y'x -ay = O.
Particular solution: yo = ax + b.
The transformation
ax + b (ax + bJ
leads to a second order equation of the form 2.1.2.62: ?z'/e + 3z'e + a~2?2z = 0.
16. tC + (ax + b)y'x + Say = 0.
The substitution a!; = ax + b leads to an equation of the form 3.1.2.35 with m = 0:
= 0.
17 • VZX + Bax
The substitution a? = ax + \b leads to an equation of the form 3.1.2.37 with n = 1:
^ + ay = 0.
18- <L + (ax ~ Ь2)УХ + аЬхУ = О-
The substitution w = y'x+by leads to an equation of the form 2.1.2.103: w'J.x — bw'x +
axw = 0.
19- tC + (oaj - Ь2)у'х + a(bx + l)y = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
(see 2.1.2.103 for the solution of the corresponding homogeneous equation).
20- vZx + (ax + b)vL + c(ax + b + °2)v = °-
The substitution w = y'x + cy leads to a second order linear equation of the form
2.1.2.12: wlx - cw'x + (ax + b + c2)w = 0.
21- VZX + (ax + b)vL + cx(c2x2 + ax + b- 3c)y = 0.
О
Particular solution: у$ = expf ^~
i p
/ CX \ I
The substitution у = expf J / z(x)dx leads to an equation of the form
2.1.2.28: zlx - 3cxzfx + CcV + ax + b - 3c)z = 0.
© 1995 by CRC Press, Inc.

22. у'ххх + ах2у'х + axy = 0.
This is a special case of equation 3.1.2.36 with n = 1.
23. V'i'xx + ax2y'x — laxy = 0.
The substitution w = xy'x — 1y leads to a second order linear equation of the form
2.1.2.7: wlx + ax2w = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
Ухх + ахУх ~ аУ = C exP
2
(see 2.1.2.103 for the solution of the corresponding homogeneous equation).
25. tC + ax2y'x + Ь(ах2 + Ь2)у = О.
The substitution w = y'x +by leads to an equation of the form 2.1.2.28: w'J.x — bw'x +
(ax2 + b2)w = 0.
26. tC + (o - 1)Ь2х2у'х + b2x(abx2 + 2o + l)y = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
У xx ~ ЬхУх + (ab2x2 + Ь)у = Сехр(
(see 2.1.2.28 for the solution of the corresponding homogeneous equation).
27. yxxx + (ax2 + b)y'x + laxy = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
у%х + (ax2 + b)y = C
(see 2.1.2.4 for the solution of the corresponding homogeneous equation).
28- tC + (ax2 - Ъ2)у'х + axA - bx)y = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
(see 2.1.2.13 for the solution of the corresponding homogeneous equation).
29. y'i'xx + (ax2 + b)y'x + c(ax2 + b + c2)y = 0.
The substitution w = y'x + cy leads to a second order equation of the form 2.1.2.13:
wxx - cw'x + (ax2 + b + c2)w = 0.
© 1995 by CRC Press, Inc.

30- y'Zx ~ (ЗЬ2ж2 + о + 3b)y'x + 2Ьх(Ь2х2 - a)y = 0.
1°. Particular solutions with a > 0:
уг = exp(—^- + x^/aj, y2 =
2°. Particular solutions with a < 0:
( i—\ (bx2\ . , .— s (bx2\
y\ = cos [Xv — a) exp I —— 1, y2 = sin(xv — a) exp I —— 1.
\ Z J \ Z J
3°. Particular solutions with a = 0:
bx2 \ / bx2
yi=exp
\ - x (——)
J' V 2 /
ЛК1 2 У
31. у^'жа; + (ax2 +Ъх + c)y'x + /гж[(о + k2)x2 + bx + с — Зк]у = 0.
/ lev \
Particular solution: yo = exp ( — J.
/ lev \ f
The substitution у = expl J / z(x) dx leads to a second order linear equa-
equation of the form 2.1.2.28:
zxx - 3kxz'x + [(a + 3k2)x2 + bx + c- 3k]z = 0.
32- VZX + (та4 + Ъх)у'х - 2(ож3 + Ъ)у = 0.
This is a special case of equation 3.1.2.38 with n = 2.
33- VZX + ахПУх - ^axn~1y = 0.
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.2.7:
wxx + axnw = 0.
34. yZx + ахПу'х + anxn~1y = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
y»x + axny = С
(see 2.1.2.7 for the solution of the corresponding homogeneous equation).
35- vZx + axm+1y'x + °(m + 3)xmy = 0.
The substitution x = t~1, у = wt~2 leads to an equation of the form 3.1.2.34 with
n = —m — 5:
w'l[t + at~m~5w't — a(m + 5)t~m~6w = 0.
пл /// i^ 2тг / i^ 2тг—1 r»
Solution:
у = ClXJ2v{u) + C2xJv{u)Yv{u) + C3xY2(u),
—;
2(n
where v = —; —, и = —; rxn+1: Jv and Yv are Bessel functions.
2(n+l) 2(n + l)
© 1995 by CRC Press, Inc.

37. у'я'хх + 2а,хпу'х + апхп гу = 0.
Solution:
у = C\w\ + C2W1W2 + C3W2,
where Wi and W2 form a fundamental set of solutions of a second order equation of
the form 2.1.2.7: 2w'x'x + axnw = 0.
38- Ух'хх + (ax2n + bxn~1)y'x - 2(ож2гг~1 + bxn~2)y = 0.
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.2.10:
on" -I- (ат^71 -V- hTn~^\ii) — fl
XX '^ \^""-" <^ UtJb I LU \J.
39. v +(ож
/cxn+1 \
Particular solution: yo = exp ( ).
V n + 1 /
/ cx \ i
The substitution у = expl ) / z(x) dx leads to a second order linear equa-
V n + 1 /J
tion of the form 2.1.2.44:
z'x'x + 3cxnz'x + [(a + 3c2)x2n + (b + Зсп)^]* = 0.
40- У'ххх + 3аУхх + 3а2У'х + °3У = 0-
Solution: у = e-ax(d + C2x + C3x2).
41- Уххх + <^y'L + <ЧУ'х + «оУ = 0.
Denote P(X) = A3 + a2X2 + aiA + a0.
1°. Let the characteristic polynomial P(X) be factorizable:
P(A) = (A-A1)(A-A2)(A-A3),
where Ai, A2, and A3 are real numbers. The following cases may take place:
a) all the roots A^ are different:
у = CxeXlX + C2eX2X + С3еХзХ
b) Xi = A2 ф A3:
у = (d + C2x)eXlX + С3еХзХ.
c) Ai = A2 = A3: see 3.1.2.40 for this case.
2°. Let P(X) = (A - Ai)(A2 + 26iA + b0), where b\ < b0. Then
у = deXlX + e~blX(C2 cos /лх + С3 sin fix), \x= Jb0 - b\.
42- Уххх + пУхх + (ЬХ + С)У'х + (пЬх + пС + Ь)У = °-
By integrating, we obtain a nonhomogeneous second-order linear equation:
y»x + (px + C)y = Ce~ax
(see 2.1.2.2 for the solution of the corresponding homogeneous equation).
© 1995 by CRC Press, Inc.

43. tC + 3oyl + 2(Ьх + а2)у'х + ЬBах + 1)у = 0.
This is a special case of equation 3.1.2.71 with n = 0, m = 1.
44. VZX + av'L + (bx2 + cx + d)y'x + a(bx2 + ex + d)y = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.2.6:
W'L + (bx2 + cx + d)w = 0.
45. tC + ay'lx + bxny'x + abxny = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.2.7:
wlx + bxnw = 0.
46- tC + Зожу^ + 3a2x2y'x + (a3x3 + b)y = 0.
о
/ ax \
The substitution у = wexpf — 1 leads to a constant coefficient equation: w'J.'xx —
3aw'x + bw = 0.
47. tC + ожу! + (оЬж + о - Ь2)у'х + aby = 0.
/2
ехрB6х — J dx.
48- VZX + 3axvL + Bо2ж2 + a + b)y'x + abxy = 0.
Solution:
у = Ciw\ + C2W1W2 + C3W2,
where Wi and W2 are linearly-independent solutions of a second order equation of the
form 2.1.2.25: w'^x + axw'x + \bw = 0.
49. y'Zx + Зожу^ + Bо2ж2 + 2Ьх + a)y'x + ЬBах2 + l)y = 0.
This is a special case of equation 3.1.2.71 with n = 1, m = 1.
50- VZX + 3axvL + 3(о2ж2 + a)y'x + (о3ж3 + Ьх + с)у = 0.
о
The substitution у = expf Jw leads to an equation of the form 3.1.2.4: w'xxx +
[(b - 3a2)x + c]w = 0.
51- VZX + Зожу^ + [2(o2 + b)x2 + a]y'x + 2bx(ax2 + l)y = 0.
This is a special case of equation 3.1.2.71 with n = 1, m = 2.
52- vZx + (ax + b)vZ + (abx + a + c)v'x + Ьсу = °-
By integrating, we obtain a nonhomogeneous second-order linear equation:
ylx + axy'x +cy = Ce~bx
(see 2.1.2.25 for the solution of the corresponding homogeneous equation).
© 1995 by CRC Press, Inc.

53. tC + (abx + о + b)y'lx + аЬ2ху'х - аЬ2у = 0.
Particular solutions: у1 = х, г/2 = е~Ьх.
54. yZL + (ах + Ь)у'^х + [(аЬ + с)х + а]у'х + с(Ьх + 1)у = 0.
By integrating, we obtain a nonhomogenous second-order linear equation:
ylx + axy'x + cxy = Ce~bx
(see 2.1.2.25 for the solution of the corresponding homogeneous equation).
55. yxxx + (ax + b + c)y'lx + (acx + bc+ s)y'x + s(ax + b)y = 0.
Particular solutions: y1 = eXlX, y<i = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + cX + s = 0.
56. y'l'xx + (ax + Ъ)у'1х + (ex + 1a)y'x + a[(c - ab)x2 + b]y = 0.
О
/ ax \
Particular solution: у$ = expf ^~j-
о /.
/ (XX \ I
The subbstitution у = expf — J / z(x) dx leads to a second order linear equa-
equation of the form 2.1.2.28:
z'x + [a2x2 + (c - 2ab)x - a]z = 0.
57. yxxx + (ax + b)y'lx + (ex + d)y'x + [acx2 + (ad + Ьс)х + с + bd]y = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
О
/ CIT* \
Ухх + (cx + d)v = Cexpl^ bxj
(see 2.1.2.2 for the solution of the corresponding homogeneous equation).
58- yZx + (ax + b)vL + (cxx2+Cx+'f)yx-k[<xx2 + (ak+C)x + k2 + bk+j}y = 0.
The substitution w = y'x — ky leads to a second order equation of the form 2.1.2.28:
wxx + (ax + b + k)w'x + [ax2 + (ak + C)x + k2 + bk + j]w = 0.
59. tC - x2y'lx + (a + b - l)xy'x - aby = 0.
The following three series, converging for any x, make up a fundamental set of solu-
solutions:
^ ab(a - 3)F - 3)... (a - 3n + 3)(b - 3n + 3) 3n
yi~1 + 2^ Cn)! X '
n=l ч '
(ol)(bl)(o4)(b4)(o3n + 2)(b3n + 2) 3n+1
Cn + l)!
„ , x 5)...(o-3n + l)(b-3n + l) 3та+2
2/3 " 2 +^- Cn + 2)! Ж '
x2 ^ (a- 2)F - 2)(a - 5)F - 5)... (a - 3n + l)(b - 3n
© 1995 by CRC Press, Inc.

60. y'xxx + axny'x[x - 2axn~2y = 0.
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.2.42:
wxx + axnw'x + axn~1w = 0.
fil ii1" 4- nrrnii" hii1 nhrrnii — fl
1°. Particular solutions with b > 0: У\= exp(—xVb), y2 = ещ>(хл/Ь).
2°. Particular solutions with b < 0: y\ = сов(жл/—b), У2 = sin(a;-\/—b).
Particular solutions: y\ = x, У2 = x .
б3- У'ххх + ахПУхх + Ьхп~1у'х — 2(o + b)xn~2y = 0.
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.2.42:
wlx + axnw'x + (a + b)xn~1w = 0.
64. y'xxx + axnyxx - (ож"-1 - Ъх2)у'х + bx(axn+1 + 3)y = 0.
x2Vb\ . fx2Vb
Particular solutions: y\ = cos ( —-— ], У2 = sin
65. y'" -\- axny" -\- (abxn -\- anxn * — b2)y' -\- abnxn *y = 0.
Particular solutions: y1 = e~bx, y2 = e~bx / exp( 2bx xn+1) dx.
J V n+1 /
xxx + axny'x'x + bxmy'x - bxrn~1y =
Particular solution: yo = x.
66. yxxx + axny'x'x + bxmy'x - bxrn~1y = 0.
67. yxxx + axny'x[x + bxmy'x + bxm-1(axn+1 + m)y = 0.
The substitution w = yxx + bxmy leads to a first order linear equation: w'x + axnw = 0.
68- VZX + ахПУхх ~ H2axn + Щу'х + Ь2(ахп + 2Ь)у = 0.
Particular solutions: y\ = еЬх, у2 = хеЬх.
69. у'1'хх + ахпу'1х + (аЬхп - Ь2 + с)у'х + с(ахп - Ъ)у = 0.
Particular solutions: y\ = eXlX, У2 = еХ2Х, where Ai and A2 are the roots of the
quadratic equation A2 + ЬХ + с = 0.
70. tC + axny'lx + (bxm - c2)y'x - c(acxn + bxm)y = 0.
Particular solution: yo = ecx.
Solution:
у = C1w\ + C2W1W2 + C3W2,
where w\ and W2 form a fundamental set of solutions of the second order linear
equation wxx + axnw'x + -jbxmw = 0.
© 1995 by CRC Press, Inc.

72- fC = (xn - a)y':x + (ax" - b)y'x + bxny.
Particular solutions: y\ = eXlX, y<i = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + aX + b = 0.
73- VZX + (°ж™ + b)v'L + (осж™ + Ьс + т)ух + (т + с2)(ахп + Ь-с)у = 0.
Particular solutions: y\ = eXlX, y<i = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + cX + m + c2 = 0.
74. y'i'xx + (axn - Ь)у?а + cxmy'x - b(abxn + cxm)y = 0.
Particular solution: yo = еЬх ¦
75- tC + (xn + а)Ух-х + (axn + bxm)y'x + abxmy = 0.
Particular solution: yo = e~ax.
76- Ух'хх + (ахП + c)vL + [acxn^ + (оп + Ь)ж"-1)у4 + Ь[сжгг-1 + (п-1)хп-2]у = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
y':x + axny'x+bxn-1y = Ce-cx
(see 2.1.2.42 for the solution of the corresponding homogeneous equation).
7. y'i'xx + (axn + bx)iC + Ь(ахп+г + 1)y'x + abxny = 0.
r, ,. , , , • ( bx2 \ ( bx2 \ f (bx2 \
Particular solutions: y\ = exp I — 1, 2/2 = exp I — 1 / exp I —— 1 ax.
78-
f ( Ьх2 \
Particular solutions: yx = е~ет, уг = e~cx I exp( 2cx — J dx.
J V 2 J
79- Уххх + (abxn + ax™-1 + b)y'lx + ab2xny'x - ab2xn~1y = 0.
Particular solutions: y\ = x, yi= e~bx.
Sfl ii1" 4- (nrrn 4- h'rrn\ti" 4- ft/ 4- n(nrrn 4- h'rrn\ti — fl
1°. Particular solutions with с > 0: y\ = cos(xy/c), У2 = sin^Xy/c).
2°. Particular solutions with с < 0: y\ = exp(—ж^/—с), уг = ехр(жу/—с).
81. У^'жа; "I" (а>хп + Ьхт)ухх + (abxn+m + bcxm + bmxm г — с2)у'х
+ с(оЬжгг"*""г — асхп + Ьтхт~1)у = 0.
_ { ( bxm+1 \
Particular solutions: yi = е~ет, уг = е~са: / ехрBсж — ) cte.
j V m +1 /
82. sctC, - о2(ож + 3)y = 0.
Particular solution: yo = жеоа:.
The substitution у = жеоа: f z(x) dx leads to a second order equation of the form
2.1.2.103: xzlx + 3(ax + l)z'x + 3a(ax + 2)z = 0.
© 1995 by CRC Press, Inc.

83. xyZx + ay'x + Ь(Ь2х + a)y = 0.
The substitution w = y'x + by leads to a second order equation of the form 2.1.2.103:
xwxx ~ bxw'x + (b2x + a)w = 0.
84- XVXL + axvL ~ ib(a + Ь2)х + а + ЗЬ2]у = О.
XL
ebx
Particular solution: y0 = xebx.
The substitution у = xebx f z(x) dx leads to a second order equation of the form
2.1.2.103: xzlx + 3(bx + 1L + [(a + 362)rr + 6b}z = 0.
85- XVXL + (b ~ °?x)y'x + aby = 0.
The substitution w = y'x + ay leads to an equation of the form 2.1.2.103: xwxx —
axw'x + bw = 0.
86- xvZx + (ax2 + bx)v'x - 2(ax + b)v = °-
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.2.2:
wxx + (ax + b)w = 0.
~ 2(ax2 + b)y = 0.
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.2.4:
wlx + (ax2 + b)w = 0.
88. xyZx + 3yl + axy = 0.
The substitution w = xy leads to a constant coefficient equation of the form 3.1.2.1:
w'xxx + aw = 0.
89. xy'Zx ~ SnyZ + axy = 0, n = 0, 1, 2, ...
Solution:
\4
с2 dx
where w is the solution of the equation 3.1.2.1: w'xxx + aw = 0.
90. 2xyx'xx + 3yxx + axy = b, а ф 0.
Solution:
A fK e**d
y=yc,
t^i" Jo V2z3 + a
where Ai, A2, and A3 are the roots of the cubic equation 2A3 + a = 0; A4 = — 00 with
x > 0 and A4 = +00 with x < 0. In addition, constants Cv are related by the constraint
л/а (Ci + C2 + C3 + C4) + 6 = 0, and the integrals are taken along straight lines.
91. xy'xxx + Sy'Z + ax2y = b.
The substitution w = xy leads to an equation of the form 3.1.2.3: w'xxx + axw = b.
92. xyxxx + Sy'Z + ax4y = bx.
The substitution w = xy leads to an equation of the form 3.1.2.5: w'xxx + ax3w = bx.
© 1995 by CRC Press, Inc.

93. xvZx + аУхх + аЬУх + ь3хУ = °-
The substitution w = у'х + by leads to a second order linear equation of the form
2.1.2.103: xwxx + (a- bx)w'x + b2xw = 0.
94. xyZx + (a + b)yZ — xy'x — ay = О, о > 0, b > 0.
Solution:
з
where 71 = —1, /?i = 72 = 0, (З2 = 1; for ж > 0, 73 = 1 and /З3 = +00; for x < 0,
73 = — 00 and /З3 = —1.
95. xyZx + ayZ + (b- c2)xy'x - c(ac + bx)y = 0.
The substitution w = y'x — cy leads to a second order equation of the form 2.1.2.103:
xwxx + (ex + a)w'x + (bx + ac)w = 0.
96. xyZx + ayZ + [(c - Ъ2)х + ab\y'x + c(a — bx)y = 0.
Particular solutions: y\ = eXlX, yi = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + ЬХ + с = 0.
97. xyZx + ayZ + bxny'x + b(a + n - \)xn~xy = 0.
The substitution w = yxx+bxn~1y leads to a first order linear equation: xw'x+aw = 0.
98. xyZx + (ax + b)yZ ~ a2by = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.2.103:
xwxx + bw'x — abw = 0.
99. xyZx + (ax + b)yZ + cxy'x - cy = 0.
The substitution w = xy'x — у leads to a second order equation of the form 2.1.2.103:
xwxx + (ax + b — l)w'x + cxw = 0.
100. xyZx + (ax + 3)yZ + (bx + 2a)y'x + (ex + b)y = 0.
The substitution w = xy leads to a constant coefficient equation: w'xxx + awxx + bw'x +
cw = 0.
- XVZX + (ax + 3)vL + a(bx + 2)vL + b[b(a - b)x + a]y = 0.
Solution:
У = —
102. xyZx + [a(b + l)x + b]yZ + a2bxy'x - a2by = 0.
Particular solutions: y\ = x, yi= e~ax.
© 1995 by CRC Press, Inc.

C -(x-2a- l)y'x + (x - l)y = 0.
Solution:
у = Cxe* + xa+1[C2Ia+1{x) + C3Ka+1(x)},
where Ia and Ka are modified Bessel functions.
104. 2xy'x"xx -Цх + а- l)y^ + B* + 6o - b)y'x + A - 2a)y = 0.
Solution:
у = Cxe* + xaex'2[C2Ia{x/2) + C3Ka(x/2)},
where Ia and Ka are modified Bessel functions.
105. 2xyZx + 3Bож + k)y'Z + 6(bx + ak)y'x + {lex + 3bk)y = 0, к > 0.
Solution:
where P(z) = z3 + iaz2 + ibz + c; Ai, A2, and A3 are the roots of this polynomial
that are assumed to be different; A4 = — 00 with x > 0 and A4 = +00 with x < 0;
4 — *~^l *~^2 ^3-
106. xy'xxx + (ax + b)vxx + [(oc + •s ~ с2)ж + bc]y'x + .s[(o — c)x + b]y = 0.
Particular solutions: y\ = eXlX, y2 = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + cX + s = 0.
107. xyZx + (ax2 + b + 2)y'Z - ab(b + l)y = 0.
This is a special case of equation 3.1.2.109 with n = 2.
108- XVXL + (ax2 + b)v'L + ±axy'x + 2ay = 0.
Integrating the equation twice, we arrive at a first order linear equation: xy'x +
{ax2 + b-2)y = C1 + C2x.
109- XVX'XX + (ахП + b + 2)УХХ ~ ab(b + l)xn~2y = 0.
The substitution w = xy'x + by leads to a second order equation of the form 2.1.2.42:
<x + аж"»; - a{b + l)xn-2w = 0.
110. xyZx + (axn + 3)yZ + Baxn-1 + bx)y'x + b(axn + l)y = 0.
Particular solutions: Vi = —cos(xVb), У2 = —sin(xVb).
X ' X '
111. xy'Zx + (axn+1 + 3)y'x'x + a(bx + 2)xny'x + b(abxn+1 + axn - b2x)y = 0.
Particular solutions:
bx\ ( bxV3 \ 1 / bx
© 1995 by CRC Press, Inc.

112. xy'Zx + (axn + 3)y'x[x + (abxn + 2axn~1 - b2x)y'x + b(axn~x - b)y = 0.
Particular solutions: y1 = —, y2 = —e~bx.
X X
113. (ax + b)y'?xx + ey'x + k(ak2x + bk2 + c)y = 0.
The substitution w = y'x + ky leads to a second order equation of the form 2.1.2.103:
(ax + b)wxx - k(ax + b)w'x + (ak2x + bk2 + c)w = 0.
114. (ax + 2)y'lxx ~ a3xy'x + 2a3y = 0.
Particular solutions: y\ = x , yi = e~ax.
115. (acx + bc- a)y'x"xx ~ c3(ax + b)y'x + ac3y = 0.
Particular solutions: y\ = ax + b, y2 = ecx.
116. (ax + b)yZx + (ex + d)y'x + s[(as2 + c)x + bs2 + d]y = 0.
The substitution w = y'x + sy leads to a second order equation of the form 2.1.2.103:
(ax + b)wxx - s(ax + b)w'x + [(as2 + c)x + bs2 + d]w = 0.
117. (ax + b)y'?xx + [(c - ak2)x + d - bk2]y'x + k(cx + d)y = 0.
The substitution w = y'x + ky leads to a second order equation of the form 2.1.2.103:
(ax + b)wxx — k(ax + b)w'x + (ex + d)w = 0.
118. (ax + b)yZx + [Ha + l)x + b2 + \\y'lx + b2xy'x - b2y = 0.
Particular solutions: y1 = x, y<i= e~bx.
119. (ax + b)y'l'xx + k(ax + b)y'^x + (ex + d)y'x + k(cx + d)y = 0.
The substitution w = y'x + ky leads to a second order equation of the form 2.1.2.103:
(ax + b)wxx + (ex + d)w = 0.
120. (ax + b)y'xxx + (ex + d)yxx + [(оЛ + c\x)x + bA + dy\y'x
+ (Л + /Lt2)[(c — а/л)х + d — Ьц]у = 0.
Particular solutions: уг = exp(six), y^ = exp(s2a;), where S\ and s^ are the roots
of the quadratic equation s2 + /is + A + /к2 = 0.
121. (ax + b)y'xxx + (ex + d)yxx — fe[Cofe + 2c)x + 3bk + 2d]y'x
+ k2[Bak + c)x + 2bk + d]y = 0.
Particular solutions: y1 = ekx, y<i = xekx.
122. (ax + b)y'xxx + (ex + d)yxx + sx(ax + b)y'x + s[cx2 + (a + d)x + b]y = 0.
The substitution w = yxx + sxy leads to a first order linear equation: (ax + b)w'x +
(ex + d)w = 0.
© 1995 by CRC Press, Inc.

123. A - x)y'x\'xx + x(ax - 2o + l)y'^x + (-ax2 + 2o - l)y'x + 2a(x - l)y = 0.
Particular solutions: y\ = x2, yi = ex.
124. (ax + b)y'xxx — (о3ж3 — Зо2ж + Ь3)у'х + abx(a2x2 — За — Ъ2)у = 0.
Particular solutions: уг = еЬх, у2 = ехр(—-
125. (о
The substitution w = yxx + sxny leads to a first order linear equation: (ax + b)w'x +
(ex + d)w = 0.
126. (ax - l)yZx + x(abxn+1 - 1bxn - a2)y'^x
+ Bbxn - a2bxn+2 + a2)y'x + 2ab(ax - l)xny = 0.
Particular solutions: y\ = x2, 2/2 = eax ¦
127. x2y'x"xx - 6y'x + ax2y + Ibx = 0.
The substitution у = х w leads to the equation 3.1.2.159 wherein w should be sub-
substituted for y.
128. x2y'x4xx + (ax2 + bx-m2 - m)y'x + (m - l)(ax + b)y = 0.
The substitution w = xy'x + (m — l)y leads to a second order equation of the form
2.1.2.103: xwxx - (m + l)w'x + (ax + b)w = 0.
129. x2y'l'xx + (ax2 + bx + c)y'x - k[(a + k2)x2 + bx + c]y = 0.
The substitution w = y'x — ky leads to a second order equation of the form 2.1.2.130:
x2wxx + kx2w'x + [(a + k2)x2 + bx + c]w = 0.
130. x2y'x"xx + (axn - b2 - b)y'x + a(b - \)xn~xy = 0.
The substitution w = xy'x + (b — l)y leads to a second order equation of the form
2.1.2.62: xwlx - (b + l)w'x + ахп~гуо = 0.
131. x2y'l'xx + 3xy'lx - 3y'x + ax2y + b = 0.
Solution: у = ( — ), where function w = w(x) satisfies a constant coefficient
ax V x I
equation of the form 3.1.2.2: w'xxx + aw = b.
132. x2y'l'xx + bxy'lx + 6y'x + ax2y = b.
The substitution w = x2y leads to a constant coefficient equation of the form 3.1.2.2:
"' b
w"' + aw = b.
133. x2y'xxx — 3(n + m)xyxx + 3nCm + l)y'x — x2y = 0, m, n = 1, 2, 3, ...
Solution:
n-l m-l
M=0 v=0 fe=l
where 6 = x , u>k are three roots of the cubic equation u>3 = 1.
ax
© 1995 by CRC Press, Inc.

134. x2yZx + Ьху'1х + 6y'x + ax3y = b.
The substitution w = x2y leads to an equation of the form 3.1.2.3: w'xxx + axw = b.
135. x2y'Zx ~ 2(n + l)xy'^x - (ax2 - 6n)y'x + 2axy = 0, n = 1, 2, 3, ...
Solution:
_ Г d + C2x4 + C3x2n+1 if a = 0,
У ~ I d (ax2 - 4n + 2) + С2ех*лР(х) + C3e~x'/KQ(x) if а ф 0,
where P and Q are some polynomials of the degree < 2n + 2.
2yZx + Zxy'L + (±a2x2a + 1 - 4a2b2)y' + 4a3x2a-1y =
Solution:
136. x2yZx + Zxy'L + (±a2x2a + 1 - 4a2b2)y'x + 4a3x2a-1y = 0.
у = С^2(ха) + C2Jb(xa)Yb(xa) + C3Y2(xa),
Y
where Jb and Yb are Bessel functions.
137. x2y'l'xx + ax2Vx-x + (bx + c)y'x + a(bx + c)y = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.2.106:
x2wxx + (bx + c)w = 0.
138. x2y'l'xx + ax2y'lx + (bxn + c)y'x + a(bxn + c)y = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.2.113:
x2w^x + (bxn + c)w = 0.
139. x2yZx ~(x + o^xy'Z + aBx + l)y'x - a(x + l)y = 0.
Solution:
у = Ciex + x^a+1^2 [C2 Ja+i BV5S) + C3Ya+1 ()]
where Ja and Ya are Bessel functions.
140. x2yZx ~ (x2 - 2x)yZ - (x2 + a2 - \)y'x + (x2 - 2x + a2 - \)y = 0.
Solution:
у = dex + ^c [C2Ia(x) + C3Ka(x)\,
where Ia and Ka are modified Bessel functions.
x2yZx ~ 2x(x - l)yZ + (x2-2x + ±- a2)y'x + (a2 - \)y = 0.
Solution:
y = C1ec + ^ex'2 [C2Ia(x/2) + C3Ka(x/2)\,
where Ia and Ka are modified Bessel functions.
1 АО tjii'" Ч^т-
Solution:
У =
where w\ and w2 form a fundamental set of solutions of a second order equation of
the form 2.1.2.103: xwxx + (a - x)wx + bw = 0.
© 1995 by CRC Press, Inc.

143. x2yxxx + ж [(о + с)х + b]yxx + [(ос + а)х2 + (be + /3)х -+
+ с(аж2 + /Зж + 7)у = 0.
The substitution w = у'х + су leads to an equation of the form 2.1.2.141 with n = 1:
x2wxx + x(ax + b)w'x + (ax2 + /3x + j)w = 0.
144. x2y'l'xx + (ax^1 - b2 - b)y'x + a(b - l)xny = 0.
The substitution w = xy'x + (b — l)y leads to a second order equation of the form
2.1.2.62: xw'^x -(b+ l)w'x + axnw = 0.
145. x2y'x"xx - 3c^"+1(n + axn+1)y'x + axn(n - n2 + 2a2x2n+2)y = 0.
/ {ТУ \ / {ТУ \
1°. Particular solutions with n ф — 1: У\= ехр( ), 2/2 = жехр( ).
Vn+1/ Vn+1/
2°. Particular solutions with n = —1: y\ = xa, г/2 = xa+1.
2п,ГГГ _i_ /л™п+1 _i и~,\л,11 _i_ \„(и оЛ™п _i „I,,/ _i_ „/„ и _i_ оЛ„гг—1», —
146. x2y'l'xx + (ax^1 + bx)y'ix + [a(b - 2)xn + c]y'x + a(c-b + 2)xn~1y = 0.
Particular solutions: y\ = xmi, y<i = xm2, where mi and mi are the roots of the
quadratic equation: m2 + (b — 3)m + c — 6 + 2 = 0.
147. x(ax + b)y'l'xx + x(cx + d)y'x - 2(cx + d)y = 0.
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.2.103:
(ax + b)wxx + (ex + d)w = 0.
148. 2x(x - l)yZx + 3Bж - l)y^ + Bax + b)y'x + ay = 0.
Solution:
у = C\W2 + C2W1W2 + C3W2,
where Wi and W2 form a fundamental set of solutions of the equation
2x(x - I)u4 + Bx - l)w'x + (-|ж + - - -)«; = 0,
which is reduced, by means of the substitution x = cos2 ?, to the Mathieu equation
2.1.6.4: 2w'^ = (a + b-2
149. (a,2X -\-aix-\-ao)yxxx-\-(bix-\-bo)yxx-\-(cix-\-co)yx — mc\y = 0, ci т^ 0,
where то is a positive integer.
A solution of this eqaution is a polynomial of the degree то, which can presented as
follows:
1 к
— J {xmlx~m~1[(ax2 + aix + ao)D3 + (bix + bo)D2+c0D\) xm,
k=o Cl
d xv+1
where D = , Ixv = with v ^ —1.
dx v + l r
© 1995 by CRC Press, Inc.

150.
This is a special case of equation 3.1.2.161.
Solution:
where n\ and n<i are the roots of the quadratic equation n2 + an + a2 — 1 = 0.
1B1- x3vZx + A - a2)xy'x + (bx3 + a2 - l)y = 0.
For a = ±1, we have a constant coefficient equation of the form 3.1.2.1. For b = 0, we
obtain the Euler equation 3.1.2.161.
If Ь ф 0 and a is a positive integer greater than 1, then
2/ = ж
fe=i
where Ai, A2, A3 are the roots of the cubic equation A3 = b and Pk(x) are polynomials
of the degree < 3(a — 1).
Denote the solution of the original equation for arbitrary (including complex) a
by ya. Then, the recurrence relation holds
Уа+з = Ьуа + Ba + 3)x-1y>; - (a + l)Ba + 3)(x-2y'a - жуа). A)
Since y±\ = e~Xx, corresponding to three values of A which satisfy the condition A3 = b,
make up a fundamental set of solutions, then formula A) makes it possible to find
all yn for any integer values of n not divisible by 3. In particular, y2 = (ж + X)e~^x
(A3 = b).
152. x3y'x"xx + Dж3 + ax)y'x - ay = 0.
Solution:
у = ClXJ2(x) + C2xJv(x)Yv(x) + C3xY2(x),
where Jv and Yv are Bessel functions; Av = 1 — a.
153- x3vZx + xlax2 + 3b(! - Ь)}У'Х + 2Ь(аж2 + Ь2 - l)y = 0.
1°. Particular solutions with a > 0: y\ = хьsin(xifa}, y^ = xbcos(xy/a).
2°. Particular solutions with a < 0: yx = ж6ехр(—Жу7—«), 2/2 = ж6ехр(жу/—«)•
3°. Particular solutions with a = 0: г/i = жь, г/2 = жь+1.
154. ж3!/^ + ж(ож2 + Ьх + с)у'х + (к - 1)(ах2 + Ьх + с + к2 + к)у = 0.
The substitution w = ху'х + (к — 1)у leads to a second order linear equation of the
form 2.1.2.126:
x2w'xX - (k + l)xw'x + (ax2 + bs + c + k2 + k)w = 0.
155. <cstC + axny'x + (b - 1)(ож"-1 + b2 + b)y = 0.
The substitution w = xy'x + (b — l)y leads to a second order linear equation of the
form 2.1.2.127:
x2wxx -(b + l)xw'x + (ax71'1 + b2 + b)w = 0.
© 1995 by CRC Press, Inc.

156. x3yZx + x(axn + b)y'x - 2(axn + b)y = 0.
The substitution w = xy'x — 2y leads to a second order linear equation of the form
2.1.2.113: x2w'^x + (axn + b)w = 0.
157. x3yZx + x(axn + b- c)y'x + (c - l)(axn + b + c2)y = 0.
The substitution w = xy'x + (c — l)y leads to a second order linear equation of the
form 2.1.2.127:
x2wxx - (c + l)xw'x + (axn + b + c2)w = 0.
158. x3yZx + (ax2n + 1 - n2)xy'x + [bx3n + a(n - l)x2n + n2 - l]y = 0.
The transformation ? = —xn, z = xn~1y leads to a constant coefficient equation:
4'^ + «4 + bz = o.
159. x3y'Zx + 6х2Ухх + (ax3 - 12)y + 2b = 0.
Solution: у = —— (—г-), where w = w(x) satisfies a constant coefficient equation
(xx V x J
of the form 3.1.2.2: w'?xx + aw = b.
160. x3yZx + ax2y'Z + bxy'x + (o - 2)by = 0.
This is a special case of equation 3.1.2.161.
Solution:
у = С1Х2~а + C2xni + C3xn\
where n\ and n2 are the roots of the quadratic equation n2 — n + b = 0.
161. x3y'Zx + ax2y'Z + bxy'x + cy = 0.
The Euler equation.
The substitution t = \n\x\ leads to a constant coefficient equation of the form
3.1.2.41: y'Ht + (a - S)y& + B - a + b)y[ + cy = 0.
162. x3y'Zx + Зах2Ух-х + 3o(o - l)xy'x + [bx3 + o(o - l)(o - 2)]y = 0.
The substitution w = xay leads to a constant coefficient equation of the form 3.1.2.1:
wxxx + bw = 0.
163. x3y'Zx + Зах2Ух-х + 3o(o - l)xy'x + [bxn + o(o - l)(o - 2)]y = 0.
The substitution w = xay leads to an equation of the form 3.1.2.7: w'xxx + bxn~3w = 0.
164. x3y'Zx + 3A - a)x2y'xlx + x[4b2c2x2c + 1 - ±v2c2 + 3o(o -
+ [4b2c2(c - a)x2c + a(±v2c2 - a2)]y = 0.
Solution:
у = ClXaJ2(u) + C2xaJv{u)Yv{u) + C3xaY2(u),
where и = bxc; Jv and Yv are Bessel functions.
© 1995 by CRC Press, Inc.

165.
x3yZx + (ax2 + Ь)у?а + 2Ba - 9)xy'x + 2(o - 6)y = 0.
Integrating the equation twice, we arrive at a first order linear equation: x3y'x +
[(a - 6)x2 + b}y = d + C2x.
166.
x2(ax
cxy'x + c(ax + b - 2)y = 0.
Particular solutions: y\ = xni, yi = xn2, where n\ and n<i are the roots of the
quadratic equation n2 — n + с = 0.
167.
x2Bax
x(a2x2 + 2abx + c)y'x + (a2bx2 + bc- 2c)y = 0.
Particular solutions: y\ = e~axxni, yi = e~axxn2, where n\ and n<i are the roots
of the quadratic equation n — n + с = 0.
168. хъу'1'хх + €ixnyZa + bxy'x + b(axn~2 - 2)y = 0.
Particular solutions: y\ = xmi, yi = xm2, where mi and mi are the roots of the
quadratic equation m? — m + b = 0.
169. хъу'1'хх + x2(axn + b)y'^x + x(axn + b- l)y'x + (axn + b - 3)y = 0.
Particular solutions: y\ = совAпж), y<i = sin(lna;).
170.
x2(axn + b + с + l)y'^x + x[ax2n + (ac + f3)xn + 7 + bc]y'x
(c - l)(ax2n + f3x
= 0.
The substitution w = xy'x + (c — l)y leads to a second order equation of the form
2.1.2.141: x2w'^x + x(axn + b)w'x + (ax2n + Cxn + j)w = 0.
171. x2(ax + b)y'xxx + (ex — bm2 — bm)y'x + (m — l)(c + am2 + am)y = 0.
The substitution w = xy'x + (m — l)y leads to a hypergeometric equation of the form
2.1.2.159: x(ax + b)wxx - (m + I)(ax + b)w'x + (c + am2 + am)w = 0.
172. x(ax2 + bx + c)y'l'xx + xy'x - 2y = 0.
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.2.166:
(ax2 + bx + c)wxx + w = 0.
173. (ax + Ъ)хъу'1'хх + (ex + d)x2y'Jx + s(ax + b)xy'x + s[(c- 2a)x + d- 2b]y = 0.
Particular solutions: y\ = xni, 2/2 = xn2, where n\ and П2 are the roots of the
quadratic equation n — n + s = 0.
174. xby'i'xx = а(ху'х - 2у).
Solution:
У =
d + C2 exp
+ d cos
x
/—a
x
xp(-^-)| ifa>0,
+ C3 sin
'—a
if a < 0.
© 1995 by CRC Press, Inc.

175. x6yZx = ay + bx2.
The transformation x = t~1, у = wt~2 leads to a constant coefficient equation of the
form 3.1.2.2: w't"t + aw + b = 0.
176. x6yZx + ax2y'x + (b- 2ax)y = 0.
The transformation x = t~1, у = wt~2 leads to a constant coefficient equation: w't"t +
aw't — bw = 0.
177. <cetC + 6ж5у^ - ay + 2bx = 0.
The substitution x = i leads to an equation of the form 3.1.2.127: i2y"/t — 6y't +
at2y - 2bt = 0.
178. (x - aK(x - b)stC - cy = 0, о ф Ь.
K(x - Ь)*у["
The transformation
x — a
x — b
w =
У
(Ж-
leads to a constant coefficient equation: (a — bK(w't"t — 3w"t + 2w't) — cw = 0.
179. (ax2 + bx + cKy'Zx = &У-
The transformation
f
/
аж2 + 6ж + с ' аж2 + 6ж + с
leads to a constant coefficient equation: w'^ + Dac — 62)w^ = kw.
180. x7y"' = ay -
The transformation x = t~x, у = wt~2 leads to an equation of the form 3.1.2.3:
w"lt + atw + 6 = 0.
181. xrvZL + (ax + Ь)У = 0-
The transformation x = t~x, у = wt~2 leads to an equation of the form 3.1.2.4:
w'llt - (bt + a)w = 0.
182. x9yZx + (a3 - 3a2x2)y = 0.
The transformation x = t~x, у = wt~2 leads to an equation of the form 3.1.2.6:
w'llt + Ca2i - a3t3)w = 0.
183. x9yZx = ay.
The transformation x = t~x, у = wt~2 leads to an equation of the form 3.1.2.5:
w'llt + at3w = 0.
184. x3/2yZx = ay.
This is a special case of equation 5.1.2.25 with n = 1.
© 1995 by CRC Press, Inc.

185. x9'2y'" = ay
This is a special case of equation 5.1.2.26 with n = 1.
186. x2{xn + a)y'?xx + x(bxm+1 + 1nxn + cx)y'^x
+ [2bmxm+1 + n(n - l)xn]y'x + bm(m - l)xmy = 0.
The twofold integration yields a first order linear equation:
(xn + a)y'x + (bxm + c)y = d + C2x.
3.1.3. Equations Containing Exponential Functions
!• VvL - aeXx(a2e2Xx + 3aXeXx + X2)y = 0.
Particular solution: yo = exP( ~TeXx) ¦
The substitution у = exp( — eXx) / z(x)dx leads to a second order equation of
the form 2.1.3.29: z'^x + 3aeXxz'x + Ca2e2Xx + 3a\eXx)z = 0.
2- У'1'хх + aeXxy'x + aXeXxy = be".
By integrating, we obtain a nonhomogeneous second-order linear equation:
(see 2.1.3.1 for the solution of the corresponding homogeneous equation).
3- tC + aeXxy'x + b(aeXx + Ъ2)у = О.
The substitution w = y'x + by leads to a second order equation of the form 2.1.3.10:
wxx — bw'x + (aeXx + b2)w = 0.
4- У'1'хх + аеХхУх + bx(aeXx + b2x2 — 3b)y = 0.
Particular solution: yo = exp I —
bx2
\
2 J
5- VZL + aeXxy'x + [о(Л - b)eXx - Ь3]у = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
y:x + by'x + (aeXx + b2)y = Cebx
(see 2.1.3.10 for the solution of the corresponding homogeneous equation).
6- tC + (aeXx - Ъ2)у'х + abeXxy = 0.
The substitution w = y'x + by leads to a second order equation of the form 2.1.3.10:
W'L - bw'x + aeXxw = 0.
© 1995 by CRC Press, Inc.

7- tC + (aeXx - Ь2)у'х + о(А - Ь)еА*у = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
y'L + Wx+aeXx=Cebx
(see 2.1.3.10 for the solution of the corresponding homogeneous equation).
8- tC + (aeXx + b)y'x + c(aeXx + b + c2)y = 0.
The substitution w = y'x + cy leads to a second order equation of the form 2.1.3.10:
<x " cw'x + (aeXx + b + <?)w = 0.
9- VvL + (ax + b)eXxy'x - aeXxy = 0.
Particular solution: yo = ax + b.
10- VvL + (ae2Xx + beXx)y'x - c(ae2Xx + beXx + c2)y = 0.
The substitution w = y'x — cy leads to a second order equation of the form 2.1.3.29:
w'L + cw'x + (ae<2Xx +beXx + °2)w = °-
11- y'JL, ~ ZaeXx{aeXx + X)y'x + aeXxBa2e2Xx - X2)y = 0.
Particular solutions: y± = exp ( — eXx ), yi = x exp ( — eXx ).
V A / \ A /
12- VvL ~ Ca2e2Xx + 3aXeXx + b)y'x + aeXxBa2e2Xx -1b- X2)y = 0.
1°. Particular solutions with b > 0:
/ a x-r fr\ ( a x-r fr\
yx = expl — елх - xVb ), y2 = expl — елх + xVb ).
V A / \ A /
2°. Particular solutions with b < 0:
2/i = сов(жл/—b) exp( — eXx ), j/2 = sin(a;-\/—b) exp( —еЛа:).
V A / \ A /
13- 1/.4» + <*y'L + beXxy'x + abeXxy = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.3.1:
yZx + W'L + (beXx + c)y'x + [b(a + X)eXx + ac]y = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
y'L + {beXx + c)y = Ce-ax
(see 2.1.3.2 for the solution of the corresponding homogeneous equation).
tC + <*y'L + (be2Xx + ceXx)y'x + a(be2Xx + ceXx)y = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.3.29:
w'xx + (be2Xx + ceXx)w = 0.
© 1995 by CRC Press, Inc.

tC + ае^У^ ~ Ь2(аеХх + b)y = 0.
The substitution w = y'x — by leads to a second order equation of the form 2.1.3.29:
wxx + (aeXX + b)w'x + KaeXX + b)W = °-
tC + aeXxy'^ - by'x ~ abeXxy = 0.
1°. Particular solutions with b > 0: У\= ехр(—жл/б), 2/2 = ехр(жл/б)-
2°. Particular solutions with b < 0: г/i = cos(a?\/—b), г/2 = sin^V-b).
tC + oe^y^ + obeAa;y; + b3y = 0.
The substitution w = y'x + by leads to a second order equation of the form 2.1.3.29:
X 2 = 0-
eXx + Щу'х + Ь2(аеХх + 2b)y = 0.
Particular solutions: y\ = еЬх, г/2 = xe x.
20- VZX + aeXxvL + ipe^x - c2)y'x - c(aceXx + be»x)y = 0.
Particular solution: г/g = ecx.
21- tC + o,eXxy'lx + (abeXx - b2 + c)y'x + c(aeXx - b)y = 0.
Particular solutions: y\ = e@lX, г/2 = е@2Х, where /?i and /?2 are the roots of the
quadratic equation /32 + b/3 + с = 0.
22. tC + aeA-y^ + [a(b + Л)еА- - Ь2]у'х + ab\eXxy = 0.
Particular solutions: г/i = e~6a:, г/2 = e~6a: / exp( 2bx TeX
23. tC + oe^y^ + Ьж"у4 + Ьж"-1 (ожеАа; + n)y = 0.
The substitution w = yxx + bxny leads to a first order linear equation: w'x + aeXxw = 0.
24. tC + axeXxy'lx + (bx2 - aeXx)y'x + bx(ax2eXx + 3)y = 0.
1°. Particular solutions with b > 0:
2/i=cos( —— j, 2/2 = sin
2°. Particular solutions with 6 < 0:
= exP 5 . № = exp .
25. yx'xx + ax2eXxy^x - 2axeXxy'x + 2aeXxy = 0.
Particular solutions: y\ = x, yi= x .
© 1995 by CRC Press, Inc.

26- tC + (oeAa; + Ъ)у'1х - ab2eXxy = 0.
The substitution w = y'x + by leads to a second order equation of the form 2.1.3.29:
wlx + aeXxw'x - abeXxw = 0.
27- tC + (oeAa; + b)y^ - c2(eAa; + b + c)y = 0.
The substitution w = y'x — cy leads to a second order equation of the form 2.1.3.29:
w" + (aeXx + b + c)w' + c(aeXx + b + c)w = 0.
28. tC + (aeXx + b)y'lx + c(aeXx + b)y'x + c3y = 0.
The substitution w = y'x + cy leads to a second order equation of the form 2.1.3.29:
w" + (aeXx + b- c)w' + c2w = 0.
29. tC + (be~ + 2o)y^ - a(be™ + a)y'x - 2o3y = 0.
Particular solutions: y± = eax, У2 = e~ax -\ .
O
'L
vZL = (eXx - a)v'
Particular solutions: y\ = e@lX, y<i = е@2Х, where /?i and /?2 are the roots of the
quadratic equation /32 + af3 + b = 0.
- s[(as + c)eXx + bs + d + s2]y = 0.
The substitution w = y'x — sy leads to a second order equation of the form 2.1.3.29:
wxx + (aeXx + b + s)w'x + [(as + c)eXx + bs + d + s2]w = 0.
32- Vx'xx + (aeXx + b)yxx + (ce2Xx + d)y'x — s(ce2Xx + aseXx + bs + d + s2)y = 0.
The substitution w = y'x — sy leads to a second order equation of the form 2.1.3.29:
W'L + (aeXx + b + s)w'x + (ce2Xx + aseXx + bs + d+ s2)w = 0.
33- У'1'хх + (aeXx + b)y'ix + (ceXx + d)v'x - keXx[k(a + k)e2Xx
+ (oA + 3feA + bk + c)eXx + X2 + bX + d]y = 0.
Particular solution: yo = expl — e x I.
V A /
/ к \ f
The substitution у = expl -г-еЛа:) / z(x) dx lead to a second order linear equation
of the form 2.1.3.29.
aeXx{aeXx +2b + 3X)y'x
+ aeXx[a(b + 2X)eXx + bX + А2] у = 0.
Particular solutions: y\ = expl —— eXx ), yi = x expl —— eXx
V A / V A j
35- VZX + (aeXx - b)y'L + ce"xyx - b(abeXx + ce»x)y = 0.
Particular solution: yo = еЬж-
© 1995 by CRC Press, Inc.

36- tC + (еХх + a)iC + (аеХх + Ъе»х)у'х + abe»xy = 0.
Particular solution: yo = е~ах.
37. tC + (ах + ЪеХх)у'1х + а(ЬхеХх + 2)у'х + аЬеХху = 0.
г, ,. , , , • ( ах2 \ ( ах2 \ f (ах2 \
Particular solutions: у!=ехр1 ^~)> 2/2 = ехр I — 1 / expl ) ах.
38. у'1'хх + (аЬхеХх + ЬеХх + а)у'^х + аЬ2хеХху'х - а2ЬеХху = 0.
Particular solutions: y\ = х, г/2 = е~ах.
39. ух'хх + ахп(ЬеХх + 2Х)у'х'х - Х(аЬхпеХх + Х)у'х - 2аХ3хпу = 0.
Particular solutions: yx = еХх, у2 = е~Хх + —.
Л
40. у'1'хх + (ахп - 2ЬеХх)Ух-х - ЬеХхBахп - ЬеХх + ЗХ)у'х
+ ЬеХх[ахп(ЬеХх - Л) + 2ЬХеХх - Х2]у = 0.
Particular solutions: y\ = ехр ( — еХх ), у2 = х ехр ( — еХх).
yZa + су'х + с(аеХх + Ъе»х)у = 0.
The substitution w = у'^х + су leads to a first order linear equation: w'x + (aeXx +
be»x)w = 0.
42- VZL + (aeXx + bei"*)yZa + [abe^x+^x + b(c + (л)е^х - c2]y'x
- aceXx + Ьце^у = 0.
Particular solutions: yx = е"ет, y2 = e~cx / ехрBсж e^) dx.
43- tC + aeXx(be»x + 2^'^ - (л(аЬе^х+^х + (л)у'х - 2а(л3еХху = 0.
Particular solutions: yi = емж, уг = е~ма: -\ .
44. sctC, + ay'lx + x(beXx + c)y'x + [b(Xx + a)eXx + ac]y = 0.
The substitution w = y'xx + (beXx +c)y yields a first order linear equation: xw'x +aw = 0.
45. xy'xxx + axeXxy'x — 2aeXxy = 0.
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.3.1:
wlx + aeXxw = 0.
46. xyZx = (eXx - ax)y'lx + (aeXx - Ьх)у'х + ЬеХху.
Particular solutions: y\ = e@lX, y2 = е@2Х, where /?i and /?2 are the roots of the
quadratic equation C2 + a/3 + b = 0.
© 1995 by CRC Press, Inc.

xyZx + (axeXx + 3)y^x + a(bx + 2)eXxy'x + b(abxeXx + aeXx - Ь2х)у = О.
Particular solutions:
bx \ ( bxV3 \ 1 / bx \
I I 1/0 = — exn I
sm
)sm{ 2
48- ^VZL + x(axeXx + b)y'lx + [a(b - 2)xeXx + c]y'x + a(c-b + 2)eXxy = 0.
Particular solutions: y\ = xni, г/2 = хП2, where n\ and П2 are the roots of the
quadratic equation n2 + (b — 3)n + c — 6 + 2 = 0.
49- x3vZL + Ьх2еХхУх-х + axy'x + a(beXx - 2)y = 0.
Particular solutions: y\ = xni, У2 = хП2, where n\ and П2 are the roots of the
quadratic equation n2 — n + a = 0.
50- хъу'1'хх + x2(aeXx + b)y'lx + x(abeXx + с - b)y'x + c(aeXx - 2)y = 0.
Particular solutions: y\ = xni, У2 = хП2, where n\ and П2 are the roots of the
quadratic equation n2 + (b — l)n + с = 0.
51. (aex + b)y'l'xx - aexy = 0.
Particular solution: yo = a-?x + b.
52. (bceax + a + c)yZx ~ (bc3eax + a3 + c3)y'x + ac(a2 - c2)y = 0.
Particular solutions: y1 = ecx, У2 = e~ax + b.
53. (aeXx + b)y'lxx + (ceXx + d)y^ + k(aeXx + b)y'x + k(ceXx + d)y = 0.
1°. Particular solutions with к > 0: У\= cos{x\fk\ y2 = sin^v^).
2°. Particular solutions with к < 0: У\= exp(—x\J—к), У2 = ехр(ж\/—/с).
54. (aex + bx)y'^'xx - aexy = 0.
Particular solution: yo = aex + bx.
55. (aex + bx2)y'l'xx - aexy = 0.
Particular solution: yo = аех + bx2.
56. (axex + b)y'l'xx + by = 0.
Particular solution: yo = ax + be~x.
57. (ax2ex + b)y'^xx + by = 0.
Particular solution: yo = ax + be~x.
© 1995 by CRC Press, Inc.

3.1.4. Equations Containing Hyperbolic Functions
!• v'vL - °3 tanh(oa;)y = 0.
Particular solution: yo = cosh(arr).
The substitution у = cosh(arr) f z(x) dx leads to a second order equation of the
form 2.1.4.44: z%x + 3atanh(arrL + 3a2z = 0.
2- v'vL ~ °3 coth(ax)y = 0.
Particular solution: yo = sinh(aa;).
The substitution у = smh(ax) f z(x) dx leads to a second order equation of the
form 2.1.4.45: z'^x + 3acoth(arrL + 3a2z = 0.
3- CL - 3a2vL + 2°3 tanh(oa;)y = 0.
Particular solutions: y\ = cosh(arr), j/2 = xcosh(ax).
4- CL - 3a2vL + 2°3 coth(oa;)y = 0.
Particular solutions: y\ = sinh(aa;), j/2 = xsmh(ax).
5- VvL + [° coshBa;) + b]y'x + a sinhBa;)y = 0.
Solution:
у = C\w\ + C2W1W2 + C3W2,
where Wi and W2 form a fundamental set of solutions of the modified Mathieu equation
2.1.4.1: 4и4'ж + [acoshBrr) + b]w = 0.
6- V*L + av'L + b coshBx)y'x + ab coshBa;)y = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.4.1:
7- У'х'хх + av'ix + b cosh2 xy'x + ab cosh2 x у = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.4.2:
wxx + b cosh2 xw = 0.
8- VvL + av'L + b sinh2 xy'x + ab sinh2 x у = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.4.5:
wxx + bsinh xw = 0.
9- VZX + av'L + \b tanh(Aa;) + c]y'x + a[b tanh(Aa;) + c]y = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.4.6:
wxx + [6tanh(Arr) + c]w = 0.
Particular solutions: y\ = cosh(Arr), 2/2 =
© 1995 by CRC Press, Inc.

tC + av'L + № coth(Aж) + c]y'x + a[b соШ(Аж) + с]у = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.4.10:
wxx + [bcoth(Xx) + c]w = 0.
Particular solutions: y\ = sinh(Aa;), 2/2 = a?sinh(Aa;).
A«)yL + Wx + ab со3Ь"(Аж)у = 0.
1°. Particular solutions with b > 0: У\= cos{x\fb), yi = sin(xVb).
2°. Particular solutions with b < 0: У\= ехр(—ж\/—~b), 2/2 = ехр(ж\/—b).
Aa;)yL + bxmy'x + bxm~x [ax со3Ьгг(Аж) + m]y = 0.
The substitution w = y'J.x + bxmy leads to a first order linear equation: w'x +
acoshn(Aa?)«; = 0.
'M)y'L + abcoshn(\x)y'x + Ь2[осо5Ь"(Аж) - Ь]у = 0.
Particular solutions: 2/1 = e~bx'2 cos I
;Аж)+3%4 + Ь2[осо5Ь"(Аж) + 2Ь]у = 0.
Particular solutions: 2/1 = ebx, у2 = xebx.
17. y'xxx + a cosh" ж yxx + (ab cosh" ж + с — Ь2)у'х + c(o cosh" ж — b)y = 0.
Particular solutions: 2/1 =exp(Aia;), 2/2 =ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + ЬХ + с = 0.
18- У'1'хх + ax cosh" ж y^, + (Ьж2 - a cosh" ж)у^, + bx(ax2 cosh" ж + 3)y = 0.
Particular solutions: y1 = cos(^x2^/b), j/2 = sin(|2;2i
A«)yL - 2ax coshn(Xx)y'x + la со3Ь"(Аж)у = 0.
Particular solutions: y\ = x, yi= x .
20- У'1'хх = (cosh" ж - a)y'x'x + (a cosh" x - b)y'x + b cosh" ж у.
Particular solutions: гд =exp(Aia;), г/2 =ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + aX + b = 0.
21- y?L + (° cosh" ж + Ьж)у^ + Ь(ож cosh" ж + 2)y^ + ab cosh" ж у = 0.
Particular solutions: г/! = exp(—убж2), г/2 = exp(—-i-бж2) exp^bx2) dx.
22- y?L + osinh"(Aaj)»4/a, + by^ + оЬ3тЬ"(Аж)у = 0.
1°. Particular solutions with b > 0: г/i = С08(жл/&), г/2 = sin(a;-\/b).
2°. Particular solutions with b < 0: г/i = exp(—ж-\/—&), 2/2 = ехр(ж\/—&)•
© 1995 by CRC Press, Inc.

23- Уххх + asintin(Xx)y'^x + bxmy'x + bxm-1[axsinhn(\x) + m]y = 0.
The substitution w = yxx + bxmy leads to a first order linear equation: w'x +
asinhn(Aa;)w = 0.
24. y'Zx + asinhn(Xx)yZ + absinh"(Лж)у'x + b2[osinh"(Aa;) - b]y = 0.
_ ,. , , ,. _bx/2 (bxV3 \ -bx/2 ¦ (bxV3 \
Particular solutions: y\ = e ' cos I —-— I y2 = e ' sin —-— I
V
bx/2 (
I I, y2 = e ' sin I.
V ^ / V ^ /
25- VZX + osinh"(Aaj)»^, - 6[2osinh"(Aaj) + Щу'х + b2[asinhn(Xx) + 2b]y = 0.
Particular solutions: y1 = еЬх, у2 = xebx.
26. y'i'xx + asinh"xy'lx + (absinh"x + c- Ь2)у'х + c(asinh"x - b)y = 0.
Particular solutions: yi=exp(Aia;), 2/2= expire), where Ai and A2 are the roots
of the quadratic equation A2 + ЬХ + с = 0.
27. y'l'^ + ax sinh" x y'^ + (bx2 - a sinh" x)y'x + bx(ax2 sinh" x + 3)y = 0.
Particular solutions: yx = cos(^x2Vb), 2/2 = sin(jfX2Vb)-
28- V'i'xx + ax2 sinhn(Xx)y'x'x - lax sinhn(Xx)y'x + lasinh"(Aa;)y = 0.
Particular solutions: y\ = x, j/2 = x .
29. y'i'xx = (sinh™ x - a)y'x-x + (a sinh" x - b)y'x + b sinh" x y.
Particular solutions: y!=exp(Aia;), у2=ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + aX + b = 0.
30. y'xxx + (a sinh" x + bx)yxx + b(ax sinh" x + 2)y'x + ab sinh" x у = 0.
r
Particular solutions: y\ = exp(—\bx ), 2/2 = exp(—\bx ) / ехр(у6ж ) dx.
31- У'х'хх — tanh x Ухх — аУх + °tanh xy = 0.
1°. Solution with a > 0: у = С\ ещ>(—Хл/а) + С2 ехр(жл/«) + Сз cosh ж.
2°. Solution with a < 0: у = C\ cos(xy/—a) + C2 sin(xy/—a) + C3 cosh ж.
32- yZL + ° tanhn(Xx)vxx + Ьхту'х + Ьхт~х [ах tanh"(Aa;) + m]y = 0.
The substitution w = yxx + bxmy leads to a first order linear equation: w'x
atanhn(Aa;)w = 0.
Уххх + otanh"(Aa;)y;/a; + ab tanh"(Xx)y'x + Ь2[а tanhn(Xx) - b]y = 0.
л j.- 1 1 j.- -bx/2 ( Ьху/Ъ \ _bx/2 . ( bxV3 \
Particular solutions: y\= e 'cos I I, y2 = e 'sin I I.
V 2 / \ 2 /
34. y^
Particular solutions: y\ = ebx, у2 = xebx.
© 1995 by CRC Press, Inc.

Particular solutions: y\ = ещ>{0\х), yi = ехр(/?2ж), where 0\ and 02 are the roots
of the quadratic equation 02 + Ь0 + с = 0.
36- Уххх + ахПУхх ~ Bож" tanh x + 3)y'x + [axnB tanh2 x - 1) + 2 tanh x]y = 0.
Particular solutions: уг = cosh ж, у^ = ж cosh ж.
37. yZx + о tanh" x J&. - Bo tanh"+1 x + 3)y'x
+ Bo tanh"+2 x - a tanh" x + 2 tanh x)y = 0.
Particular solutions: y\ = cosh ж, y<i = ж cosh ж.
38. y'l'xx + ax tanh" x y'J.x + (bx2 - a tanh" x)y'x + bx(ax2 tanh" x + 3)y = 0.
Particular solutions: y\ = cos(-^2vt>), У2 = sm{^x2\b).
39- CL + ax2 tanhn(Xx)v'L ~ 2ax tanhn(Xx)y'x + 2o tanh"(Aa;)y = 0.
Particular solutions: y\ = ж, У2 = x2.
40- VZL = (tanh" ж - а)у'^х + (о tanh" x - b)y'x + b tanh" x y.
Particular solutions: у1=ехр(А1ж), у2=ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + aX + b = 0.
41. y'^ + (a tanh" x + bx)yxx + b(ax tanh" x + 2)y'x + ab tanh" x у = 0.
Particular solutions: y\ = exp(—-^bx2), yi = exp(—\bx2^) I ехр(у6ж2) dx.
42- y'Zx + Hitanhx - b) - b\y'lx + [a(b2 - 1)ж" - l]y'x
+ b[axn(l - b tanh ж) + l]y = 0.
Particular solutions: y\ = ebx, yi = cosh ж.
43. y'Zx + [Лtanh(Лж)(oж" - 1) - ax^^y'^ - a\2xny'x + a\2xn-1y = 0.
Particular solutions: yi = ж, %J = cosh(Ax).
44. y'Zx + (° tanh"+1 x-ab tanh" ж - )^
+ [a(b2 - 1) tanh" ж - l]y'x + b(-obtanh"+1 ж + о tanh" ж + 1)у = 0.
Particular solutions: yi = ebx, 2/2 = cosh ж.
45. y'l'xx — coth ж y^ — ay'x + о coth ж у = 0.
1°. Solution with a > 0: у = Ci exp(—x^fa) + C<i ехр(жл/«) + Сз sinh ж.
2°. Solution with a < 0: у = Ci сов(жу/—«) + Сг вш(жу/—«) + Сз sinh ж.
46. у'ххх + (° c°th ж - ab - b)y'lx + (ab2 -a- l)y'x + b(-ab coth ж + a + l)y = 0.
Particular solutions: y\ = ebx, yi = sinh ж.
© 1995 by CRC Press, Inc.

47. y'XXx + осо*Ь"(Аж)у^,/а, + bxmy'x + bxm г[ах соШ"(Аж) + т]у = 0.
The substitution w = yxx + bxmy leads to a first order linear equation: w'x +
acothn(Aa;)w = 0.
48. yxxx + a co^hn (\x)yxx + аЬсоШ"(Аж)у^, + Ь2[асоШ"(Аж) — b]y = 0.
Particular solutions: y\ = e~ba:'2cosl I, yi= e~fa'2sin I.
\ 2 J \ 2 J
49. y^ + а соШ^АжЭу^ - b[2a coth"(Aж) + 3b]y'x + b2[a coth"(Aж) + 2% = 0.
Particular solutions: y\ = еЬх, у2 = xebx.
50. y'Zx "I" ax c°th" ж yxx + (Ьж2 — a coth" ж)у^, + bx(ax2 coth" ж + 3)y = 0.
Particular solutions: yi = cos(-jX2Vb), У2 = sin(-jX2Vb).
51. у -\- ax coth (Аж)у — 2ttжcoth (Аж)у -\- 2<xcoth (Аж)у ^ 0.
Particular solutions: y\ = x, У2 = x2.
52- Ух'хх + ахПУхх — Bож" coth ж + 3)y'x + [ож"B coth2 ж — 1) + 2 cothж]y = 0.
Particular solutions: y\ = sinha;, 2/2 = a; sinh a;.
53. у'1'хх + a coth" ж yxx — Ba coth"+1 ж + 3)y^,
+ Ba coth"+2 ж - о coth" ж + 2 coth ж)у = 0.
Particular solutions: y\ = sinha;, j/2 = a; sinh a;.
54. y^ + (o coth" ж + bx)y'lx + b(ax coth" x + 2)y'x+ab coth" ж у = 0.
г
Particular solutions: y1 = exp(—-jbx2), У2 = exp(—-jbx2) / exp(-i-6a;2) da
J
dx.
55- VvL + [Acoth(Aa;)(oa;" - 1) - аж"^ - оА2ж"у^ + аХ2хп~1у = 0.
Particular solutions: y\ = x, 2/2 = sinh(Aa;).
56. жу^ + ж[осоэЬBж) + Ь\у'х — 2[осоэЬBж) + b]y = 0.
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.4.1:
wxx + [acoshBa;) + b]w = 0.
57'• XVZX +x(a cosh2 ж + b)y'x - 2(o cosh2 ж + b)y = 0.
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.4.2:
wxx + (a cosh2 x + b)w = 0.
58- XVXL +x(a sinh2 ж + b)y'x - 2(o sinh2 ж + b)y = 0.
The substitution w = xy'x — 2y leads to a second order equation of the form 2.1.4.5:
wxx + (a sinh x + b)w = 0.
© 1995 by CRC Press, Inc.

59. xy'xxx = (cosh" x — ax)yxx + (a cosh" x — bx)y'x + b cosh" x y.
Particular solutions: yi=exp(Aia;), у2=ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + aX + b = 0.
60. x2y'x"xx + (ax2 cosh" x + bx)y'x'x + [a(b - 2)x cosh" x + c]y'x
+ a(c - b + 2) cosh" x у = 0.
Particular solutions: y\ = xmi, y<i = xm2, where mi and m.2 are the roots of the
quadratic equation m2 + F — 3)m + с — 6 + 2 = 0.
61- x2vxL + (ax2 sinh" x + bx)y'L + [°(b - 2)x sinh" x + °]у'х
+ a(c - b + 2) sinh" x у = 0.
Particular solutions: y\ = xmi, y<i = xm2, where mi and m.2 are the roots of the
quadratic equation m2 + (b — 3)m + c — 6 + 2 = 0.
62. x2y'x"xx + (ax2 tanh" x + bx)y'x'x + [a(b - 2)x tanh" x + c]y'x
+ a(c - b + 2) tanh" x у = 0.
Particular solutions: уг = xmi, y^ = xm2, where mi and m.2 are the roots of the
quadratic equation m2 + (b — 3)m + c — 6 + 2 = 0.
63. x2y'x"xx + (ax2 coth" x + bx)y'x'x + [a(b - 2)x coth" x + c]y'x
+ a(c - b + 2) coth" x у = 0.
Particular solutions: уг = xmi, y^ = xm2, where mi and m.2 are the roots of the
quadratic equation m2 + (b — 3)m + c — 6 + 2 = 0.
64. x3y'x"xx+x2(a cosh" x+b)y'x'x+x(abcoshn x+c-b)y'x + c(a cosh" x-2)y = 0.
Particular solutions: уг = xmi, y^ = xm2, where mi and m.2 are the roots of the
quadratic equation m2 + F — l)m + с = 0.
65. x3y'x"xx+x2(a sinh" x + b)y'x'x+x(absinhn x + c-b)y'x + c(a sinh" x-2)y = 0.
Particular solutions: y\ = xmi, y<i = xm2, where mi and m.2 are the roots of the
quadratic equation m2 + F — l)m + с = 0.
66. x3y'x"xx+x2(a tanh" x+b)y'x'x+x(ab tanh" x+c-b)y'x+c(a tanh" x-2)y = 0.
Particular solutions: y\ = xmi, y<i = xm2, where mi and m.2 are the roots of the
quadratic equation m2 + F — l)m + с = 0.
67. x3y'x"xx+x2(a coth" x+b)y'x'x+x(abcothn x+c-b)y'x+c(a coth" x-2)y = 0.
Particular solutions: y\ = xmi, y<i = xm2, where mi and m.2 are the roots of the
quadratic equation m2 + F — l)m + с = 0.
68. cosh" x y'?xx + ay'^ + aby'x + b2(a-b cosh" x)y = 0.
_ ,. , , ,. _bx/2 (bx\ph \ _b /2
Particular solutions: y\=e ox' cos I —-— I, y2 = e M/
V 2 /
sin
I.
\ 2 /
© 1995 by CRC Press, Inc.

69. cosh" ж yZx + ay'^x + b cosh" ж y'x + aby = 0.
1°. Particular solutions with b > 0: г/i = cos(xvb), г/2 = sin(xvb).
2°. Particular solutions with b < 0: г/i = ехр(—ж\/—~b), г/2 = ехр(ж\/—&)•
70. cosh" x y'l'xx + ay'lx - bBa + 3b cosh" x)y'x + Ь2(а + 2b cosh" x)y = 0.
Particular solutions: y\ = еЬх, г/2 = xebx.
71. cosh" x y'l'xx + y'lx + [(b - a2) cosh" x + a]y'x + b(l - a cosh" x)y = 0.
Particular solutions: гд = eXlX, г/2 = еЛ2а:, where Ai and A2 are the roots of the
quadratic equation A2 + aX + b = 0.
72. cosh"(Xx)yZx + ах2Ух-х - 2axy'x + lay = 0.
Particular solutions: г/i = x, г/2 = x2.
73. cosh" x yZx + (a cosh" x + ax + l)y'^x + a2xy'x - a2y = 0.
Particular solutions: г/i = x, г/2 = e~
o —ax
74. cosh" x y'Jxx + (ax cosh" x + l)y'^x + a(x + 2 cosh" ж)у4 + ay = 0.
/ аж2 \ / ax2 \ f / ax2 \
Particular solutions: г/i = exp I — 1, г/2 = exp I — 1 / exp I —— 1 ax.
75. x cosh" x yxxx + C cosh" x+x)y'x'x + (ax cosh" a;+2)y4 + o(cosh" x+x)y = 0.
Particular solutions: г/i = —cosfrr-v/^), J/2 = —sinfrr-v/^)-
X У ' X У '
76. ж3 cosh" x yxxx + ax2yxx - 2x cosh" x y'x + 2B cosh" x — a)y = 0.
Particular solutions: г/i = x~ , г/2 = x .
77. ж3 cosh" x y'xxx + ax2yxx - 6x cosh" x y'x + 6B cosh" x — a)y = 0.
Particular solutions: г/i = x~ , г/2 = x .
78. ж3 cosh" ж y'l'xx + ax2y'lx + x(a - cosh" x)y'x + 0@ - 3 cosh" x)y = 0.
Particular solutions: г/i = cos(ln |ж|), г/2 = sin(ln |ж|).
79. ж3 cosh" ж у^ + ж2(cosh" ж + a)y'lx +
ж [о - (b + 1) cosh" x]y'x + bB cosh" ж - o)y = 0.
Particular solutions: г/i = x~ , г/2 = ж .
80. sinh" ж у^ + oy^ + oby^ + b2(o - b sinh" ж)у = 0.
Particular solutions: y1 = e~bx^2 cos ( —-— j, г/2 = e~hxl2 sin ( —-— j.
V 2 / \ 2 /
© 1995 by CRC Press, Inc.

81. sinh" ж yZx + ayZ + Ь sinh" xy'x + aby = 0.
1°. Particular solutions with b > 0: г/i = cos(xvb), г/2 = sin(xvb).
2°. Particular solutions with b < 0: г/i = ехр(—ж\/—~b), г/2 = ехр(ж\/—~b)-
82. sinh" xyZx + ayZ - bBa + 3bsinh" x)y'x + Ь2(а + 2bsinhnx)y = 0.
Particular solutions: y\ = еЬх, г/2 = xebx.
83. sinh" x y'Zx + y'L + [(Ь - о2) sinh" x + a]y'x + b(l - a sinh" x)y = 0.
Particular solutions: гд = eXlX, г/2 = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + aX + b = 0.
84. sinh"(Xx)yZx + ax2yZ - 2axy'x + 2ay = 0.
Particular solutions: г/i = x, г/2 = x2.
85. sinh" x y'Zx + (a sinh" x + ax + l)y'Z + a2xy'x - a2y = 0.
Particular solutions: г/i = x, г/2 = e~
o — ax
86. sinh" x y'Zx + (ax sinh" x + \)y'lx + a(x + 2 sinh" x)y'x + ay = 0.
/ ax2 \ / ax2 \ f / ax2 \
Particular solutions: г/i = exp I — 1, г/2 = exp I — 1 / exp I —— 1 ax.
87. y^a (
f/) —
X
Particular solutions: г/i = —cos(xy/a), г/2 =
x
88. ж3 sinh" ж yZx + ax2yZ - 2x sinh" xy'x + 2B sinh" ж - o)y = 0.
Particular solutions: y\ = x~ , yi = x .
89. ж3 sinh" ж y^ + ax2yZ ~ 6ж sinh" ж y^, + 6B sinh" ж - o)y = 0.
Particular solutions: y\ = ж~ , г/2 = x .
90. ж3 sinh" ж у^ + ax2yZ + x(a — sinh" ж)у^, + o(o - 3 sinh" ж)у = О.
Particular solutions: г/i = cos(lna;), г/2 = sin(lna;).
91. ж3 sinh" ж yZx + x2(sinh" ж + a)yZ +
x[a — (b + 1) sinh" ж]у^ + bB sinh" ж - a)y = 0.
Particular solutions: y\ = x~ , г/2 = ж .
92. tanh" ж yZx + Wxx + aby'x +b2(a-b tanh" ж)у = 0.
_ . , , . -h-r/2 ( bX\f?,\ -br/2 ¦ ( bX\f?,\
Particular solutions: г/i = e ' cos I —-— I, y2 = e ' sin I —-— I.
V 2 / \ 2 /
© 1995 by CRC Press, Inc.

93. tanh" ж yZx + ay'^x + Ь tanh" ж y'x + aby = 0.
1°. Particular solutions with b > 0: г/i = cos(xvb), г/2 = sin(xvb).
2°. Particular solutions with b < 0: г/i = ехр(—ж\/—~b), г/2 = ехр(ж\/—&)•
94. tanh" x yZx + ay'Z - bBa + 3btanh" x)y'x + Ь2(а + 2btanh" x)y = 0.
Particular solutions: y\ = еЬх, г/2 = xebx.
95. tanh" x y'l'xx + V'L + Kb ~ °2) tanh" x + a]y'x + b(l - a tanh" x)y = 0.
Particular solutions: гд = eXlX, г/2 = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + aX + b = 0.
96. tanhn(Xx)yZx + ax2y'Z - 2axy'x + lay = 0.
Particular solutions: г/i = x, г/2 = x2.
97. tanh" x yZx + (a tanh" x + ax + \)y'lx + a?xy'x - a2y = 0.
Particular solutions: г/i = x, г/2 = e~
o — ax
98. tanh" x y'l'xx + (ax tanh" x + V)y^x + a(x + 2 tanh" x)y'x + ay = 0.
/ ax2 \ / ax2 \ f / ax2 \
Particular solutions: г/i = exp I — 1, г/2 = exp I — 1 / exp I —— 1 ax.
99. x tanh" x yxxx + C tanh" x+x)y'x'x + (ax tanh" a;+2)y^,+o(tanh" x+x)y = 0.
Particular solutions: г/i = —cosfrr-v/^), J/2 = —
X У ' X
100. ж3 tanh" x y'l'xx + ax2y'x'x - 2x tanh" x y'x + 2B tanh" x - а)у = 0.
Particular solutions: г/i = x~ , г/2 = x .
101. ж3 tanh" x y'l'xx + ax2y'x'x - 6x tanh" x y'x + 6B tanh" x - а)у = 0.
Particular solutions: г/i = x~ , г/2 = x .
102. ж3 tanh" x y'l'xx + ax2y'x'x + x(a - tanh" x)y'x + 0@ - 3 tanh" x)y = 0.
Particular solutions: г/i = cos(ln |ж|), г/2 = sin(ln |ж|).
103. ж3 tanh" ж y'Zx + ж2(tanhrг ж + a)y'lx +
x[a - (b + 1) tanh" x]y'x + bB tanh" x - a)y = 0.
Particular solutions: г/i = x~ , г/2 = ж .
104. coth" ж у^ + oy^ + aby'x + Ь2(а-Ь coth" ж)у = 0.
Particular solutions: y1 = e~bx^2 cos ( —-— j, г/2 = e~hxl2 sin ( —-— j.
V 2 / \ 2 /
© 1995 by CRC Press, Inc.

105. coth" ж yZx + ay'^x + b coth" xy'x + aby = 0.
1°. Particular solutions with b > 0: г/i = cos{x\fb), г/2 = sm(xVb).
2°. Particular solutions with b < 0: г/i = ехр(—ж-\/—&), 2/2 = ехр(ж\/—Ь).
106. coth" ж y^ + oy^ - bBo + 3bcoth" ж)у^ + Ь2(а + 2bcoth" ж)у = 0.
Particular solutions: y1 = еЬх, у2 = xebx.
107. coth" ж yZx + V'L + Kb ~ °2) coth" x + a]vL +b(l-a coth" x)y = 0.
Particular solutions: y\ = eXlX, г/2 = eX2X, where Ai and A2 are the roots of the
quadratic equation A + aX + 6 = 0.
108. соШ^АжЭу^ + ax2yZa - 2axy'x + lay = 0.
Particular solutions: yi = x, У2 = x ¦
109. coth" ж yZx + (a coth" x + ax + l)y'Jx + a?xy'x - a2y = 0.
Particular solutions: гд = x, г/2 = e~ax.
110. coth" ж yZx + (ax coth" ж + l)y'^x + о(ж + 2 coth" ж)у^ + ay = 0.
/ ax2 \ ( ax2 \ f (ax2 \
Particular solutions: г/i = exp ( — J, г/2 = exp I — 1 / exp I —— 1 ax.
\ Zi / \ Zi / j \ Zi /
111. ж coth" ж Уххх + C coth" x+x)y'^x + (ax coth" ж + 2)y4 + o(cothrг ж+ж)у = 0.
Particular solutions: г/i = —cos(x\/a), г/2 = —sm(xy/a).
112. ж3 coth" ж y'^ + ax2y%x - 2x coth" ж y'x + 2B coth" ж - o)y = 0.
Particular solutions: y\ = x~ , У2 = x .
113. ж3 coth" ж y^'^ + ax2y^x - 6x coth" ж y^ + 6B coth" ж - o)y = 0.
Particular solutions: г/i = ж~ , г/2 = x .
114. ж3 coth" ж y^ + ож2у^ + ж(о - coth" ж)у^ + o(o - 3 coth" x)y = 0.
Particular solutions: г/i = cos(lna;), г/2 = sin(lna;).
115. ж3 coth" ж у^ + ж2(cothrг ж + o)y^ +
ж [a - F + 1) coth" x]y'x + bB coth" ж - o)y = 0.
Particular solutions: y\ = x~ , г/2 = x .
© 1995 by CRC Press, Inc.

3.1.5. Equations Containing Logarithmic Functions
!• tC + « 1п"(Аж)у^ + Ьу'х + ab \nn(Xx)y = 0.
1°. Particular solutions with b > 0: г/i = cos(a?vt>), г/2 = sin(a;vt>).
2°. Particular solutions with b < 0: г/i = exp(—ж\/—б), г/2 = ехр(ж\/—&)•
2- tC + о In" ж y'lx + ab In" ж y'x + b2 (a In" ж - b)y = 0.
_hW2 /tev^A _hT/2 /
Particular solutions: г/i = e ' cos I —-— I, y2 = e ' sin
\ ^ / \
3- y^'L + о In" ж у^ + (Ьж + с) у'х + (abx In" ж + ас In" ж + b)y = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
Ухх + (bx + c)y = C exp (-a In™
(see 2.1.2.2 for the solution of the corresponding homogeneous equation).
Лж)у^ - Ь[2а 1п"(Лж) + 3b]y^ + Ь2[а 1п"(Лж) + 2b]y = 0.
Particular solutions: г/i = ebx, у2 = хеЬх.
5- VZX + о In" ж y'lx + (ab In" ж + с - Ь2)у'х + с(а In" х - Ь)у = 0.
Particular solutions: г/i =exp(Aia;), г/2 =ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + ЬХ + с = 0.
6- tC + о 1п"(Лж)у^ + Ьж™у; + Ьж—1 [ож 1п"(Лж) + т]у = 0.
Assuming ад = y'xX+bxmy, we obtain a first order linear equation: w'x+a In™ (Аж)ад = 0.
7- tC + (o In" ж + b)y^ + cy'x + c(a In" +b)y = 0.
1°. Particular solutions with с > 0: y\ = cos(x\fc), г/2 = sin (ж-у
2°. Particular solutions with с < 0: г/i = exp(—ж^/—с),г/2 = ехр(жу/—с).
8- y^'L = On" ж - o)yl + (о In" ж - Ъ)у'х + Ып" ж у.
Particular solutions: г/i =exp(Aia;), г/2 =ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + aX + b = 0.
9- VvL + ax ln" x V'L + (bx2 ~ °ln" x)vL + bx(ax2 In" ж + 3)y = 0.
Particular solutions: г/i = cos(-ja;2v6), г/2 = sin(-^-
VZX + (oln" x + bx)iC + b(ax In" ж + 2)y^ + obln" ж у = 0.
г
Particular solutions: г/i = exp(—-^bx2), г/2 = exp(—-^bx2) I ещ>{-^Ьх2) dx.
© 1995 by CRC Press, Inc.

И- tC + (ах In" ж + Ъ)у'^х + а(Ьх - 1) In" х у'х - аЫпп х у = 0.
The substitution w = у'х + by leads to a second order linear equation of the form
2.1.5.5: wxx + axlnnxw'x -alnnxw = 0.
12- VvL + (abx In" ж + о In" x + Ъ)у'1х + ab2x In" xy'x- ab2 In" x у = 0.
Particular solutions: y\ = x, yi = e~ x.
tC + ax2 1п"(Лж)у1 - 2ах 1п"(Лж)у4 + 2о 1п"(Аж)у = О.
Particular solutions: y\ = ж, J/2 = ж .
14. жу^ + ожу^ - Ь(Ьж In2 ж + 1)у4 - ab(bx In2 ж + 1)у = 0.
The substitution w = у'х + ay leads to an equation of the form 2.1.5.7: xwxx
(b2xln2x + b)w = 0.
. xyZx + а 1п"(Лж)у^ + Ьжу^ + ab 1п"(Лж)у = 0.
1°. Particular solutions with b > 0: У\= cos{x\fb\ yi = sin(xVb).
2°. Particular solutions with b < 0: yi = exp(—ж\/—б), 2/2 = ехр(ж-\/—&
16- ЖУ^ + ож In ж у^ + (abx In ж - Ь2ж + а)у'х + аЪу = 0.
Particular solutions: yx = е~Ьх, у2 = e~bx f x-^e{a+2b)x dx.
17- жу^ = (In" ж - ax)y'lx + (a In" ж - Ьж)у^ + Ып" ж у.
Particular solutions: г/i =exp(Aia;), г/2 =ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A + aX + 6 = 0.
18- xyZL + (ax ln" x + $)y'L + Bo ln" x + bx)vL + Чах In" ж + l)y = 0.
Particular solutions: У\ = —cos(rrv6), 2/2 = —sin(a;v6).
X X
Particular solutions: Vi = —> 2/2 = —e~bx.
X X
19.
20- xyZx + [a(b - 1пж)ж" + 2]y^ + aa;"-1^ - ож"-2у = 0.
Particular solutions: 2/1 = x, y2 =lnx — b+ 1.
21- ж2у^ + о 1п"(Лж)у4'а; + Ьж2у^ + ab 1п"(Лж)у = 0.
1°. Particular solutions with b > 0: 2/1 = cos(a?v6), 2/2 = sin(a;v6).
2°. Particular solutions with b < 0: 2/1 = exp(—ж\/—&), 2/2 = ехр(ж\/—&)•
© 1995 by CRC Press, Inc.

22. x2y'Zx + x2(alnx + Ь)у%я + 2аху'х - ay = 0.
Integrating the equation twice, we obtain a first order linear equation:
y'x + (alnx + b)y = d+ C2x.
23. x2y'l'xx ~ Зах[ах 1п2(Аж) + l]y'x + [2a2x2 1п3(Аж) + l]y = 0.
\ f 1 \ f 1
Particular solutions: y\ = exp a / 1п(Аж) dx\, y2 = xexp a I 1п(Аж) dx .
L J J L J J
24. x2yZx + x2(a Inx + Ьх)у'^х + 2х(Ьх + а)у'х - ay = 0.
Integrating the equation twice, we obtain a first order linear equation:
y'x + (a In x + bx)y = Ci + C2x.
25. a;2tC + (aa;2 In"ж + Ьж)У;/а; + [a(b-2)x\nnx + c}yfx + a(c-b+2) \nnxy = 0.
Particular solutions: у г = xmi, y<i = xm2, where mi and mi are the roots of the
quadratic equation m2 + (b — 3)m + c — 6 + 2 = 0.
26. sb8»^ + ж2(о In ж + Ь)^ + 2ожу^ - ax = 0.
Integrating the equation twice, we obtain a first order linear equation:
xy'x + (a In ж + b - 2)y = Ci + С2ж.
27. «3y4'L + о lnn(Xx)y'x'x + bx3y'x + ab 1п"(Лж)у = 0.
1°. Particular solutions with b > 0: y\ = cos(xVb), j/2 = sin(жv/&)•
2o. Particular solutions with 6 < 0: ух = ехр(—жл/—b), 2/2 = ехр(жл/—b).
28. x3yZx + ax2 In" ж y^ - 2жу^ + 2B - о In" x)y = 0.
Particular solutions: уг = ж, j/2 = x2.
29. ж3у^ + ax2 In" ж y^ - бжу^ + 6B - о In" x)y = 0.
Particular solutions: y\ = x~2, yi = ж3.
30- x3vZx + ж2(о In ж + Ьж)у^ + 2ж(Ьж + a)y'x - ay = 0.
Integrating the equation twice, we obtain a first order linear equation:
xy'x + (a In ж + bx - 2)y = CX + C2x.
31- x3VxL + x2(a ln" x + b)v'L + x(abIn" ж + с - b)y'x + c(a In" ж - 2)y = 0.
Particular solutions: уг = xmi, y2 = xm2, where m\ and m^ are the roots of the
quadratic equation m2 + (b — l)m + с = 0.
32. In"жy'i'xx + ay'ix + aby'x + b2(a - bin"x)y = 0.
_ ,. , , ,. _bx/2 (bx\ph \ _b /2 ¦
Particular solutions: y\=e ox' cos I —-— I, y2 = e M/ sin
V 2 /
© 1995 by CRC Press, Inc.
\ _b /2 ¦ (bx\ph \
I, y2 = e M/ sin —-— I.
\ 2 J

33. In" x yZx + +(ax In" x + l)y'^x + a(x + 2 In" x)y'x + ay = 0.
/ ax2 \ / ax2 \ f / ax2 \
Particular solutions: yi = exp I — 1, y<i = exp I — 1 / exp I —— 1 ax.
34. In" x y'l'xx + (a In" x + ax + 1)^ + a2x y'x - a2y = 0.
Particular solutions: уг = x, j/2 = e~ax.
35. 1пп(\х)Ух-хх + ах2Ух-х - 2axy'x + 2ay = 0.
Particular solutions: y\ = x, j/2 = x .
36. In" x y'i'xx + у?а + [(a - b2) In" x + b]y'x + o(l - Ь In" x)y = 0.
Particular solutions: y\ = eXlX, yi = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + bX + a = 0.
37. In" x y'l'xx + ay'lx - bBa + 3b In" x)y'x + Ь2(а + 2b In" x)y = 0.
Particular solutions: yx = ebx, yi = xebx.
38. lnn(Xx)yZx + ay'lx + Ь lnn(Xx)y'x + aby = 0.
1°. Particular solutions with b > 0: У\= cos{x\fb), yi = sin(xVb).
2°. Particular solutions with b < 0: У\= ехр(—ж\/—b), y<i = ехр(ж\/—b).
3.1.6. Equations Containing Trigonometric Functions
!• VvL + °3 tan(aa;) у = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
С
(see 2.1.6.29 for the solution of the corresponding homogeneous equation).
2- VvL ~ °3 tan(aa;) у = 0.
Particular solution: yo = cos(arr).
The substitution у = cos(ax) f z(x) dx leads to a second order linear equation of
the form 2.1.6.29: z'^ - 3tan?z^ - 3z = 0, where ? = ax.
3- yZx + a3cot(ax)y = 0.
The substitution x = t -\ leads to a linear equation of the form 3.1.6.2: y'?t —
ZiQj
a3 tan(ai) у = 0.
4- VvL ~ °3 cot(aa;) у = 0.
The substitution x = t -\ leads to a linear equation of the form 3.1.6.1: y'^t +
a3 tan(ai) у = 0.
© 1995 by CRC Press, Inc.

5- VZX + 3а2У'х + 2°3 tan(oaj) у = 0.
Particular solutions: y\ = cos(ax), У2 = xcos(ax).
6- VZL + ay'x + *(« ~ A2) tan(Aaj) у = 0.
Particular solution: yo = cos(Arr).
The substitution у = cos(Arr) f z(x) dx leads to a second order equation of the
form 2.2.6.29: z'^ - 3tan?z? + (aX~2 - 3)z = 0, where С = Аж.
7- vZx + 3a2vL - 2a3 cot(ax) у = 0.
Particular solutions: yi = sin(aa;), j/2 = xsin(ax).
8- CL + ° cosBa;)y; - b[a cosBa;) + Ь2]у = О.
The substitution w = ebx^2(yfx — by) leads to the Mathieu equation 2.1.6.4: w^x +
[acosBrr) + -f b2]w = 0.
9- VvL + [° cosBa;) + b]y'x - a sinBa;) у = 0.
Solution:
у = C\W2 + C2W1W2 + C3W2,
where Wi and W2 is the fundamental set of solutions of the Mathieu equation 2.1.6.4:
4«4/a, + [acosBrr) + b]w = 0.
ж) + Ъ]у'х - la sinBaj) у = 0.
By integrating, we obtain a nonhomogeneous Mathieu equation:
lCx + [acoeBx)+b]y = C.
<L + [° со5(Лж) + b]y^ - c[a со5(Лж) + b + c2]y = 0.
The transformation ^ = -jАж, ад = есх/2{у'х — су) leads to the Mathieu equation 2.1.6.4:
ад^ + 4A[acosBC) + b + f с2]ад = О.
ж) - Ь2]у'х -а[Ь со3Bж) + 2 ЗтBж)]у = 0.
By integrating and substituting ад = yebx^2, we obtain a nonhomogeneous Mathieu
equation: w%x + [acosBrr) - \b2]w = Ce3bx/2.
13- V'ZX + a sin ж y^, - b(a sin ж + Ь2)у = 0.
The substitution ад = ebx^2(y'x — by) leads to an equation of the form 2.1.6.3: v/xx +
(a sin x + \b2)w = 0.
14- yZL + °sin2 x v'x - b(°sin2 x + ь2)у = °-
The substitution ад = ebx^2(yfx — by) leads to an equation of the form 2.1.6.5: w'xx +
(a sin2 x + -f 62)ад = 0.
© 1995 by CRC Press, Inc.

a;) + Щу'х + а\ соэ(Аж) у = 0.
By integrating, we obtain a nonhomogeneous second-order linear equation:
y'x!x + [asin(Xx)+b}y = C
(see 2.1.6.3 for the solution of the corresponding homogeneous equation).
v'Zx + [asin(Xx) - Ь2]у'х + o[Acos(Aa;) - bsin(\x)]y = 0.
By integrating and substituting w = yebx'2, we obtain a nonhomogeneous second-order
linear equation:
<x + [asin(Xx) - \b2}w = CeZhxl2
(see 2.1.6.3 for the solution of the corresponding homogeneous equation).
aj) + b]y'x - c[a5т(Аж) + b + c2]y = 0.
The substitution w = ecx^2(yfx — су) leads to an equation of the form 2.1.6.3: w^x
[asin(Aa;) + b + \c2)w = 0.
V'i'xx — 3o[osin2(ba;) + bcos(ba;)]y^, + osin(ba;)[b2 + 2o2 sin2(ba;)]y = 0.
Particular solutions: y\ = exp —— costbx) \, yi = x exp —— costbx) .
L b J L b J
19. Ух'Хх "I" ° tan2 ж у'х — b(a tan2 ж + Ь2)у = 0.
The substitution w = ebx^2(y'x — by) leads to an equation of the form 2.1.6.10: wxx +
(a tan2 x + ^b2)w = 0.
20- y'vL + Мап2(Аж) + b]y'x - c[a tan2^) + b + c2]y = 0.
The transformation ? = Xx, w = ecx'2(y'x — су) leads to an equation of the form
2.1.6.10: w'^ + X~2(atan2? + b + f c2)w = 0.
21. y^ + a cot2 ж y^ — b(a cot2 ж + Ь2)у = 0.
The substitution w = ebx^2(yfx — by) leads to an equation of the form 2.1.6.12: wxx +
(a cot2 x + ^b2)w = 0.
22. y'" -\- ay" -\- (bcos 2ж + c)y' + o(bcos 2ж + c)y = 0.
The substitution w = y'x + ay reduces the original equation to the Mathieu equation
2.1.6.4: wxx + (bcos2x+ c)w = 0.
23. y'XXx "I" av'xx "I" [Ьв1п(Аж) + c]y'x + а[Ьэт(Аж) + c]y = 0.
The substitution w = y'x + ay leads to a second order linear equation of the form
2.1.6.3: w'^ + [6sin(Arr) + c]w = 0.
24- VZX + av'L + b sin2 (Xx)v'x + ab sin2 (Аж) у = 0.
The substitution w = y'x + ay leads to a second order linear equation of the form
2.1.6.5: w'^ + 6sin2(Arr) w = 0.
© 1995 by CRC Press, Inc.

25- VZX + av'L + (btan2 x + c)y'x + a(b tan2 ж + c)y = 0.
The substitution w = y'x + ay leads to a second order linear equation of the form
2.1.6.10: w'x'x + (btan2x + c)w = 0.
26- У^
Particular solutions: y\ = cos(Arr), уг = жсов(Аж).
27- tC + ay'lx + (b cot2 x + c)y'x + a{b cot2 x + c)y = 0.
The substitution w = y'x + ay leads to a second order linear equation of the form
2.1.6.12: 2
28- yZ
Particular solutions: y\ = sin(Aa;),
29. <L + о со3"(Лж)у^ + by^ + ab со3"(Лж) у = 0.
1°. Particular solutions with b > 0: yi = С08(жл/&), 2/2 = sm(x-\/b).
2°. Particular solutions with b < 0: г/i = exp(—ж\/—&), У2 = ехр(ж-\/—b).
30- <L + ocos"(Aa;)y;/a; + obcos"(Aa;)y; + Ь2[осо3"(Лж) - b]y = 0.
Particular solutions: yx = e~bx^2 cos ( —-— j, y2 = e~bx^2 sin ( —-— j.
V 2 / \ 2 /
31- VvL + acosn(Xx)v'L - b[2acosn(\x) + 3b]y'x + Ь2[аcosn(\x) + 2b]y = 0.
Particular solutions: y\ = ebx, у2 = xebx.
32. y^ + a cos" x у'1х + (ab cos" x + с - Ь2)у'х + c(a cos" x - b)y = 0.
Particular solutions: yi=exp(Aia;), у2=ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + ЬХ + с = 0.
33- VvL + ° cosn(Xx)v'L + ЬхГПУ'х + bxm~x [ax cos"(A«) + m]y = 0.
Assuming w = yxx + bxmy yields a first order linear equation: w'x + a cosn(Aa?) w = 0.
34- vZx + (° cos" x + b)y'L + cvL + c(° cos" х + ь)у = 0.
1°. Particular solutions with с > 0: y\ = cos {x\fc), У2 = 8^
2°. Particular solutions with с < 0: y\ = exp(—ж^/—с), уг = ехр(жу/—с)
35- y^'L = (cos" ж - a)y'L + (° cos" Х-Ь)у'х + Ь cos" ж у.
Particular solutions: yi=exp(Aia;), у2=ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + aX + b = 0.
36. y'Zn + (a cos" ж + bx)y'x'x + b(ax cos" ж + 2)y'x + ab cos" ж у = 0.
г
Particular solutions: y\ = exp(— ^bx2), yi = exp(— \bx2) I ещ>{^Ьх2)<1х.
J
© 1995 by CRC Press, Inc.

37- y'Zx + ax cos" x V'L + (bx2 - ° cos" X)V* + bx(ax2 cos" ж + 3)y = 0.
Particular solutions: y\ = cos(ya;2v6), 2/2 = sin(-jrr2v6).
38. у'ххх ~1~ (чЬх cos" ж + о cos" ж + Ь)ухх -\- ab x cos" ж ух — ab cos" ж у = 0.
Particular solutions: 2/1 = х, yi = е~ х.
39- Уа-'L + ож2 cosn(Xx)y'vX ~ 2ож со5"(Лж)у^, + 2о со5"(Лж) у = 0.
Particular solutions: уг = х, у2 = х .
40. fC + a sin"(AaJ)»^ - by'x - ab 31п"(ЛЖ) у = 0.
1°. Particular solutions with b > 0: yi = exp(—жл/Ь), 2/2 = ехр(жл/&).
2°. Particular solutions with b < 0: 2/1= cos(a?\/—b), 2/2 = sxn(x-°i/—b).
4 + b2[osin"(Aaj) - b]y = 0.
_Ъх/2 .
-bx/2 (V
Particular solutions: y1 = e ' cos —-— , y2 = e
Xx)y'^x - b[2asinn(Xx) + 3b]y'x + b2[asinn(Xx) + 2b]y = 0.
Particular solutions: y1 = ebx, 2/2 = xebx.
43- y'i'xx + a sin" x y'^x + (ab sin" x + с - Ъ2)у'х + c(a sin" x - b)y = 0.
Particular solutions: 2/1 =exp(Aia;), 2/2 = expire), where Ai and A2 are the roots
of the quadratic equation A2 + ЬХ + с = 0.
44. y'i'xx + a sinn(Xx)y'^x + bxmy'x + bxm~x [ax sin"(Aa;) + m]y = 0.
Assuming w = y'J.x + bxmy yields a first order linear equation: w'x + a sin™ (Аж) w = 0.
45- tC + (o sin" x + b)y'^x + cy'x + c(a sin" +b)y = 0.
1°. Particular solutions with с > 0: yi = cos (x^fc), 2/2 = вш(жу/с)-
2°. Particular solutions with с < 0: у\ = ехр(—Жу/—с), 2/2 = ехр(жу/—с).
46- y^'L = (sin" ж - а)у'х-х + (о sin" х - Ь)у'х + Ь sin" x у.
Particular solutions: 2/1 =exp(Aia;), 2/2 =ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + aX + b = 0.
47. yxxx + (a sin" x + bx)y'x'x + b(ax sin" x + 2)y'x + ab sin" x у = 0.
Particular solutions: 2Л = exp(— -j&rr2), 2/2 = exp(— \bx2} I ехр(^Ьх2) dx.
48- y'i'xx + ax sin" ж y^, + (Ьж2 - о sin" ж)у^ + bx(ax2 sin" ж + 3)y = 0.
Particular solutions: 2/1 = со8(уЖ2л/&), 2/2 = sin(jj2v/b).
© 1995 by CRC Press, Inc.

49- Ух'хх + (°^ж sin" x + ° sin" ж + Ь)У4'а; + °Ь2ж sin" ж y'x — ab2 sin" x у = 0.
Particular solutions: 2/1 = ж, г/2 = е~Ьх.
50- CL + ож2 s^n(Xx)v'L ~ 2ax sinn(Xx)v'x + 2o вт"(Аж) у = 0.
Particular solutions: уг = x, 2/2 = x .
<L L ^ у = 0.
1°. Solution with a > 0: у = C\ exp(—Xifa} + C^ ехр(жу/«) + Сзсов(Аж).
2°. Solution with a < 0: у = C\ cos(xy/—a) + C^ sin^/^a) + C3 cos(Arr).
52- <L + <* tan(^)yL + ^ + A(oA + b - A2) tan(Aa;) у = 0.
Particular solution: yo = cos(Arr).
С f
The transformation ж = —, у = сов(Аж) zdx leads to a second order equation
of the form 2.1.6.55:
(y - З) tan?4 + (-^- - 3 - ^ tan2 ф = 0.
53- VZL + a tan"(^)yL + by'x + ob tan"(Aa;) у = 0.
1°. Particular solutions with b > 0: yi = С08(жл/&), 2/2 = sin(xy/b).
2°. Particular solutions with b < 0: г/i = exp(—ж\/—&), 2/2 = ехр(ж\/—Ь).
54- y^'L + otan"(Aa;)yL + obtan"(Aa;)y; + b2[otan"(Aa;) - b]y = 0.
_ ,. , , ,. _bx/2 (bx^/i \ _b /2 . (Ьх^Д \
Particular solutions: y1 = e 'cos I I, 2/2 = e 'sin I I.
V 2 / \ 2 /
55. |C + otan"(Aa;)y4/a; - b[2otan"(Aaj) + 3b]y^ + Ь2[аtan"(Aaj) + 2b]y = 0.
Particular solutions: 2/1 = e6a:, 2/2 = же6а:.
56. tC + о tan" ж y^ + (ob tan" ж + с - Ь2)у^ + c(a tan" x - b)y = 0.
Particular solutions: 2/1 =exp(Aia;), 2/2 = expire), where Ai and A2 are the roots
of the quadratic equation A2 + ЬХ + с = 0.
57. tC + о tan^^y^ + Ьж^у^ + bxm~1[ax tan"(Aaj) + m]y = 0.
Assuming ад = у^'д. + 6жт2/ yields a first order linear equation: w'x + a tan™ (Аж) ад = 0.
58- tC + (a tan" ж + b)y^ + cy'x + c(a tan" ж + b)y = 0.
1°. Particular solutions with с > 0: 2/i = cos (x^/c), 2/2 = sin (ж^
2°. Particular solutions with с < 0: уг = ехр(—Жу/—с), 2/2 = ехр(жу/—с).
59. tC = (tan" ж - о)у4'а; + (о tan" ж - Ь)у'х + Ь tan" ж у.
Particular solutions: 2/1 =ехр(А1ж), 2/2 =ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A + aX + 6 = 0.
© 1995 by CRC Press, Inc.

60- tC + (о tan" ж + bx)y'x'x + Ь(ах tan" x + 2)у'я + ab tan" x у = 0.
Particular solutions: у\ = ехр(— у&ж2), г/2 = ехр(— у&ж2) I ещ>{\Ьх2} dx.
61- y^'L + ож tan" x V'L + (bx2 - °tan" X)V'X + bx(ax2 tan" x + 3)y = 0.
Particular solutions: y\ = cos(ya;2v6), 2/2 = sin(-ja
62. yx'xx + (abx tan" ж + о tan" ж + b)yxx + оЬ2ж tan" ж у^, — ab2 tan" ж у = О
Particular solutions: y1 = x, г/2 = e~bx.
63- Уа-'L + ож2 tan"(Aaj)»^, - 2ож tan"(Лж)y4 + 2о tan"(Aaj) у = 0.
Particular solutions: y1 = х, г/2 = ж2.
64- y4/L[(+)+]
Particular solutions: г/i = еах, г/2 = cos ж.
65- Уа-'L - (°b tan" x + b tan"+1 ж + а)у'1х + [Ь(а2 + 1) tan" х + 1]у'х
+ a(ab tan"+1 х-Ь tan" ж - 1)у = 0.
Particular solutions: г/i = еах, г/2 = cos ж.
66- Уа-'L + ^ап(Аж)(аж" + 1) + ож"]»^ - оЛ2ж"у^ + оА2»"» = 0.
Particular solutions: г/i = ж, г/2 = сов(Аж).
67- Уа-'L - Л СО*(Лж)Уа.'а; - °Уа- + °Л COt(Aflj) У = 0.
1°. Solution with a > 0: у = С\ ехр(—Ху/а) + C<i ехр(жу/«) + Сзвш(Аж).
2°. Solution with а < 0: у = С\ сов(жу/—а) + Сг вш(жу/—а) + Сзsin(Aa;).
68. j/^sb + a co^n (^х)Ухх + abcot"(Aa;)j/^ + Ь2[асс^"(Аж) — b]y = 0.
ЪхуД\
^ , , _ь.г/2 (Ьх\/Ъ\
Particular solutions: y1 = e ' cos —-— , y2 = e
69. y'i'xx + ocot"(Лж)ya./a; - b[2ocot"(Лж) + 3b]y'x + b2[ocot"(Лж) + 2b]y = 0.
Particular solutions: y\ = ebx, г/2 = xebx.
70- Уа-'L + ° cot" x V'L + (ab cot" x + c-b2)y'x + c(a cot" x-b)y = 0.
Particular solutions: уг =exp(Aia;), г/2 =ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + ЬХ + с = 0.
САж^ + Ьж^у^ + bxm~1 [ax cot"(Aaj) + m]y = 0.
Assuming ад = y'xX + bxmy yields a first order linear equation: w'x + a cotn(Aa;) ад = 0.
© 1995 by CRC Press, Inc.

72- tC + (о cot" ж + Ъ)у'1х + cy'x + c(a cot" ж + b)y = 0.
1°. Particular solutions with с > 0: y\ = cos(xy/c), y2 = sin(xy/c).
2°. Particular solutions with с < 0: y\ = exp(—ж^/—с), yi = ехр(жу/—с).
7з. y'i'xx = (cot™ x - a)v'L + (o cot" ж - ь)у4 + b cot" ж у-
Particular solutions: y!=exp(Aia;), у2=ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + aX + b = 0.
74- Ух'хх + ax cot" x У'хХ + (ba;2 "" ° cot" ж)у4 + bx(ax2 cot" ж + 3)y = 0.
Particular solutions: y\ = cos(ya;2v6), 2/2 = sin(-jrr2v6).
75- y^'L + (abx cot" ж + о cot" x + b)y'lx + ab2x cot" xy'x- ab2 cot" ж у = 0.
Particular solutions: y1 = x, yi= e~bx-
76- V'i'xx + ax2 cotn(Xx)y'L - 2ax cotn(Xx)y'x + la cot"(Лж) у = 0.
Particular solutions: y\ = x, yi= x .
77'• *y?L + ж(°cos 2ж + Ь)У'Х ~ 2(ocos 2ж + b)y = 0.
The substitution w = xy'x — 2y leads to the Mathieu equation 2.1.6.4:
wxx + (a cos 2x + &)w; = °-
78. xy'xxx + x(a cos2 ж + b)y^, - 2(o cos2 ж + Ъ)у = 0.
The substitution w = xy'x — 2y leads to a second order linear equation of the form
2.1.6.6: w%x + (a cos2 x + b)w = 0.
79. xyx'xx = (cos" ж — ax)yxx + @ cos" ж — bx)y'x + b cos" ж у.
Particular solutions: yi=exp(Aia;), 2/2= expire), where Ai and A2 are the roots
of the quadratic equation A2 + aX + b = 0.
80. xyf^fxx + (ax cos" ж + 3)y^, + Bo cos" ж + Ьж)у^ + b(ax cos" ж + l)y = 0.
Particular solutions: Wi =—cos(rrv6), V2 = —sinfrrvb).
ж v ' x y '
81- *y?L + ж[о5т(Лж) + Ь]у'х - 2[о5т(Лж) + Ъ]у = 0.
The substitution w = xy'x — 2y leads to an equation of the form 2.1.6.3: wxx +
[asin(Aa;) + b]w = 0.
82- xv'xL = (sin" x - ax)v'L + (°sin" x - Ьх)у'х + b sin" x у-
Particular solutions: yi=exp(Aia;), у2=ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A + aX + 6 = 0.
© 1995 by CRC Press, Inc.

83- *tC + (ах sin" ж + 3)у^ + Bа sin" ж + Ьх)у'х + Ь(ах sin" х + 1)у = 0.
Particular solutions: Vi = —cos(xVb), г/2 =—вт(жл/&).
ж v ' x '
84- ^'L + x(a tan2 ж + Ь)у4 - 2(o tan2 ж + b)y = 0.
The substitution ад = ж$4 — 2г/ leads to an equation of the form 2.1.6.10: wxx +
(a tan2 ж + b)w = 0.
85- xyZx = (tan™ x - ax)y'^x + (a tan" ж - Ъх)у'х + Ь tan" ж у.
Particular solutions: y!=exp(Aia;), 2/2= ехр^ж), where Ai and A2 are the roots
of the quadratic equation A2 + aX + b = 0.
86. xy'?xx + (ax tan" ж + 3)y'x'x + Ba tan" ж + Ьж)у^ + b(ax tan" ж + l)y = 0.
Particular solutions: yi = —cos(rrv6), 2/2 = —sin(a;v6).
X X
87- ожу^ + [1 - Л(о + 1)ж cot(Лж)]y;/a; - Л2жу^ + Л2у = 0.
Particular solutions: y\ = х, г/2 = sin(Aa;).
88- XVZX + ж(° cot2 ж + ь)у4 - 2(° cot2 ж + b)y = 0.
The substitution ад = ху'х — 2у leads to an equation of the form 2.1.6.12: wxx
(a cot2 ж + b)w = 0.
89- «y^'L + (ax cot" x + 3)yL + Bo cot" ж + bx)v'x + Чах cot" ж + l)y = 0.
Particular solutions: У\ = —cos^vfr), 2/2 = —sin(a;vb).
90. x2yx'xx = (cos" ж — ax2)yxx + (a cos" ж — Ьж2)у^, + b cos" ж у.
Particular solutions: г/i =ехр(А1ж), г/2 =ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + aX + b = 0.
91- x2vZx = (sin™ x ~ a^b'L + (o sin" x ~ bx2)vL + b sin" x V-
Particular solutions: гд =ехр(А1ж), г/2 =ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A2 + aX + b = 0.
92- «2y?L = (tan™ x ~ ax<2)v'L + (°tan" ж - Ьж2)у4 + ъ tan" ж У-
Particular solutions: у1=ещ>(Х\х), г/2 =ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A + aX + 6 = 0.
93. x2y'x"xx = (cot" ж - ож2)у^ + (о cot" ж - Ьж2)у^ + b cot" ж у.
Particular solutions: г/i =ехр(А1ж), г/2 =ехр(А2ж), where Ai and A2 are the roots
of the quadratic equation A + aX + 6 = 0.
© 1995 by CRC Press, Inc.

94. x3y'x"xx + ax2 cosn{Xx)y'x[x + bxy'x + b[a cos"(Aa;) - 2]y = 0.
Particular solutions: y\ = xmi, уг = xm2, where mi and тг are the roots of the
quadratic equation m — m + b = 0.
95- *stC» + ax2 sinn(Xx)yfx1x + bxy'x + b[a sin"(Aa;) - 1\y = 0.
Particular solutions: y\ = xmi, уг = xm2, where mi and тг are the roots of the
quadratic equation m2 — m + b = 0.
96- *StC» + «ж2 tan"(Aa;)y;/a; + Ьжу^ + b[o tan"(Aaj) - 2]y = 0.
Particular solutions: yi = xmi, y^, = xm2, where mi and m.2 are the roots of the
quadratic equation m — m + b = 0.
97. ж3у4/4а, + ax2 cotn(Xx)y'x'x + bxy'x + b[a cot"(Aa;) - 2]y = 0.
Particular solutions: y\ = xmi, yi = xm2, where mi and тг are the roots of the
quadratic equation m — m + b = 0.
98. cos2 x y'l'xx + a cos2 x y'lx + by'x + aby = 0.
The substitution x = ? + -|- leads to an equation of the form 3.1.6.112: sin ^yl'L +
a sin2 С y'^ + Щ + aby = 0.
99. cos" x y'l'xx + ay'lx + aby'x + Ь2(а-Ь cos" x)y = 0.
_ , . , , , . _bx/2 ( ЬХ\[2, \
Particular solutions: y\=e ox' cos I —-— I, y2 = e
V 2 /
_b /2
sin
\
I.
\ 2 J
100. cos" x y'l'xx + ay'lx + Ь cos" x y'x + aby = 0.
1°. Particular solutions with b > 0: y\ = cos(xVb), y<i = sin(xVb).
2°. Particular solutions with b < 0: У\= ехр(—ж\/—b), уг = ехр(ж\/—b).
101. cosnxyZx + ay'lx - bBa + 3bcosnx)y'x + Ь2(а + 2bcosnx)y = 0.
Particular solutions: y\ = еЬх, у2 = xebx.
102. cos" x y'l'xx + y'ix + [(b - a2) cos" x + a]y'x + b(l - a cos" x)y = 0.
Particular solutions: уг = eXlX, уг = eX2X, where Ai and Аг are the roots of the
quadratic equation A2 + aX + b = 0.
103. cos™(\x)yZx + ax2y'x-x - 2axy'x + 2ay = 0.
Particular solutions: y\ = x, У2 = x .
104. cos" x y'l'xx + (a cos" x + ax + l)y'^x + a2xy'x - a2y = 0.
Particular solutions: y\ = x, уг = e~ax.
© 1995 by CRC Press, Inc.

105. cos" ж у'ххх + (ax cos" ж + l)yxx + a(x + 2 cos" x)y'x + ay = 0.
/ ax2 \ ( ax2 \ f (ax2 \ ,
Particular solutions: yi=expl ^~)> 2/2 = exp I — 1 / exp I ) ax.
106. ж cos" ж y'xxx + C cos" ж + ж)^ + (ax cos" ж + 2)у4 + o(cos" ж + ж)у = 0.
Particular solutions: У\ = —cos(x\fa), г/2 = —sin(xy/a).
X X '
107. ж3 cos" ж у'^'хх + ax2y'lx - 2x cos" ж y^ + 2B cos" ж - o)y = 0.
Particular solutions: yx = x~ , У2 = x .
108. ж3 cos" ж 2/^..,, + ах2УхХ — 6ж cos" ж Уж + 6B cos" ж "" а)У = °-
Particular solutions: y\ = ж~ , уг = x .
109. ж3 cos" ж у^з, + ах2ухх + х(а — cos" ж)у^ + а(а — 3 cos" ж)у = 0.
Particular solutions: уг = cos(ln |ж|), уг = sin(ln |ж|).
110. ж3 cos" ж yZx + ж2(cos" ж + а)у'1х +
ж [о - (Ь+ 1)соэ"ж]у4 +ЬBсоэ"ж - а)у = 0.
Particular solutions: y1 = х~ , уг = х .
111. sin2 ж у^з, + 3 sin ж cos ж у^ + [cos 2ж + 4v(v + 1) sin2 ж]у^
+ 2iv(iy + 1) sin 2ж у = 0.
Solution:
у = Ciy2 + С2У1У2 + СзУг»
where yi, уг form a fundamental set of solutions of the Legendre equation 2.1.2.148,
with argument x of functions y\ and уг substituted by cos x.
112. sin2 ж y^ + о sin2 ж y^ + by^ + aby = 0.
The substitution w = y'x + ay leads to a second order equation of the form 2.1.6.108:
sin2 x wxx +bw = 0.
113. sin" ж y'l'^ + ay'lx + aby'x + Ь2(а- bsin" ж)у = 0.
Particular solutions: yi = e~ ' cos I I ¦»« = р~ож/ <
114. sin" ж yZx + ay'^x + b sin" ж y'x + aby = 0.
1°. Particular solutions with b > 0: У\= cos{x\fb), yi = sin(xVb).
2°. Particular solutions with b < 0: У\= ехр(—ж\/—b), yi = ехр(ж\/—b).
115. sin" ж y^ + ay'^x - bBa + 3b sin" x)y'x + Ь2(а + 2b sin" x)y = 0.
Particular solutions: y\ = ebx, у2 = xebx.
© 1995 by CRC Press, Inc.

116. sin" ж у'ххх + yxx + [(b — a2) sin" ж + a]y'x + b(l — a sin" ж)у = О.
Particular solutions: г/i = eXlX, г/2 = eX2X, where Ai and A2 are the roots of the
quadratic equation A + aX + 6 = 0.
117. s\TLn(Xx)y'x\'xx + ax2y'lx - 2axy'x + 2ay = 0.
Particular solutions: уг = х, г/2 = x .
118. sin" ж y'xxx + (a sin" ж + ax + l)yxx + a2xy'x — a2y = 0.
Particular solutions: г/i = x, г/2 = e~ax.
119. sin" x Ух-'хх + (ax sin" x + 1^ + a(x + 2 sin" x)y'x + ay = 0.
т, / ax2 \ ( ax2 \ f (ax2 \
Particular solutions: yi=expl ^~)> 2/2 = expl — 1 / expl ) ax.
\ Zi J \ Zi J j \ Zi J
120. x sin" x y'l'xx + C sin" x + x)y'lx + (ax sin" x + 1)y'x + o(sin" x + x)y = 0.
Particular solutions: г/i = —cos^v^), 2/2 = —sin(xy/a).
X X
121. ж3 sin" x y'l'xx + ax2y'lx - 2x sin" xy'x + 2B sin" x - a)y = 0.
Particular solutions: y\ = x~ , г/2 = x .
122. ж3 sin" x y'^ + ax^y'^ - 6ж sin" xy'x + 6B sin" x - a)y = 0.
Particular solutions: г/i = x~2, г/2 = x3.
123. ж3 sin" ж y'xxx + ax2yxx + x(a — sin" ж)у^, + 0@ - 3 sin" ж)у = О.
Particular solutions: г/i = cos(lna;), г/2 = sin(lna;).
124. ж3 sin" ж CL+«2(sin" x+a)y'x'x+x[a-(b+l) sin" ж]у^+ЬB sin" x-a)y = 0.
Particular solutions: г/i = x~ , г/2 = x .
125. tan" ж y^ + ay'lx + aby'x + Ь2(а-Ь tan" ж)у = 0.
л j.- 1 1 i.- -Ьж/2 ( Ьх^Д \ _b /2 . ( Ьх^Д \
Particular solutions: г/i = e 'cos I I, г/2 = e 'sin I I.
V 2 / \ 2 /
126. tan" ж y^ + oy^ + b tan" ж y^ + oby = 0.
1°. Particular solutions with b > 0: г/i = cos(a?v6), г/2 = sin(a;v6).
2°. Particular solutions with b < 0: г/i = exp(—ж\/—&), г/2 = ехр(ж\/—&
127. tan" ж у^ + oy^ - bBa + 3b tan" ж)у^ + Ь2(а + 2b tan" ж)у = 0.
Particular solutions: г/i = ebx, г/2 = xebx.
© 1995 by CRC Press, Inc.

128. tan" ж у'1'хх + y'lx + [(Ь - a2) tan" ж + а]у'х + ЬA - a tan" х)у = 0.
Particular solutions: y\ = eXlX, г/2 = еХ2Х, where Ai and A2 are the roots of the
quadratic equation A + aX + 6 = 0.
129. tann(Xx)yfx-xx + ax2y'^x - 2axy'x + 2ay = 0.
Particular solutions: y\ = x, г/2 = x .
130. tan" x y'J'xx + (a tan" x + ax + \)yxx + a2xy'x - a2y = 0.
Particular solutions: уг = x, 1/2 = e
xx
~ax
131. tan" x yZx + (ax tan" x + l)y'^x + a(x + 2 tan" x)y'x + ay = 0.
/ ax2 \ / ax2 \ f / ax2 \
Particular solutions: yi = exp I — 1, y<i = exp I — 1 / exp I —— 1 ax.
\ Zi / \ Zi / j \ Zt /
132. x tan" x y'l'^ + C tan" x + x)y'x[x + (ax tan" x + 2)y'x + o(tan" x + x)y = 0.
Particular solutions: Vi = —cosfrr-v/a), 2/2 = —sinfrr-v/tt)-
X У ' X У '
133. ж3 tan" x y'?xx + ax2y'lx - 2x tan" x y'x + 2B tan" x - a)y = 0.
Particular solutions: y\ = x~ , yi = x .
134. ж3 tan" x y'?xx + ax2y'lx - 6x tan" x y'x + 6B tan" x - a)y = 0.
Particular solutions: y\ = x~ , yi = x .
135. ж3 tan" x y'?xx + ax2y'x'x + x(a - tan" x)y'x + 0@ - 3 tan" x)y = 0.
Particular solutions: уг = cos(ln |ж|), у<± = sin(ln |ж|).
136. ж3 tan" ж yZx + ж2 (tan" ж + a)y'^x +
x[a - (b + 1) tan" x]y'x + bB tan" x - a)y = 0.
Particular solutions: y\ = x~ , yi = x .
137. cot" ж y'lxx + ay^x + aby'x + Ь2(а- bcot" x)y = 0.
^ , , -h-r/i (bxV% \ -h-r/i ¦ (bxV% \
Particular solutions: y1 = e ' cos I —-— I, y2 = e ' sin I —-— I.
\ ^ / \ ^ /
138. cot" ж yZx + ay'^x + Ь cot" ж y'x + aby = 0.
1°. Particular solutions with b > 0: У\= cos{x\fb), y2 = sin(xVb).
2°. Particular solutions with b < 0: У\= ехр(—ж\/—b), yi = ехр(ж\/—b).
139. cot" ж Ух-'хх + аУх-х - bBa + 3b cot" x)y'x + Ь2(а + 2b cot" x)y = 0.
Particular solutions: y\ = ebx, у2 = xebx.
© 1995 by CRC Press, Inc.

140. cot" ж yZx + V'L + Kb - °2) cot" x + aWx +b(l-a cot" x)y = 0.
Particular solutions: y1 = eXlX, 1/2 = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + aX + b = 0.
141. cotn(Xx)yfx'xx + ах2Ух-х - 2axy'x + lay = 0.
Particular solutions: y\ = x, yi = x .
142. cot" x yZx + (a cot" x + ax + l)y'^x + a2xy'x - a2y = 0.
Particular solutions: y\ = x, J/2 = e~ax¦
143. cot" x yZx + (ax cot" x + l)y'^x + a(x + 2 cot" x)y'x + ay = 0.
/ ax2 \ ( ax2 \ f (ax2 \ ,
Particular solutions: yi = exp I — 1, y<i = exp I — 1 / exp I —— 1 ax.
144. x cot" x y'xxx + C cot" x + x)yxx + (ax cot" x + 2)y'x + o(cot" x + x)y = 0.
Particular solutions: Vi = —cosfrr-v/^), 2/2 = —sin
X У ' X
145. ж3 cot" x y'^ + ax^y'^ - 2x cot" xy'x + 2B cot" x - a)y = 0.
Particular solutions: y\ = x~ , yi = x .
146. ж3 cot" x y'^ + ax^y'^ - 6ж cot" xy'x + 6B cot" x - a)y = 0.
Particular solutions: y\ = x~2, У2 = x .
147. ж3 cot" ж y'l'xx + ax2y'lx + x(a - cot" x)y'x + o(o - 3 cot" x)y = 0.
Particular solutions: y\ = cos(lna;), j/2 = sin(lna;).
148. ж3 cot" ж y'l'xx + ж2(cot" ж + a)y'lx +
x[a - (b + 1) cot" x]y'x + bB cot" ж - a)y = 0.
Particular solutions: y\ = x~ , yi = x .
3.1.7. Equations Containing Inverse Trigonometric Functions
!• VvL + av'L + by'x + СУ = arcsinfc ж.
This is a special case of equation 5.1.5.9.
2- У'х'хх + arcsinfc ж yxx + ay'x + a arcsinfc ж у = 0.
1°. Particular solutions with a > 0: Уг= сов(хл/а), У\ = sin(
2°. Particular solutions with a < 0: У\= exp(—Xy/—a), y\ =
The substitution w = y'^.x + ay leads to a first order linear equation: w'x
arcsin xw = 0.
© 1995 by CRC Press, Inc.

3- V'i'xx + arcsinfc x yxx + axny'x + axn x (x arcsinfc x + ri)y = 0.
The substitution w = yxx + axny leads to a first order linear equation: w'x +
arcsin fe xw = 0.
4- V'i'xx + arcsinfc x yxx + a arcsinfc x y'x + o2(arcsinfc x — а)у = 0.
Particular solutions: уг = e~ax^2 cos( —-—"¦ 1 ¦»« — e~ax/2
(
5- Vx'xx + arcsinfc x yxx — oB arcsin x + 3a)y'x + o2(arcsin x + 2o)y = 0.
Particular solutions: уг = eax, yi = xeax.
6* Vx'xx "I" arcsinfc x yxx + (o arcsin x + b — a2)y'x + b(arcsin x — a)y = 0.
Particular solutions: уг = eXlX, yi = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + aX + 6 = 0.
7. y"' = (arcsinfc x — a)y'' + (o arcsinfc x — b)y' + b arcsinfc x y.
The substitution w = yxx + ay'x + by leads to a first order linear equation: w'x =
arcsin xw.
8. v'i'xx "I" x arcsinfc x y'^ + (ax2 — arcsinfc x)y'x + ax(x2 arcsinfc x + 3)y = 0.
Particular solutions: y\ = cos I —-— 1, 2/2 = sin
9. v"' + (arcsinfc x + ax)y'' + a(x arcsinfc x + 2)y' + a arcsinfc x у = 0.
( ax2 \ ( ax2 \ f / ax2 \
Particular solutions: уг = exp I 1, 2/2 = exp I 1 / exp I 1 dx.
10. y'i'xx + x2 arcsinfc x yxx — 2x arcsinfc x y'x + 2 arcsinfc x у = 0.
Solution:
у = Cix + С2ж2 + C3(x2 / х~3фdx- x х~2фdx),
where ф = exp(— f x2 arcsinfe x dx).
11. y'i'xx + (o.x arcsinfc a;+arcsinfc х + а)у'^х + а2х arcsinfc xy'x — a2 arcsinfc x у = 0.
Particular solutions: y\ = x, J/2 = e~ax.
12. y'xxx + arccosfc x y'ix + ay'x + a arccosfc x у = 0.
1°. Particular solutions with a > 0: У\= cos^x^/a), y\ = sin^ya).
2°. Particular solutions with a < 0: У\= exp(—xy/—a), y\ = ещ>(хл/—а).
The substitution w = yxx + ay leads to a first order linear equation: w'x +
arccosfe xw = 0.
© 1995 by CRC Press, Inc.

13. y'xxx + arccosfc x yxx + axny'x + axn г (x arccosfc x + n)y = 0.
The substitution w = yxx + axny leads to a first order linear equation: w'x +
arccosfe x w = 0.
ауД ^ -^anCAx,
14. у'ххх + arccosfc ж ухх + о arccosfc xy'x -\- o2(arccosfc ж — а)у = 0.
х I, у2 = е~ах12
15. у^а; + arccosfc ж у^'з, — оB arccosfc ж + Зо)у^ + o2(arccosfc ж + 2о)у = 0.
Particular solutions: у1 = еах, y<i = хеах.
16. у'ххх + arccosfc ж у^, + (о arccosfc x -\- b — а2)у'х + b(arccosfc ж — о)у = 0.
Particular solutions: y\ = eXlX, yi = еХ2Х, where Ai and A2 are the roots of the
quadratic equation A2 + aX + b = 0.
17'• Vx'xx = (arccosfc ж - a)yxx + (a arccosfc ж - b)y'x + b arccosfc ж у.
The substitution w = yxx + ay'x + by leads to a first order linear equation: w'x =
arccosfe x w.
18. y"' + ж arccosfc ж у" + (ож2 — arccosfc ж)у' + ож(ж2 arccosfc ж + 3)у = 0.
Particular solutions: y\ = cos I —-— 1, 2/2 = sin I
19. y'xxx + (arccosfc ж + ax)yxx + о(ж arccosfc ж + 2)y^, + a arccosfc ж у = 0.
Particular solutions: y\ = exp ( J, yi = exp ( J / exp ( J dx.
20. y'xxx + x2 arccosfc ж yxx — 2x arccosfc ж y'x + 2 arccosfc ж у = 0.
Solution:
у = C\x + C2X2 + Сз(х2 / x~zipdx — x I x~2ipdx\,
J J
where ф = exp(— J x2 arccosfe x dx).
21. Ух'хх-\-(о.х arccosfc ж+агссоэ'8 x-\-d)yxx-\-a2 x arccosfc xy'x — a? arccosfc ж у = 0.
Particular solutions: y\ = x, 2/2 = e~ax.
22- Vxxx + arctanfc ж y^x + ay'x + a arctanfc ж у = 0.
1°. Particular solutions with a > 0: Уг= сов(хл/а), У\ = sin(
2°. Particular solutions with a < 0: У\= exp(—Xy/—a), y\ =
The substitution w = yxx + ay leads to a first order linear equation: w'x +
arctanfe xw = 0.
© 1995 by CRC Press, Inc.

23. y'l'xx + arctanfc x yxx + axny'x + ахп~г(х arctanfc x + ri)y = 0.
The substitution w = yxx + axny leads to a first order linear equation: w'x +
arctanfe xw = 0.
24. yx'xx + arctanfc x yxx + a arctanfc xy'x-\- o2(arctanfc x — а)у = 0.
Particular solutions: y1 = e~ax^2 cos(—-—xj, уг = e~°^2sin(—-—xj.
25. у'1'хх + arctanfc x yxx — oB arctanfc x + 3a)y'x + o2(arctanfc x + 2a)y = 0.
Particular solutions: y\ = eax, y<i = xeax.
26- VvL + arctanfc x y'lx + (a arctanfc x + b- a2)y'x + b(arctanfc x - a)y = 0.
Particular solutions: y\ = eXlX, yi = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + aX + b = 0.
27. y'xxx = (arctanfc x — a)yxx + (o arctanfc x — b)y'x + b arctanfc x y.
The substitution w = y'J.x + ay'x + by leads to a first order linear equation: w'x =
arctanfe xw.
28. y'xxx + x arctanfc x yxx + (ож2 — arctanfc x)y'x + ax(x2 arctanfc x + 3)y = 0.
i x у/а \ (х
Particular solutions: y\ = cos I —-— , yi = sin I
29- VvL + (arctanfc x + ax)y'^x + a(x arctanfc x + 2)y'x + a arctanfc x у = 0.
T-. / aX<2 \ ( aX<2 \ f ( aX<2 \
Particular solutions: y\ = exp I 1, yi = exp I 1 / exp I 1 ax.
30. y'xxx + x2 arctanfc x yxx — 2x arctanfc x y'x + 2 arctanfc x у = 0.
Solution:
у = dx + C2x2 + C3(x2 / x~3ipdx-x х~2фdx\,
where ip = exp(— J x2 arctanfe x dx).
31. y'xxx + (ax arctanfc x + arctanfc x + a)yxx
+ a2x arctanfc x y'x — a2 arctanfc x у = 0.
Particular solutions: y\ = x, yi= e~ax.
32. yx'xx + arccotfc x yxx + ay'x + a arccotfc x у = 0.
1°. Particular solutions with a > 0: У\= cos(xy/a), y\ = sin(xyfa).
2°. Particular solutions with a < 0: y\ = exp(—Xy/—a), y\ =
The substitution w = yxx + ay leads to a first order linear equation: w'x +
arccotfe xw = 0.
© 1995 by CRC Press, Inc.

33. y'l'xx + arccotfc x yxx + axny'x + ахп~г(х arccotfc x + ri)y = 0.
The substitution w = yxx + axny leads to a first order linear equation: w'x +
arccotfe xw = 0.
34. y'xxx "I" arccotfc ж y'xx + a arccotfc ж y'x + o2(arccotfc ж — a)y = 0.
x )
«УЗ \ -ax/2 ¦ ( aV^
ж , 2/2 = e OI/ism —— ж I.
/ \ z J
35. У^'жа; "I" arccotfc ж y^, — oB arccotfc ж + 3o)y^, + o2(arccotfc ж + 2o)y = 0.
Particular solutions: y1 = eax, y<i = xeax.
36. y'xxx + arccotfc x yxx + (o arccotfc x + b — a2)y'x + b(arccotfc x — а)у = 0.
Particular solutions: y\ = eXlX, y<i = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + aX + b = 0.
37. y'xxx = (arccotfc x — a)yxx + (o arccotfc x — b)y'x + b arccotfc x y.
The substitution w = yxx + ay'x + by leads to a first order linear equation: w'x =
arccotfe x w.
38. y'xxx + x arccotfc x yxx + (ож2 — arccotfc x)y'x + ax(x2 arccotfc ж + 3)y = 0.
T3 *.- 1 1 *.- {x2i/a\ . (x2i/a
Particular solutions: y\ = cos I I, yi = sin I
\ 2 /
I, yi sin I
\ 2 / \ 2
39- V'JL* + (arccotfc ж + ax)y'x[x + a(x arccotfc ж + 1)y'x + a arccotfc ж у = 0.
т-. / aX<2 \ ( aX<2 \ f ( aX<2 \
Particular solutions: y\ = exp I — 1, yi = exp I — 1 / exp I —— 1 ax.
40. Уххх + ж arccot ж yxx — 2x arccot ж yx + 2 arccot ж у = 0.
Solution:
where ф = exp(— / x2 arccotfe x dx).
41. Ух'хх~*г(ах arcc°tfc ж+arccotfc x-\-a)yxx-\-a2x arccotfc xy'x — a2 arccotfc ж у = 0.
Particular solutions: y\ = x, yi= e~ax.
42- хУ'х'хх + (ax2 + b)y'L + 4ахУ'х + 2°У = arcsinfc ж.
Twice integrating yields a first order linear equation:
/f f \
( / arcsinfe x dx \ dx.
43. xy'x'xx "I" (x arcsinfc ж + 3)yxx -\- B arcsinfc ж + ax)y'x + о(ж arcsinfc ж + l)y = 0.
Particular solutions: У\ = —cosfrrv^), 2/2 = —
X ' X
© 1995 by CRC Press, Inc.

44. xyxxx "I" (ж arccosfc х + 3)ухх + B arccosfc ж + ах)у'х + о(ж arccosfc ж + 1)у = 0.
Particular solutions: yi = —cos(xy/a), г/2 = —sin(xy/a).
X X
45- хУп'хх + (x arctanfc x + ^)yxx + B arctanfc x + ax)y'x + a(x arctanfc x + l)y = 0.
Particular solutions: Уг =—cos(xy/a), \ji = —sm(xy/a).
X X
46. xyx'xx + (ж arccotfc ж + 3)yxx + B arccotfc ж + ax)y'x + о(ж arccotfc ж + l)y = 0.
Particular solutions: %j\ = —cos^v^), 2/2 = —sa\{x\f(i).
X X
47. x3yxxx + [(о + 6)ж2 + Ь]ухх + 2Bо + 3)жу^, + 2оу = arcsinfc ж.
Twice integrating yields a first order linear equation:
/( f \
I / arcsinfe x dx I dx.
\J J
48. x3y'xxx + x2 arcsinfc жу^ — 2жу^ + 2B — arcsinfc ж)у = 0.
Particular solutions: y\ = x~ , У2 = x .
49. хъу'ххх + ж2 arcsinfe ж ухх - 6ху'х + 6B - arcsinfe ж)у = 0.
Particular solutions: y\ = х~ , уъ = х .
50- х3у'х"хх + ж2 arcsinfc ж у^ + ж^гсзт*1 ж - 1)у'х + (arcsinfc ж - 3)у = 0.
Particular solutions: y\ = cos(lnrr), y<i = sin(lna;).
51- x3vZx + я2 (arcsinfc ж + Л)у'1х + ж(агс51п'г ж - о - 1)у'х - (arcsinfc ж - 2)у = 0.
Particular solutions: y\ = х~^, уъ = х"^.
52. ж3у4/'''
Particular solutions: y\ = xni, y<i = хП2, where n\ and П2 are the roots of the
quadratic equation n2 + (a — l)n + b = 0.
53. x3yxxx + x2 arccosfc ж yxx — 2xy'x + 2B — arccosfc ж)у = 0.
Particular solutions: y\ = ж, уг = x2.
54. x3y'xxx + ж2 arccosfc ж ухх — 6xy'x + 6B — arccosfc ж)у = 0.
Particular solutions: y\ = x~ , yi = x .
55. x3yxxx + x2 arccosfc ж yxx + ж(агссоэ'г ж — l)y'x + (arccosfc ж — 3)y = 0.
Particular solutions: y\ = cos(lna;), y<i = sin(lna;).
© 1995 by CRC Press, Inc.

56. x3yxxx + x2(arccosk x + l)yxx + a;(arccosfc x — а — l)y'x — (arccosfc x — 2)y = 0.
Particular solutions: y\ = х~^а, yi = xv<x.
57. x3y'xxx+x2(arccosfc a;+o)y^,/a,+a;(oarccosfc x+b—a)y^,+b(arccosfc x — 2)y = 0.
Particular solutions: y\ = xni, yi = xn2, where n\ and n<i are the roots of the
quadratic equation n+(a — l)n + 6 = 0.
58. x3y'xxx + x2 arctanfc x yxx — 2xy'x + 2B — arctanfc x)y = 0.
Particular solutions: yx = x~ , У2 = x .
59. x3yx'xx + x2 arctanfc x yxx — 6xy'x + 6B - arctanfc x)y = 0.
Particular solutions: y^ = x~2, уъ = x3.
60. x3y'xxx + x2 arctan x y'xx + «(arctan x — l)y'x + (arctan x — 3)y = 0.
Particular solutions: уг = cos(lna;), y^ = sin(lna;).
61. x3y'xxx + x2(arctanfc x + l)yxx + a;(arctanfc x — a — l)y'x — (arctanfc x — 2)y = 0.
Particular solutions: y\ = x~^a, yi = x^a.
62. x3y'xxx + a;2(arctanfc x + a)y'^x + x(a arctanfc x + b — a)y'x
+ b(arctanfc x — 2)y = 0.
Particular solutions: y\ = xni, У2 = хП2, where n\ and П2 are the roots of the
quadratic equation n2 + (a — l)n + b = 0.
63. x3y'xxx + x2 arccotfc x yxx — 2xy'x + 2B — arccotfc x)y = 0.
Particular solutions: y\ = x~ , yi = x .
64. x3y'xxx + x2 arccotfc x yxx — 6xy'x + 6B - arccotfc x)y = 0.
Particular solutions: y\ = x~2, У2 = x3.
65. x3y'xxx + x2 arccotfc x yxx + a;(arccotfc x — l)y'x + (arccotfc x — 3)y = 0.
Particular solutions: y\ = cos(lna;), y<i = sin(lna;).
66. x3y'xxx + x2(avccotk x + l)yxx + a;(arccotfc x — а — l)y'x — (arccotfc x — 2)y = 0.
Particular solutions: y\ = х~^а, У2 = х^а.
67. a;3y^a,+a;2(arccotfc a;+o)y^,/a,+a;(oarccotfc x+b—a)y'x+b(avccotk x — 2)y = 0.
Particular solutions: y\ = xni, yi = xn2, where n\ and n<i are the roots of the
quadratic equation n+(a — l)n + 6 = 0.
© 1995 by CRC Press, Inc.

3.1.8. Equations Containing Combinations of Exponential, Logarithmic,
Trigonometric, and Other Functions
!• CL + aeXxv'L + BoeAa: tan ж + 3)y'x + [aeXxB tan2 x + 1) +2tan#]y = 0.
Particular solutions: y\ = cos ж, уг = ж cos ж.
2- tC + oe^y^ + C - 2oeAa; cot x)y'x + [aeXxB cot2 x + 1) - 2 cot ж]у = О.
Particular solutions: уг = sin ж, j/2 = ж sin ж.
3- У'1'хх + ° cosh" x y'lx + Ba cosh" x tan ж + 3)y^,
+ [o cosh" ж B tan2 ж + 1) + 2tanж]y = 0.
Particular solutions: y\ = cos ж, y<i = ж cos ж.
4- CL + ° cosh" x V'L + C - 2o cosh" ж cot x)y'x
+ [a cosh" ж B cot2 ж + 1) — 2 cot ж]у = О.
Particular solutions: y\ = sin ж, yi = ж sin ж.
5- У'1'хх + ° sinh" x V'L + Bo sinh" ж tan ж + 3)y'x
+ [оэтЬ"ж B tan2 ж + 1) + 2taпж]y = 0.
Particular solutions: y\ = cos ж, y<i = ж cos ж.
6- У'1'хх + ° sinh" ж yxx + C - 2o sinh" ж cot ж)у^,
+ [о sinh" ж B cot2 ж + 1) - 2 cot ж]у = 0.
Particular solutions: y^ = sin ж, y<i = ж sin ж.
7- tC + о tanh" ж y^ + Bo tanh" ж tan ж +
+ [o tanh" ж B tan2 ж + 1) + 2taпж]y = 0.
Particular solutions: y\ = cos ж, уг = ж cos ж.
8- CL + ° tanh" ж y'lx + C - 2o tanh" ж cot ж)у^
+ [о tanh" ж B cot2 ж + 1) - 2 cot ж]у = 0.
Particular solutions: y\ = sin ж, y<i = ж sin ж.
9- lC + ° coth" ж y'L + Bo coth" ж tan ж + 3)y'x
+ [a coth" ж B tan2 ж + 1) + 2taпж]y = 0.
Particular solutions: y\ = cos ж, уг = ж cos ж.
10- VZX + ° coth" x y'L + C - 2o coth" ж cot ж)у^
+ [о coth" ж B cot2 ж + 1) — 2 cot ж]у = 0.
Particular solutions: уг = sin ж, у^ = ж sin ж.
tC + a In" ж y^ - Ba In" ж tanh ж +
+ [o In" ж B tanh2 ж - 1) + 2tanl^]y = 0.
Particular solutions: y\ = cosh ж, y<i = ж cosh ж.
© 1995 by CRC Press, Inc.

12. tC + о In" x vlx - Bo In" ж coth ж + 3)y^
+ [o In" ж B coth2 ж - 1) + 2соШж]у = 0.
Particular solutions: y\ = sinha;, y<i = жвшЬж.
13. tC + о In" ж y^ + Bo In" ж tan ж + 3)t?
+ [o In" ж B tan2 ж+ 1) + 2tanж]y = 0.
Particular solutions: y\ = cos ж, уг = ж cos ж.
x yL+C~2a ln" ж cot x)y'x + [a In" ж Bcot2 ж+1)-2 cot x]y = 0.
Particular solutions: y\ = sin ж, y<i = ж sin ж.
- v'lL + a cos" x v'L - Bo cos" x tanh x
+ [a cos" ж B tanh2 ж - 1) + 2 tanh ж] у = 0.
Particular solutions: y\ = cosh ж, y<i = ж cosh ж.
I6- VZX + ° cos" x V'L ~ Ba cos" ж coth ж + 3)y^
+ [o cos" ж B coth2 ж - 1) + 2 coth ж] у = 0.
Particular solutions: y\ = вшпж, yi = жвшпж.
VZ* + a sin" x V'L - Bo sin" ж tanh ж + 3)y^
+ [a sin" ж B tanh2 ж — 1) + 2 tanh ж] у = 0.
Particular solutions: y\ = cosh ж, уг = ж cosh ж.
i8- v'iL + ° sin" x V'L ~ Bo sin" ж coth ж + 3)y^
+ [a sin" ж B coth2 ж — 1) + 2 coth ж] у = 0.
Particular solutions: y\ = втпж, y<i = жвшпж.
19- v'iL + a tan" x V'L ~ Bo tan" ж tanh ж + 3)y^
+ [o tan" ж B tanh2 ж - 1) + 2 tanh ж] у = 0.
Particular solutions: y\ = cosh ж, y<i = ж cosh ж.
20. y'iL + °tan" x V'L ~ Bo tan" ж coth ж + 3)y^
+ [o tan" ж B coth2 ж - 1) + 2 coth ж] у = 0.
Particular solutions: y\ = втпж, yi = жвшпж.
21- V'IL + ° cot" x V'L ~ Ba cot" ж tanh ж + 3)y^
+ [o cot" ж B tanh2 ж - 1) + 2 tanh ж] у = 0.
Particular solutions: y\ = cosh ж, уг = ж cosh ж.
22. y'lL + ° cot" x V'L ~ Bo cot" ж coth ж + 3)y^
+ [o cot" ж B coth2 ж - 1) + 2 coth ж] у = 0.
Particular solutions: y\ = втпж, y<i = жвшЬж.
© 1995 by CRC Press, Inc.

23- tC + (Ъеах + 2а) cosh" ж у'^х - а(Ьеах cosh" ж + о)»; - 2о3 cosh" ж у = 0.
Particular solutions: ух = еах, yi = е ах Л .
а
24- tC + (Ьеаа; + 2о) sinh" ж у^ - o(beaa; sinh" ж + а)|? - 2о3 sinh" ж у = 0.
Particular solutions: ух = еах, yi = е ах Л .
а
25- CL + (ЬепХ + 2о) tanh" ж у^, - o(beaa: tanh" ж + о)у^ - 2о3 tanh" ж у = 0.
Particular solutions: yx = еах, yi = е ах Л .
а
26- V'JL* + (ЪепХ + 2°) coth" ж у'^х - а(Ьеах coth" ж + а)у'х - 2о3 coth" ж у = 0.
Particular solutions: y1 = еах, У2 = е ах Н .
а
+ (beaa; + 2о) In" ж у^ - o(beaa; In" ж + о)у^ - 2о3 In" ж у = 0.
Particular solutions: y1 = еах, У2 = е~ах -\ .
а
28- VZX + (°1п" х ~ 2Ьех)Ух-х - ЬехBа In" ж - Ьех + 3)у'х
+ Ьех[а\ппх (Ьех - 1) + 2Ьех - 1]у = 0.
Particular solutions: y\ = ещ>{Ъех), у2 = хещ>(Ъех).
29- VZL + (° cos" х ~ 2beX)v'L - ЬехBа cos" ж - Ьех + 3)у'х
+ bex[acosnx(bex - 1) + 2Ъех - 1]у = 0.
Particular solutions: y\ = ещ>{Ъех), у2 = хещ>(Ъех).
30- У'1'хх + (ЬепХ + 2о) cos" х У'1Х - а(Ьеах cos" ж + а)у'х - 2о3 cos" ж у = 0.
Particular solutions: ух = еах, У2 = е~ах -\ .
а
31- VZX + (оsin" ж - 2bea:)yL - Ье^Сгоsin" х - Ьех + 3)у'х
+ Ьех[а sin" ж (be* - 1) + 2Ьех - 1]у = 0.
Particular solutions: yi = exp(bex), y2 = xexp(bex).
32- vZx + (beax + 2°)sin" x v'L - a(beax sin" x + a)v'x - 2°3 sin" x у = о-
Particular solutions: № = eoa:, уг = e~oa: -\ .
a
33- y4/L-[eAa:(tanж+o)+o]y4/a;+[(o2+l)eAa;+l]y;+o[eAa;(otanж-l)-l]y = 0.
Particular solutions: y\ = eax, У2 = cos ж.
34- VZX + itanx (axeXx + 1) + aeXx\y'lx - axeXxy'x + aeXxy = 0.
Particular solutions: y\ = x, J/2 = cos ж.
© 1995 by CRC Press, Inc.

35- tC + (о tan" x - 2bex)y'x'x - bexBa tan" x - bex + 3)y'x
+ bex[atannx(bex - 1) + 2bex - l]y = 0.
Particular solutions: уг = exp(bex), y^ = xexp(bex).
36- tC + (Ьеах + 2°)tan"хv'L - а(Ьеахtan"х + а)у'х - 2°3tan"xy = o.
Particular solutions: yx = еах, уг = е~ах -\ .
Particular solutions: y\ = е~ах, yi = sin ж.
38- VZX + iaeXx ~ cot х (ахеХх + 1)]у'^х - ахеХху'х + аеХху = 0.
Particular solutions: уг = х, г/2 = sin ж.
39- VvL + (° cot" х ~ 2beX)v'L - ЬехBа cot" х - Ьех + 3)у'х
+ Ьех[а cot" х (Ьех - 1) + 2Ъех - 1}у = 0.
Particular solutions: y\ = ещ>{Ъех), уг = хещ>(Ъех).
40. у'1'хх + (Ьеах + 2а) cot" x y'Jx - а(Ьеах cot" х + а)у'х - 2о3 cot" x у = 0.
Particular solutions: yx = еах, уъ = е~ах -\ .
(X
X (tan X + O) + °]у! + K°2 + !) COSh" Ж + Wm
-\- o[cosh" x (otana; — 1) — l]y = 0.
Particular solutions: гд = еах, уг = cos ж.
42- <L + [cosh™ x (cot ж + o) + a\y'lx + [(a2 + 1) cosh" x + l]y'x
-\- o[cosh" x A — a cot ж) + l]y = 0.
Particular solutions: гд = e~ax, уг = sin ж.
43- tC - Isinh" a; (tan a; + o) + o]»^ + [(o2 + 1) sinh" x + l]y'x
+ o[sinh" x (otana; — 1) — l]y = 0.
Particular solutions: y\ = eax, уг = cos ж.
44- VZL + Isinh" x (cot ж + o) + o]»^,, + [(o2 + 1) sinh" x + l]y'x
+ o[sinh" x A — a cot x) + l]y = 0.
Particular solutions: уг = е~ах, уг = sin ж.
45- VZL ~ [tanh™ x (tanж + o) + a\y'lx + [(a2 + 1) tanh" x + l]y'x
+ о [tanh" x (otana; — 1) — l]y = 0.
Particular solutions: y\ = eax, yi = cos ж.
© 1995 by CRC Press, Inc.

46- tC + [tanh™ x (cot x + a) + a]y'^x + [(a2 + 1) tanh" ж + l]y'x
+ a[tanh" x A - a cot ж) + l]y = 0.
Particular solutions: y\ = e~ax, 2/2 = sin ж.
47- tC ~ [coth™ ж (tan x + a)+ a\y'lx + [(a2 + 1) coth" x + l]y'x
+ o[coth"a; (tana; - 1) - l]y = 0.
Particular solutions: y\ = eax, У2 = cos ж.
48- vZx + [coth™ x (cot x + °) + °\v'L + K°2 + !)coth" ж + !]у4
+ o[coth" ж A - о cot ж) + l]y = 0.
Particular solutions: уг = e~ax, yi = sin ж.
« (tanhж - b) - %^ + [o(b2 - 1) tan" x-l]y'x
+ b[atan"^(l - b tanh ж) + l]y = 0.
Particular solutions: y\ = ebx, yi = cosh ж.
50. y'l'^ + (a tan" x + b tanh x)y'x[x + cy'x + c(a tan" x + b tanh x)y = 0.
1°. Partical solutions with с > 0: y\ = сов(жу/с), 2/2 = вш(жу/с)-
2°. Partical solutions with с < 0: г/i = exp(—ж^/—с), 2/2 = ехр(жу/—с).
51- <L + [оtan" ж (coth ж — Ь) — b]y^ + [o(b2 - 1) tan" ж - l]y^
+ b[otan"ж(l - b coth ж) + l]y = 0.
Particular solutions: y\ = ebx, 2/2 = sinhi.
52. y^ + (o tan" x + b coth ж)у4'а, + cy'x + c(a tan" ж + b coth ж)у = 0.
1°. Partical solutions with с > 0: 2/i = сов(жу/с), 2/2 = вш(жу/с)-
2°. Partical solutions with с < 0: 2/i = exp(—ж^/—с), 2/2 = ехр(жу/—с).
53- <L + [о cot" ж (tanh яв — Ь) — Ь]у^ + [о(Ь2 - 1) cot" ж - 1]|?
+ Ь[осо^жA - Ь tanh ж) + 1]у = 0.
Particular solutions: 2/1 = еЬж, 2/2 = cosh ж.
54- CL + ° cot" ж tanhr" ж У^ - Ьу'х - ab cot" ж tanh ж у = 0.
1°. Particular solutions with b > 0: 2/1 = exp(—жл/&), 2/2 = ехр(жл/&).
2°. Particular solutions with 6 < 0: 2/1 = сов(ж-\/—&), 2/2 = sin(x\/—b).
55- tC + [o cot" « (cothж - b) - b]y^ + [a(b2 - 1) cot" x-l]y'x
+ Ь[асо^жA - b coth ж) + l]y = 0.
Particular solutions: 2/1 = еЬж, 2/2 = sinha;.
© 1995 by CRC Press, Inc.

56. y?L + (о cot" x + b coth x)y'x[x + cy'x + c(a cot" x + b coth x)y = 0.
1°. Partical solutions with с > 0: y\ = cos(x^/c), Ц2 = sm(x^/c).
2°. Partical solutions with с < 0: y\ = exp(—ж^/—с), уг = ехр(жу/—с).
ж (tanh * - Ь) - %1 + [°(ь2 - 1Iп" х
+ b[a In" ж A - Ъ tanh ж) + 1]у = 0.
Particular solutions: у\ = еЬх, yi = cosh ж.
58- VvL + о In" ж tanh™ ж у'1х - Ьу'х - ab In" ж tanh ж у = 0.
1°. Particular solutions with b > 0: yi = exp(—xVb), y2 = exp(xVb).
2°. Particular solutions with b < 0: yi = cos(a;-\/~b)) 2/2 = sm
« (coth ж - b) - b]y^ + [°(b2 - 1)ln" ж - l]t?
+ Ь[о1п"жA - b coth ж) + l]y = 0.
Particular solutions: y\ = e , j/2 = sinha;.
60- VvL + (o In" ж + b coth x)y'^x + cy'x + c(a In" ж + b coth ж)у = 0.
1°. Partical solutions with с > 0: y\ = cos(xyfc), yi = ш\{х\[с).
2°. Partical solutions with с < 0: y\ = exp(—Жу/—с), г/2 = ехр(жу/—с).
61- tC - Iln" ж (tanх + о) + °]у! + [(°2 +1Iп" ж + i]i/i
+ о[1п"ж^апж- 1) - 1]у = 0.
Particular solutions: yi = еах, уг = cos ж.
62. tC + [In" ж (cot ж + а) + a\y'ix + [(а2 + 1) In" ж + l]t?
+ o[ln" ж A - о cot ж) + 1]у = 0.
Particular solutions: y\ = е~ах, уг = sin ж.
63. у^ + [о cos" ж (tanh ж - Ъ) - Ъ]у'^х + [а(Ь2 - 1) cos" ж - 1]у'х
+ Ь[осоэ"жA - Ь tanh ж) + 1]у = 0.
Particular solutions: y\ = еЬх, уг = cosh ж.
64. у^а, + о cos" ж tanh ж у^ + Ьу'х + ab cos" ж tanh ж у = 0.
1°. Partical solutions with b > 0: y\ = сов(жл/&), Уг = sin(a;-\/b).
2°. Partical solutions with 6 < 0: yx = exp(—жу7—b), y2 = ехр(жл/—&)•
65. y^ + [ocos" ж (coth ж - b) - bjy^ + [o(b2 - 1) cos" ж - l]y^,
+ Ь[осо5"жA - b coth ж) + 1]у = 0.
Particular solutions: y\ = еЬх, уг = sinhi.
© 1995 by CRC Press, Inc.

66. y'l'xx + (о cos" x + b coth x)y'x[x + cy'x + c(a cos" x + b coth x)y = 0.
1°. Partical solutions with с > 0: y\ = cas(xyfc), г/2 = sm{x\fc).
2°. Partical solutions with с < 0: уг = ехр(—ж^/—с), г/2 = ехр(жу/—с).
67- tC + [°sin" ж (tanhж - ь) - b\v'L + [°(ь2 -1)sin" x - i]t?
+ Ь[аэт"жA - b tanh ж) + 1]у = 0.
Particular solutions: y\ = е ж, 2/2 = cosh ж.
68. у"' + о sin" ж tanh ж у" + byi + ab sin" ж tanh ж у = 0.
1°. Partical solutions with Ь > 0: y\ = cos(xVb), y<i = sm(xVb).
2°. Partical solutions with b < 0: y\ = exp(—xy/—6), j/2 = ехр(ж\/—&)•
69- <L + [o sin" « (coth ж - b) - b]y^ + [o(b2 - 1) sin" ж -
+ Ь[оэт"ж A — b coth ж) + 1]у = 0.
Particular solutions: y\ = еЬх, У2 = sinha;.
70- V'JL* + (° sin" x + b coth x)y'lx + cy'x + c(a sin" x + b coth ж)у = 0.
1°. Partical solutions with с > 0: у\ = cos{x^/c^, y^ = sin^^/c)-
2°. Partical solutions with с < 0: yi = exp(—Жу/—с), 2/2 = exp(a?v/—с).
71- sctC, + [ож2еАа;(Ь - In ж) + 2]у^ + ахеХху'х - аеХху = 0.
Particular solutions: y\ = х, г/2 = In ж — Ъ + 1.
72. (еАа; - 1)у^ - (оеАа; + tan х)у'х'х + (еХх + а2)у'х + а(а tan ж - еХх)у = 0.
Particular solutions: гд = еоа:, г/2 = cos ж.
73. о cosh" ж у^ + [tan ж (о cosh" ж + ж) + \\у'1х - ху'х + у = 0.
Particular solutions: г/i = ж, г/2 = cos ж.
74. о cosh" ж у^а, + [1 — cot ж (о cosh" ж + х)]ухх — ху'х + у = 0.
Particular solutions: г/i = ж, г/2 = sin ж.
75. о sinh" ж у^ + [tan ж (о sinh" ж + ж) + 1]у'^х - ху'х + у = 0.
Particular solutions: г/i = ж, г/2 = cos ж.
76. о sinh" ж у^ + [1 - cot ж (о sinh" ж + ж)^ - жу^ + у = 0.
Particular solutions: г/i = ж, г/2 = sin ж.
77. о tanh" ж у^'хх + [tan ж (о tanh" ж + ж) + 1]у'^х - ху'х + у = 0.
Particular solutions: г/i = ж, г/2 = cos ж.
© 1995 by CRC Press, Inc.

78. о tanh" ж yZx + [1 - cot ж (a tanh" x + x)]y'^x - xy'x + у = 0.
Particular solutions: уг = ж, г/2 = sin ж.
79. a coth" x yZx + [tan x (a coth" x + x) + l]y'Jx - xy'x + у = 0.
Particular solutions: y\ = ж, г/2 = cos ж.
80. a coth" x yZx + [1 - cot ж (о coth" ж + x)]y'^x -xy'x+y = 0.
Particular solutions: y\ = x, J/2 = sin ж.
81. a In" ж y^ + [tanh ж (ж - о In" ж) - l]y'^x - xy'x + у = 0.
Particular solutions: y\ = x, J/2 = cosh ж.
82. a In" ж yZx + [coth ж (ж - о In" ж) - \\y'lx - xy'x + у = 0.
Particular solutions: yi = ж, г/2 = sinhi.
83. a In" ж y^ + [tan ж (о In" ж + ж) + 1]у?я - ху'х + у = 0.
Particular solutions: y\ = ж, г/2 = cos ж.
84. о In" ж yZx + [1 - cot ж (о In" ж + ж)]у^ - жу^ + у = 0.
Particular solutions: г/i = ж, г/2 = sin ж.
85. о cos" ж у?4з, + [tanh ж (ж - о cos" ж) - 1]ухх — ху'х + у = 0.
Particular solutions: г/i = ж, г/2 = cosh ж.
86. о cos" ж у^ + [coth ж (ж - о cos" ж) - \\у'1х - ху'х + у = 0.
Particular solutions: г/i = ж, г/2 = вшЬж.
87. ож cos" ж з/^а, + Bо cos" ж - ж2 In ж + Ьх^у'^ + ху'х - у = 0.
Particular solutions: г/i = ж, г/2 = In ж — 6 + 1.
88. о sin" ж у^ + [tanh ж (ж - о sin" ж) - 1]у'^х - ху'х + у = 0.
Particular solutions: г/i = ж, г/2 = cosh ж.
89. о sin" ж у^ + [coth ж (ж - о sin" ж) - 1]у^ - жу^ + у = 0.
Particular solutions: уг = ж, г/2 = втЬж.
90. ож sin" ж у^з, + Bо sin" ж — ж2 In ж + Ьх2)ухх + ху'х — у = 0.
Particular solutions: г/i = ж, г/2 = In ж — Ъ + 1.
91. о tan" ж yZx + [tanh ж (ж - a tan" ж) - 1]у^ - жу^ + у = 0.
Particular solutions: г/i = ж, г/2 = cosh ж.
© 1995 by CRC Press, Inc.

92. о tan" ж yZx + [coth ж (ж - a tan" x) - l]y'Jx - xy'x + у = 0.
Particular solutions: г/i = x, г/2 = sinha:.
93. ax tan" x y'^xx + Ba tan" x - x2 In ж + Ьж2^ + жу^ - у = 0.
Particular solutions: уг = x, г/2 = In ж — b + 1.
94. о cot" ж y^ + [tanh ж (ж - о cot" ж) - l]y'^x -ху'х+у = 0.
Particular solutions: y\ = x, J/2 = cosh ж.
95. a cot" ж y^ + [coth ж (ж - о cot" ж) - l]y^ - жу^ + у = 0.
Particular solutions: y\ = x, г/2 = sinha;.
96. ож cot" ж у^ + Bo cot" ж - ж2 In ж + Ьж2)у^ + жу^ - у = 0.
Particular solutions: y\ = х, г/2 = In х — Ь + 1.
3.1.9. Equations Containing Arbitrary Functions
Notation: f = f(x), g = g(x), and h = h(x) are arbitrary function of argument x;
a, b, c, n, and A are parameters.
i- tC + fvi - Ы + o3)y = o.
Particular solution: г/о = с°ж-
The substitution w = y'x — ay leads to a second order linear equation: wxx + aw'x +
2
2- v'JL* + fv'x + ax(f + °2a;2 - 3a)v = °-
т, / ax2 \
Particular solution: г/о = exp I — 1.
2 /•
/ ax \ I
The substitution у = expf — 1 / z(x) dx leads to a second order linear equa-
\ A / J
tion: zxx - 3axz'x + (/ + 3a2x2 -3a)z = 0.
3- y'vL + (/ - °2)у4 + °/у = °-
Particular solution: г/о = е~ах¦
The substitution w = y'x + ay leads to a second order linear equation: wxx — aw'x +
fw = 0.
4. Ух'хх "b xfy'x — 2/y ^ 0.
Particular solution: yo = x .
The substitution ад = xy'x — 2г/ leads to a second order linear equation: wxx +
xfw = 0.
5- tC + (ож + b)/y4 - о/у = 0.
Particular solution: yo = ax + b.
© 1995 by CRC Press, Inc.

6- tC + (/ - a2x2)y'x + ax(f - 3a)y = 0.
О
/ (xx
Particular solution: у$ = expl
\
О
(xx
о /.
/ (XX \ I
The substitution у = expf —-— J / z(x) dx leads to a second order linear equa-
tion: zxx - 3axz'x + Ba2x2 - 3a + f)z = 0.
7- tC + (/ - a2x2n)y'x - a[xnf + Запх2^1 + n(n - l)xn-2]y = 0.
Particular solution: yo = expf — xn+1).
Vn+l /
The substitution у = exp( — xn+1 ) / z(x) dx leads to a second order linear
Vn + l J J
equation: z'^x + 3axnz'x + Ba2x2n + Запж™ + f)z = 0.
8- VZL + ay'L + bvL + cy = f(x).
This is a special case of equation 5.1.5.9.
9- tC + av'L + fv'x + afV = °-
The substitution w = y'x+ ay leads to a second order linear equation: wxx + fw = 0.
Particular solution: yo = eax ¦
The substitution w = y'x — ay leads to a second order linear equation: wxx +
(/ + a)w'x + a(f + a)w = 0.
11- У'ХХХ + fv'L + аУх + °/У = °-
1°. Particular solutions with a > 0: yi = cos^v^), 2/2 = sin^v^)-
2°. Particular solutions with a < 0: yi = exp(—a?v/—«), 2/2 = exp(a?v/—«)•
The substitution ад = yxx + ay leads to a first order linear equation: w'x + fw = 0.
12- VZX + fy'L + axny'x + axn~1(xf + n)y = 0.
The substitution ад = yxx + axny leads to a first order linear equation: w'x + fw = 0.
13- y'i'xx + fy'L + afy'x + °3y = °-
The substitution w = y'x + ay leads to a second order linear equation: wxx +
(/ — a)qw'x + a2w = 0.
14- v'x"xx +fy'L + afy'x + a2(f - °)y = °-
Particular solutions: y\ = e~ax'2 cos I —-—x I, У2 = e~ax'2 sin I —-—x I.
V 2 / \ 2 /
15. y'l'xx + fy'L + 9У'Х + h = 0.
The substitution w = y'x leads to a second order linear equation: wxx+fw'x+gw+h = 0.
© 1995 by CRC Press, Inc.

16. tC + fy'L - oB/ + 3a)y'x + o2(/ + 2o)y = 0.
Particular solutions: y1 = eax, У2 = xeax.
17. vZL + fv'L + xgy'x -gy = o.
The substitution w = xy'x — у leads to a second order linear equation: xwxx +
(xf - l)w'x + x2gw = 0.
18- У'ххх + fy'L + (g~ a2)y'x - a(af + g)y = 0.
Particular solution: yo = eax.
The substitution w = y'x — ay leads to a second order linear equation: wxx +
(/ + a)w'x + (af + g)w = 0.
19- У'ххх + fv'L + (af + b- a2)y'x + b(f - a)y = 0.
Particular solutions: y\ = eXlX, уг = eX2X, where Ai and A2 are the roots of the
quadratic equation A2 + aX + b = 0.
20. y'Lx + (f - a)y'L - a2fy = 0.
Particular solution: yo = eax.
The substitution w = y'x—ay leads to a second order equation: wxx+fwx+afw = 0.
21- Уххх = (f ~ a)y'L + (af ~ b)y'x + bfy.
Particular solutions: y\ = exp(Aia;), уг = ехр(А2ж), where Ai and A2 are roots of
the quadratic equation A2 + aX + 6 = 0.
The substitution w = yxx + ay'x + by leads to a first order linear equation: w'x = fw.
22. y'xxx + (f ~ a)yxx + gy'x - 0@/ + g)y = 0.
Particular solution: yo = eax.
23. y'xxx + (f + a)y'L + (af + g)y'x + аду = 0.
Particular solution: yo = e~ax.
24. y'Lx + xfy'L + (ax2 - f)y'x + ax(x2f + 3)y = 0.
fx2Ja\ ,
Particular solutions: y\ = cos I —-— , У2 = sin
25. yZx + (ax + b)fy'lx + xfy'x - If у = 0.
Particular solution: yo = x2 + 2ax + b.
26. y'i'xx + (f + ax)y'lx + a(xf + 2)y'x + afy = 0.
т, / ax2 \ ( ax2 \ f (ax2 \ ,
Particular solutions: yi=expl ^~)> 2/2 = expl — 1 / expl ) ax
\ Zi J \ Zi J j \ Zi J
© 1995 by CRC Press, Inc.

27. Уххх + x2fyxx — 2xfyx -\- 2/y = 0.
Particular solutions: y\ = x, y2 = x .
Solution:
у = dx + C2x2 + C3(x2 / ж^dx- x х~2фdx),
J J
where ip = exp(— Jx2f dx).
28- tC + (/ + ax)y'x1x + (g + 2a)y'x + a[xg + A - ож2)/]у = О.
/ ax2 \
Particular solution: yo = exp I J.
The substitution w = y'x + axy leads to a second order linear equation: wxx +
fw'x + (g — axf)w = 0.
29. y'xxx + (axf + f + a)y'x'x + a2xfy'x - a2fy = 0.
Particular solutions: y\ = x, уг = e~ax.
30- <L + (°ж2 + Ьж + c)/yL - 2o/y = 0.
Particular solution: yo = яж + Ьх + с.
31- <L + *(*/ + fl)l/™ - 9У'Х - 2/y = 0.
Particular solution: yo = x . The substitution w = жу^, — 2у leads to a second order
linear equation: wxx + x(xf + g)w'x + xfw = 0.
32. tC - ж(ож + b)/yl + (Ь - о2)/У; + 2о/у = 0.
Particular solution: yo = х2 + ах + — (а2 — Ь).
Z
33. tC - [Bж + о)/ + (ж2 + ож + b)g]y^ + 2fy'x + 2gy = 0.
Particular solution: yo = х2 + ах + Ь.
34. ху'^хх + 3Ух-х + х(ах2 + l)fy'x - (ах2 - 1)/у = 0.
Particular solution: yn = ах -\ .
х
35. xyZx + (ах2 + Ь)у^ + 4аху'х + 2оу = /.
Integrating the equation twice, we obtain a first order linear equation:
f ( f \
xy'x + (ax2 + b — 2)y = C\ + C2x + /I I f dxj dx.
j \j /
36. sctC, + xfy'x - [(ax + 1)/ + a3x + 3o2]y = 0.
Particular solution: yo = xeax.
© 1995 by CRC Press, Inc.

37. xyZx + x(f - 2a)yZ + x(g + a2)y'x - [a(ax + 2)/ + (ax + l)g]y = 0.
Particular solution: г/о = xeax.
38- *y*L + (xf + 3)v'L + B/ + ax)v'x + a(xf + !)y = o-
Particular solutions: Vi = —cos(x\/a), y2 = —sin(x\/a).
X X
39. xyZx + (xf + 3)y'Z + (ax + 2)fy'x + a(axf + f- a2x)y = 0.
Particular solutions: Vi = —e ax' cos I —-—x ), yi = —
x \ 2 ) x
40- *vZm + (xf + 3)V'L + (axf + 2/ - a2x)y'x + a(f - a)y = 0.
Particular solutions: y\ = —, 2/2 = —e~ax.
xx
41- xy*L + (xf + 3)V'L + B/ + axn^)y'x + axn(xf + n + l)y = 0.
The substitution w = xy leads to an equation of the form 3.1.9.12: w'^'xx + fwxx +
axnw'x + axn~1(xf + n)w = 0.
42. xyZx + (x2f + a + 2)yZ - a(a + l)fy = 0.
Particular solution: yo = x~a.
The substitution w = xy'x + ay leads to a second order linear equation: wxx +
xfw'x - (a + l)fw = 0.
43. xyZx + [x2(ax2 + 1)/ + 3}у'^ - 2fy = 0.
Particular solution: yo = ax -\ .
x
44. xyZx + [x(ax2 - 1)/ + x2(ax2 + l)g + 3]y'Z - 2fy'x - 2gy = 0.
Particular solution: w0 = ax -\ .
x
45. (ax)yZx+[()f]yxx
Particular solutions: y\ = x , г/2 = eax.
46- x2y'l'xx + (xf -a2- a)y'x + (a - l)fy = 0.
Particular solution: г/о = x ~a.
The substitution г<; = xy'x + (a — 1)г/ leads to a second order linear equation:
ж«&. - (a + l)w'x + fw = 0.
47. x2yZx + [x(ax + 1)/ - 6}y'x + fy = 0.
Particular solution: г/о = a -\ .
x
© 1995 by CRC Press, Inc.

48. x2y'Zx + xfyxx -\- [ж(ож + 1)<7 + 2/ — 6]y'x + 9У = 0.
Particular solution: yo = a -\ .
x
ztQ T24i'" 4- тГт^/тг 4-1^^4- 4l»i" 2 f 11 — fl
Particular solution: yg = a H .
ж
50. ж2у^ + x(xf + a)yZ + [(a - 2)xf + b]y'x + (b - a + 2)fy = 0.
By integrating, we obtain the nonhomogeneous Euler equation 2.1.8.15:
х<2Ухх + (a- 2)xy'x + (b - a + 2)y = Cexpf- / /dx\.
51. {ax + b)xyZx + (аж + /3)у'^ + xy'x + y = f.
By integrating, we obtain a second order linear equation:
r
(ax + b)xylx + [(a - 2d)x + C - b]y'x + (x + 2a-a)y= fdx + C.
J
52. x(x + l)yZx + x(f-x- 3)y'x -(x + l)fy = 0.
Particular solution: yo = xex.
53. x3yZx + xfy'x + (o - 1) (/ + o2 + a)y = 0.
Particular solution: yo = x1~a.
The substitution w = xy'x + (a — l)y leads to a second order linear equation:
x2wxx - (a + l)xw'x + (f + a2 + a)w = 0.
54. x3yZx + ax2yZ + bxy'x + cy = f(x).
The nonhomogeneous Euler equation.
The substitution t = In |ж| leads to an equation of the form 3.1.9.8.
y'Ht + (a - S)y? + (b-a + 2)y't + cy = /(±e*).
55. x3yZx + (a + 2)x2yZ + xfy'x + afy = 0.
Particular solution: yo = x~a.
56. x3y'Zx + [(a + 6)x2 + b]yZ + 2Bo + 3)xy'x + lay = f(x).
Integrating the equation twice, we obtain a first order linear equation:
x3y'x + (ax2 + b)y = C1 + C2x+ f( f f dxj dx.
57. x3y'Zx + x2^2^1 - 3a)yZ + 2o(o + l)Bo + l)y = f(x).
Integrating the equation twice, we obtain a first order linear equation:
x~lay'x + (ax-2a-" +b)y = d + C2x+ / / fx~za-6dx dx.
© 1995 by CRC Press, Inc.

58. x3yZx + *?fy'L ~ 2xy'x + 2B - f)y = 0.
Particular solutions: y\ = ж, yi = x .
59. x3y'Zx + x2fy'L ~ 6xy'x + 6B - f)y = 0.
Particular solutions: y\ = x~2, У2 = x3.
60. x3yZx + x2fy'Z + x(f - l)y'x + (/ - 3)y = 0.
Particular solutions: уг = cos(lna;), y<± = sin(lna;).
Z + x(f-a- l)y'x - a(f - 2)y = 0.
Particular solutions: y\ = x~^a, У2 = ж •
62. x3y'Zx + x2(f + a)y'Z + x(af + b - a)y'x + b(f - 2)y = 0.
Particular solutions: уг = xni, y<i = хП2, where n\ and n<i are the roots of the
quadratic equation n2 + (a — l)n + b = 0.
63. x3y'Zx + x2(f + a)y'lx + x[g + (o - l)f]y'x + (o - 2)gy = 0.
Particular solution: yo = x2~a. The substitution w = xy'x + (a — 2)y leads to a second
order linear equation: x2wxx + xfw'x + gw = 0.
64. x3y'Zx + x2(f + 2ax)yZ + xBaxf + a2x2 + b)y'x + (a2x2f + bf- 2b)y = 0.
Particular solutions: yx = e~axxni, yi = e~axxn2, where n\ and n<i are the roots
of the quadratic equation n2 — n + b = 0.
65. x6y'Zx + x2fy'x + (a3 + af- 2xf)y = 0.
Particular solution: yo = x2ea'x ¦
66. tC + aeXxyZ ~ 3X2y'x + 2X3y = f(x).
Integrating the equation twice, we obtain a first order linear equation:
e~Xxy'x + (a + 2\e~Xx)y = C1 + C2x+ f(f fe~Xx dx\ dx.
2e2Xx)y' - aeXx(f + 3a\eXx + X2)y =
67. tC + (/ - a2e2Xx)y'x - aeXx(f + 3a\eXx + X2)y = 0.
Particular solution: yo = expl — e x 1.
V Л /
Л
The substitution у = expf —eXx J / z(x) dx leads to a second order linear equa-
equation: z'?x + 3aeXxz'x + (/ + 2a2e2Xx + 3a\eXx)z = 0.
68. tC + [A + 6e~)/ - a2]y'x + afy = 0.
Particular solution: yo = e~ax + b.
© 1995 by CRC Press, Inc.

69. yZx + (f + a)y'ix + [af + A + Ьеах)д}у'х + аду = О.
Particular solution: yo = e~ax + b.
70. tC + (be™ + 2o)/y^ - a(be~/ + o)y4 - 2a3 f у = 0.
Particular solutions: yx = eax, y2 = e~ax -\ .
a
Уххх + (/ - 2oeAa;)y;/a; - oeAa;B/ - oeAa; )^
+ aeXx[(aeXx - A)/ + 2aXeXx - A2]y = 0.
Particular solutions: y± = exp ( — eXx ), yi = x exp ( — еЛа:
V A / \ A
72- tC + (/ - aeXx)y'lx + (g - 2a\eXx)y'x - aeXx[(aeXx + \)f + g + A2]y = 0.
Particular solution: yo = exP( ~TeXx) ¦
The substitution у = exp( — eXx) / z(x) dx leads to a second order equation:
zlx + (/ + 2аеЛа;L + BаеЛа;/ + д + a2e2Xx + a\eXx)z = 0.
73-
Particular solutions: y\ = ecx, y2 = e~ax -\ .
с
74. eXxyZx + BXeXx + Ce»x + 7)^ + (X2eXx + 2f3fie^)yx + f3fi2e^y = f(x).
Integrating the equation twice, we obtain a first order linear equation:
eXxy'x + (Ce»x +1)y = C1+C2x+ f(ff dx] dx.
75- Уххх + fy'L + 9У'Х ~ A[A/ + tanh(Aa;)(g + A2)]y = 0.
Particular solution: yo = cosh(Aa;).
The substitution у = cosh(Arr) f z(x) dx leads to a second order equation:
z'x'x + [/ + 3Atanh(Arr)]4 + [g + ЗА2 + 2A/tanh(Arr)]z = 0.
76. y"' + fy" — A[2/ tanh(Aa;) + ;
+ A2{[2 tanh2(Aa;) - 1]/ + 2Atanh(Aa;)}y = 0.
Particular solutions: y\ = cosh(Arr), 2/2 =
77. tC + fy'L - A[2/ cothCAx) + 3A]y4
+ A2{[2coth2(Aa;) - 1]/ + 2Acoth(Aa;)}y = 0.
Particular solutions: y\ = sinh(Aa;), y<i = a;sinh(Aa;).
78- yZx+[()f}
Particular solutions: y\ = еах, y<i = cosh ж.
© 1995 by CRC Press, Inc.

79- yZx + [()f]
Particular solutions: y\ = eax, y<i = sinha;.
80. yZx + [Atanh(Xx)(xf - 1) - f]y'Z - X2xfy'x + A2/y = 0.
Particular solutions: y\ = x, yi = cosh(Aa;).
- tC + [Acoth(Xx)(xf - 1) - /]yl - А2ж/у^ + A2/y = 0.
Particular solutions: yi = x, y<i = sinh(Aa;).
82. xyZx + [x2(a - Inx)f + 2]yZ + xfy'x - fy = 0.
Particular solutions: уг = x, y<± = In ж — a + 1.
83. yZx + fVx + tan x(f - l)y = 0.
Particular solution: yo = cos ж.
The substitution у = cos x f z(x) dx leads to a second order linear equation: z'J.x —
3tanxz'x + (f -3)z = 0.
84. y'Zx + fVx + cot x(l - f)y = 0.
Particular solution: yo = sin ж.
85. y'Zx + fv'L + 9VX + A[A/ + tan(AaO(<7 - A2)]y = 0.
Particular solution: yo = сов(Аж).
The substitution у = сов(Аж) J z(x) dx leads to a second order linear equation:
zxx + [/ - 3Atan(Aa;)]4 + [g- ЗА2 - 2A/tan(Aж)]z = 0.
86. y^ 22
Particular solutions: y\ = сов(Аж), уг = жсов(Аж).
87. yZ 2
Particular solutions: y\ = вт(Аж), уг = жвт(Аж).
88. y'i'xx ~ [(о +tanx)f + °\УХХ + [(°2 + 1)/ + l]t? + °[(°tanx ~ 1)/ - l]l/ = 0-
Particular solutions: y\ = еах, у2 = cos ж.
89. y'l'xx + Kcot x + a)f + a]yZa + [(о2 + 1)/ + 1]у'х + о[A - о cot ж)/ + 1]у = 0.
Particular solutions: уг = е~ах, у^ = sin ж.
90. yZx + [/ + Xtan(^)(xf + l)]yZ ~ >?*fy'x + А2/У = 0.
Particular solutions: y\ = ж, Уч= сов(Аж).
91. VZX + [/ - Xcot(Xx)(xf + l)]yZ ~ X2xfy'x + A2/y = 0.
Particular solutions: y\ = ж, Уч= вт(Аж).
© 1995 by CRC Press, Inc.

92. asin(Xx)yxxx + Ьухх + ЗоЛ2 sin(\x)y'x + 2оЛ3 соэ(Аж)у = /(ж).
Integrating the equation twice, we obtain a first order linear equation:
rr)y^, + [b - 2a\cos(Xx)]y = Сг + C2x + /I fdxjdx.
J \J J
93. sin(Xx)yfx1xx + [a+ BЛ + 1) сс^Аж)^ - (А2 + 2Л) sin(\x)y'x
— А2 соэ(Аж)у = /(ж).
Integrating the equation twice, we obtain a first order linear equation:
f ( f \
sin(Xx)y'x + [a + cos(Xx)}y = d + C2x + П fdxj
J \J /
dx.
94. (/ - l)yZx -Ы + Х Ып(Хх)]Ух-х + (A2/ + a2)y'x + aX[a Ып(Хх) - A/]y = 0.
Particular solutions: уг = eax, У2 = cos(Arr).
Integration yeilds a second order linear equation: yxx + fy = f g dx + C.
96. tC + 2fy'x + fxy = 0.
Solution: у = C\w\ + C2W1W2 + C3W2,, where W\ and W2 are linearly-independent
solutions of the second order linear equation 2wxx + fw = 0.
97. yZx + fLvL + /B/i - Р)У = 0.
Integration yeilds a second order linear equation: yxx + fy'x + f2y = Cexp(J / dx).
98. tC + (a - \)py'x - [fx'x - Ba + l)ffx + af*]y = 0.
Integration yeilds a second order linear equation:
Уxx + fy'x + (af2 ~ f'x)y = Cexp( / / dx).
99. yZx + (/ - a2)v'x + (fx ~ аЛУ = 0-
The substitution w = yxx + ay'x + fy leads to a first order linear equation: w'x —
aw = 0.
loo- yxxx + fy'L + 9УХ + (fg + g'x)y = o.
Integration yeilds a second order linear equation: yxx + gy = Cexp(— J f dx).
101. yZx + 3fyxx + (fL + 2/2 + 29)y'x + B/9 + g'x)y = 0.
Solution:
у = C\W2 + C2W1W2 + C3W2,
where w\ and W2 is the fundamental set of solutions of the second order linear equation
w'xx + fw'x + -2-9W = 0.
© 1995 by CRC Press, Inc.

102. y'Zx + (/ + 9)V'L + (fL + f9 + h)y'x + (h'x + gh)y = 0.
Integration yeilds a second order equation: yxx + fy'x + hy = Cexp(— Jgdx).
y'Zx + (/ + 9)V'L + (?g'x + fg + h)y'x + (g'^ + fg'x + gh)y = 0.
The substitution w = y'x + gy leads to a second order equation: wxx + fw'x + hw = 0.
Ю4. /tc - r:xxV = o.
Particular solution: yo = /•
The substitution у = f J z dx leads to a second order equation: fzxx + 3fxz'x +
3f'Jxz = 0.
J УXXX ' v XXX ** *3 '
Integration yeilds a second order equation: fyxx — fxy'x + fxxy = f gdx + C.
Ю6. tC = f(x)y.
The transformation x = i, у = wt~^ leads to an equation of the similar form:
wttt — ~l J \ — lw-
3.2. Equations of the Form y?x = Ax*yP\y'xf'
3.2.1. Preliminary Comments. Classification Table
The value of the insignificant parameter A is in many cases definined in the form of a
function of two (one) auxiliary coefficients a and b:
A = ip(a, b) A)
and the corresponding solutions are represented in the parametric form
x = f1(T,C1,C2,C3,a), у = /2(т,С1,С2,С3,Ь), B)
where т is a parameter, C\, C2, and C3 are arbitrary constants, /1 and /2 are some functions.
Having fixed the auxiliary coefficient sign a > 0 (or b > 0), the coefficient b should be
expressed in terms of both A and a with the help of
b = ф{А, а).
Substituting this formula into B), we obtain a solution of the equation under consideration
(where the concrete numerical value of the coefficient a may be chosen arbitrarily). The
case a < 0 (or b < 0), which may lead to the branch of the solution or to a different domain
of determining the variables x and у in B), should be considered in a similar manner.
The following Table 3.1 represents all solvable equations whose solutions are outlined
in Subsections 3.2.2—3.2.4. The two-parameter families (in the space of parameters a, [3, 7,
and 6), one-parameter famililies and isolated points are represented in a consecutive fashion.
Equations are arranged in accordance with the growth of 6, the growth of 7 (for identical 6),
the growth of C (for identical 6 and 7), and the growth of a (for identical 6, 7, and /3). The
number of the equation sought is indicated in the last column in this table.
© 1995 by CRC Press, Inc.

TABLE 3.1
Solvable equations of the form у™ = Ахау'3{у'хУ {
6
arbitrary
arbitrary
{6 ±2)
7 + 4/3+5
7 + 2/3+3
37 + 7
2G + 2)
З7 + 7
2G + 2)
arbitrary
F Ф 1, 2)
arbitrary
Fф2)
3/3 + 4
2/3 + 3
arbitrary
(<^f)
arbitrary
(^1)
arbitrary
arbitrary
{6 ±2)
3/3 + 4
2/3 + 3
-1
-1
0
0
0
0
0
0
0
7
arbitrary
arbitrary
G + -1)
arbitrary
G Ф -1)
arbitrary
G + -2)
arbitrary
G + -2)
-1
-1
0
0
1
1
1
3
3
3
arbitrary
G + -1)
arbitrary
-2/3-5
-13
-13
-7
-7
/3
0
0
arbitrary
(/3 ^ -1)
1
2
1
-1
0
arbitrary
(/3^-1)
1
2
arbitrary
(/3^-i)
-1
1
arbitrary
(/3^-1)
7
5
0
0
-tG + 5)
arbitrary
(/3 Ф -2)
1
3
0
1
a
arbitrary
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Equation
3.2.4.15*
3.2.4.1
3.2.4.174
3.2.4.10
3.2.4.7
3.2.4.175
3.2.4.11
3.2.4.8
3.2.4.87
3.2.4.2
3.2.4.13
3.2.4.4
3.2.4.9
3.2.4.168
3.2.4.164
3.2.4.3
3.2.4.171
3.2.4.5
3.2.4.153
3.2.4.155
3.2.4.141
3.2.4.145
* given are formulae of reducing to the generalized Emden—Fowler equation
© 1995 by CRC Press, Inc.

TABLE 3.1 Continued
Solvable equations of the form y'?,w = Ахау13(ухУ(у^хУ
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
-4
-4
-3
-3
-3
-3
7
3
7
3
7
3
7
3
7
3
7
3
7
3
7
3
9
5
9
5
-1
-1
0
0
0
0
0
0
0
0
0
0
0
0
a
1
2
0
-2
-1
0
1
10
3
7
3
4
3
5
6
1
2
0
1
2
13
5
1
-2
0
7
2
7
2
5
2
5
2
-2
4
3
4
3
5
4
5
4
7
6
7
6
1
2
a
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
1
0
4
3
0
3
2
0
5
3
0
-3
Equation
3.2.4.127
3.2.4.123
3.2.4.95
3.2.4.30
3.2.4.26
3.2.4.91
3.2.4.76
3.2.4.42
3.2.4.52
3.2.4.133
3.2.4.131
3.2.4.48
3.2.4.38
3.2.4.70
3.2.4.64
3.2.4.60
3.2.4.22
3.2.4.18
3.2.2.2
3.2.3.3
3.2.2.3
3.2.3.4
3.2.2.6
3.2.3.5
3.2.2.4
3.2.3.7
3.2.2.8
3.2.3.6
3.2.2.5
3.2.3.8
© 1995 by CRC Press, Inc.

TABLE 3.1 Continued
Solvable equations of the form y'J,'xx = Axayl3(yx)~1\yxx)
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
1
2
1
2
1
2
1
2
2
3
4
5
1
1
1
h^
h^
h^
h^
8
7
7
0
0
0
0
0
1
2
2
3
3
5
5
5
5
0
3
3
3
3
0
-4
arbitrary
-3
-3
-1
1
1
1
3
a
1
2
0
0
1
1
7
2
0
arbitrary
CM-2)
-2
-5
20
7
15
7
0
5
2
15
8
20
13
5
4
0
7
6
1
2
-1
1
2
1
-1
arbitrary
-1
1
3
4
a
3
2
0
arbitrary
0
arbitrary
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Equation
3.2.3.9
3.2.2.7
3.2.3.1
3.2.2.1
3.2.3.2
3.2.4.35
3.2.4.165
3.2.4.161
3.2.4.85
3.2.4.82
3.2.4.105
3.2.4.117
3.2.4.111
3.2.4.101
3.2.4.74
3.2.4.114
3.2.4.120
3.2.4.108
3.2.4.104
3.2.4.157
3.2.4.137
3.2.4.140
3.2.4.32
3.2.4.24
3.2.4.177
3.2.4.14
3.2.4.17
3.2.4.21
3.2.4.159
© 1995 by CRC Press, Inc.

TABLE 3.1 Continued
Solvable equations of the form у™ = Axay'3{y'x)'i {
6
8
7
6
5
5
4
5
4
5
4
9
7
9
7
9
7
13
10
27
20
18
13
7
5
7
5
7
5
7
5
7
5
7
5
7
5
7
5
7
5
10
7
22
15
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
7
3
0
-4
3
3
9
4
0
0
0
0
0
-7
5
2
13
7
1
3
0
1
3
3
11
0
0
arbitrary
-3
-3
0
0
0
1
3
a
i
2
2
1
2
1
2
0
1
1
3
1
5
2
2
3
7
2
1
1
1
1
1
1
0
1
1
5
2
2
iG-i)
1
2
1
-2
l
2
1
1
-2
a
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Equation
3.2.4.151
3.2.4.109
3.2.4.57
3.2.4.148
3.2.4.144
3.2.4.66
3.2.4.169
3.2.4.62
3.2.4.80
3.2.4.121
3.2.4.68
3.2.4.54
3.2.4.45
3.2.4.78
3.2.4.72
3.2.4.40
3.2.4.50
3.2.4.126
3.2.4.130
3.2.4.135
3.2.4.43
3.2.4.115
3.2.4.173
3.2.4.100
3.2.4.97
3.2.4.99
3.2.4.84
3.2.4.93
3.2.4.28
3.2.4.98
© 1995 by CRC Press, Inc.

TABLE 3.1 Continued
Solvable equations of the form у™ = Axay'3{y'x)'i {
6
3
2
3
2
23
15
11
7
8
5
8
5
8
5
8
5
8
5
8
5
8
5
8
5
8
5
21
13
33
20
17
10
12
7
12
7
12
7
7
4
7
4
7
4
9
5
13
7
13
7
2
2
2
2
7
3
3
l
3
-4
1
3
3
3
3
3
3
3
3
-6
1
3
-4
l
3
3
3
0
0
1
1
3
l
2
0
arbitrary
G Ф -1)
-1
-1
-1
a
i
2
0
1
2
1
2
1
-4
7
4
10
7
2
1
2
0
1
5
l
2
1
2
1
2
1
2
13
8
1
2
5
2
1
1
1
2
1
1
0
arbitrary
(/3^0)
-1
0
a
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Equation
3.2.4.94
3.2.4.29
3.2.4.116
3.2.4.47
3.2.4.125
3.2.4.55
3.2.4.46
3.2.4.79
3.2.4.73
3.2.4.41
3.2.4.51
3.2.4.129
3.2.4.136
3.2.4.69
3.2.4.122
3.2.4.81
3.2.4.170
3.2.4.67
3.2.4.63
3.2.4.56
3.2.4.147
3.2.4.143
3.2.4.110
3.2.4.160
3.2.4.152
3.2.4.12
3.2.4.139
3.2.4.176
3.2.4.16
© 1995 by CRC Press, Inc.

TABLE 3.1 Continued
Solvable equations of the form у'?хх = Axayl3(yx)'> ' (yxxf
6
2
2
2
li
5
7
3
5
2
5
2
5
2
5
2
5
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
7
0
3
3
0
4
3
-4
li
4
27
13
3
2
1
arbitrary
G + -3)
-2/3-5
arbitrary
-9
-6
-6
17
3
33
7
21
5
-4
11
3
23
7
-3
-3
-3
-3
5
3
5
3
4
3
a
-2
-2
0
5
2
1
2
1
2
1
1
1
1
1
arbitrary
ifi + -2)
-7-2
2
l
2
1
2
5
3
2
7
5
1
2
5
3
2
-2
-1
l
2
1
5
3
1
2
1
2
a
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Equation
3.2.4.33
3.2.4.25
3.2.4.19
3.2.4.138
3.2.4.158
3.2.4.75
3.2.4.113
3.2.4.119
3.2.4.107
3.2.4.103
3.2.4.86
3.2.4.6
3.2.4.172
3.2.4.106
3.2.4.59
3.2.4.166
3.2.4.77
3.2.4.118
3.2.4.65
3.2.4.37
3.2.4.44
3.2.4.112
3.2.4.96
3.2.4.23
3.2.4.90
3.2.4.83
3.2.4.53
3.2.4.149
3.2.4.150
© 1995 by CRC Press, Inc.

TABLE 3.1 Continued
Solvable equations of the form у'?хх = Axayl3(yx)'> ' (yxxf
6
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
7
-1
2
0
0
0
0
1
1
1
1
1
1
1
1
3
3
3
3
3
3
3
5
7
9
5
1
P
-2
5
3
5
2
5
3
1
2
1
-4
5
2
-2
5
3
-1
1
2
1
2
2
-7
-4
-2
5
3
7
5
1
2
0
5
3
-7
1
1
a
0
0
0
0
0
-3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Equation
3.2.4.31
3.2.4.134
3.2.4.128
3.2.4.132
3.2.4.89
3.2.4.88
3.2.4.142
3.2.4.124
3.2.4.27
3.2.4.49
3.2.4.20
3.2.4.34
3.2.4.162
3.2.4.102
3.2.4.154
3.2.4.146
3.2.4.92
3.2.4.39
3.2.4.61
3.2.4.58
3.2.4.36
3.2.4.71
3.2.4.156
3.2.4.167
3.2.4.163
3.2.2. Equations of the Form у"' = Ay13
1-
Solution: у = jrAx3 + С2ж2 + Cxx + Co
© 1995 by CRC Press, Inc.

2- tC = Ay~V\
Solution in the parametric form:
x = aCf Г [de2<TT + C2e-CTT sin(VSar)] ~3/2 dr + C3,
у = ЬС1 [Сге2<тт + С2е~ат sin(v/3 ат)] ~\
where A = -8a69/2(j3.
3. tC = Ay-5/2.
Solution in the parametric form:
x = ad[ /(t3 - 3t + C2)/2 dr + C3, у = бС^т3 - Зт +
where A = -6a67/2.
4- tC = Ay-4/3.
Solution in the parametric form:
ж =
l f R~1BtI t Rf dr + C3, y = bCfBrI
where R = л/±Dт3 - 1), I = f tR'1 dr + C2, A = ±18a67/3.
Solution in the parametric form:
where R = a/±Dt3 - 1), I = JtR'1 dr + C2, A = т18а613/6.
> In the solutions of equations 6-7, the following notation is used:
CiJi/3(t) + C2Y1/3(t) for the upper sign,
Ii/3(t) + C2K1/3(t) for the lower sign,
where J\/3 andY\j3 are Bessel functions, I\/3 andKi/3 are modified Bessel functions.
( CiJi/3(t
\ CiIi/3(t)
6- tC = Ay~\
Solution in the parametric form:
x = aCi It'1 Z~2 dr + Cz, у = &Ci
where A = ±±a~43.
' * Уххх — АУ
Solution in the parametric form:
x = aC1 jZdr + C3, y = bC2T2/3Z2,
where A = =F-|-a~363/2.
8- VZX = Ay'5/4.
This is a special case of equation 3.2.4.171 with 7 = 0.
© 1995 by CRC Press, Inc.

3.2.3. Equations of the Form y"xx = AxayP
See Subsection 3.2.2 for the case a = 0.
1- tC = ^ab-
^absolution: у = Af(x)
/0*0 =
, where
r.a+3,
—ж In |ж| + ж
if а ф -1, -2, -3;
if a = -1;
if a = -2;
if a = -3.
2-
3-
See equation 3.1.2.7.
Solution in the parametric form:
-i
where / = C1e2<TT + C2e-<TTsin(v/3(TT), A = 8a-%9/2a3.
4-
Solution in the parametric form:
= aC\ \ /(t3 - 3t + C2)/2 dr + C3~\
у = bCf(r3 - Зт
where A = 6a67/2.
t3 - 3t
/2 dr + C
> In the solutions of equations 5-6, the following notation is used:
- 1), I=jtR-1 dr + C2.
R =
5-
tC =
Solution in the parametric form:
x = aC\
\ \ j R-1 BtJ
C
у = ЪС!Bт1 TRflf R~1{2tI t Rf dr + C3\ ,
where A =
© 1995 by CRC Press, Inc.

6- tC =
Solution in the parametric form:
у = ЪС1{2т1 т Д) [ I R~1BtI т R)~5/2 dr + С3~\
where A = т18а-4/3&13/6.
Solution in the parametric form:
X == CtO-^ / T -2 j (XT ~r O3 ? у == t)Oi J \ I T Z J ^'
where 2 = С2 + ^т2+4Вт1/2, / = e^p(Jz~^2dr), A=^Ba-^2b9^.
8. y"' =
Solution in the parametric form:
1-1 г /¦ -1—2
x = d\ jZdr + C3j , y = bT2'zZ2
where
_ ±_4 ,3/2 ^ _ Г Ci ji/3(t) + C2Y1/3(t) for the upper sign,
~ 3 ' ~ I Cli-fi/3(V) +С^/з(т) for the lower sign,
(Ji/з and У1/3 are Bessel functions, I1/3 and ЛГ1/3 are modified Bessel functions).
9- tC r
Solution in the parametric form:
аС3ехрB f Pdr^, x = bC3P2exp^2 f Pdr^,
where P = Р(т, Ci, C2) is the general solution of the second Painleve transcendent:
P"t = ±tP + 2P3, A = ±\aT
3.2.4. Equations with I7I + |?| ф О
Solution in the parametric form:
„^-1_л—1 / _i л /_ . -X+JL\ ~
ж =
^6-1 J г-1'2 (l ± r^) S~2 dr + C3,
У =
where A = ±— -2S~
2 — 0
© 1995 by CRC Press, Inc.

2- tC = AyPy'x(y'L) \ /3^-1, бф1.
Solution in the parametric form:
x = act6 / I / A ± r^1) ^ dr + C21 dr + C3, y =
where A =
l — o
> In the solutions of equations 3-10, the following notation is used:
R= \Л±тт+1, Е= f (l±Tm+1)~1/2dT + C2, F = RE-t.
Solution in the parametric form:
х = аСГ I t-^2R-4t + C3, y =
where m = —-—, A =
4. у -
Solution in the parametric form:
where m =- , A = -8ma26 ±- —
6-2 |_ (m + l)b\
Чт + С3, y = bC2m+2R,
2 1 1/"»
5- VZX = АуР(у'хГ2е-\ (З ф -2.
Solution in the parametric form:
x =
^-3JT-l/2E-3/2R-ldT + C3t y
where m = -/3 - 3, A = ±^(-1)т(т + 1)а2т63-
-263"т
6- y^'L = Ау^(у^)-2/3-5(у^K, /З ^ -2.
Solution in the parametric form:
x =
-im+3 / р-3/2п-1р^,/1 ,, hri2m+2^T^~l
^ j hj к t ат + Сз, у = oG1 rii ,
where m = (i, A = T^(m + l)a~2m-2bm+3.
© 1995 by CRC Press, Inc.

7- tC У
x
Solution in the parametric form:
= aCf+m+2 f E-^R-1 dr + C3, y = bC$F,
2G + 2) 2mb-1 Г (m + 1N1 ^+2" Г 4а2
where m = — -, A = ±- — ±
7+1 m + 2 I 2a
8- Ух'хх = АУ (Ухх) 2/3+3 » /^ 7^ -3/2.
Solution in the parametric form:
_ n(~rrr? + 2m-\ I „m-p-m-2 p-1 j , /-i .^(m+1J m+1 ттт-m-l
ж = aG]^ I т hi к ат + Сз, у = oG]^ т ii
в -, m
where m = - — , A=(m + ,)а~Ч ^
3/3+4
9- VZX = Ay4y'xf(y'L) 2/3+3 » /3 Ф -3/2.
Solution in the parametric form:
x = aCf+n~x f ET^F-^R-1 dT + C3, y = bc[m-1)(m+2)Em+2,
2/3+3 m
2/3+3 m 2 ±L[
where m = —?——, ^=7 т^гаЧ ^+1 \±
3+1 ( + 2K |
/3+1 (m + 2K |_ 4a
Solution in the parametric form:
x = aCf+2m~7 I r^R^E^Fdr + C3, у = bC
2
1-7 , 2(т + 3),1/2Г a 1 m+i Г (m + lVb 1 ™+з
where m=- L, A = ^ -^-61/2 ±- —- ±-
1 + 7' m + 1 [ (m+l)b\ [ 2a2
ч<5 с / 2
Solution in the parametric form:
/6 r s
т s~2 ехр(+т2) dr + Сг, у = ЬС2 / т s~2 ехр(+2т2) dr + C3,
J
© 1995 by CRC Press, Inc.

Solution in the parametric form:
x = ad / t^ exp(=FT2) dr + C2,
where A = ±G + l)<af
3-1
у = ЬС1 т !+t ехр(тт )dr + C3,
13. у = Ay у (у ) , 6 ф 1.
Solution in the parametric form:
x =
ad / техр(тт2) 1тН ехр(тт2) dr+C2
where A =
l — o
Solution in the parametric form:
/
т
-1/2
ехр(тт2) dr + C2\ dr + C3, у = Ът
J
Solution in the parametric form:
x =
у = bCl+2S~a-3 J Y(r)^
+ C3,
where X = Х(т), Y = V(r) is the general solution of the generalized Emden—Fowler
equation
Y'xx =
A = Ba7+2<5-«-36i-7-<5 _
Solution:
У =
I-A
B - A)d
-?¦ exp(Cirr) + C3
{dx + C2) 1-^4 +C3 if A ^ 1, А ф 2,
if A = 1,
if A = 2.
<L = ау 1у'хУхх-
Solution:
X =
J {
+ С*)-1'2 dy + C
© 1995 by CRC Press, Inc.

Solution in the parametric form:
x = aC1 I ехр(т|т2) dr + C2, y = bC2 f ехр(тт2) dr + C3,
where A = ^a~%2.
Solution in the parametric form:
//¦
т~х'2 exp(=FT2) dr + d, У = bd / exp(=Fr2) dr +
J
where A = ±Aa4b~4.
> In the solutions of equations 20-25, the following notation is used:
E= ехр(тт2) dr + C2, F = 2тЕ ± ехр(тт2).
20. У':'ХХ = АУ-1У'ЖХK-
Solution in the parametric form:
x = ad f т ехр(тт2)?;-1/2 dr + C3, y = ЬС\ ехр(тт2),
where A = ±\a4~2.
Solution in the parametric form:
x = d E~1/2 dr + C3, y = Ьт,
where A = =f26~2.
22. Ух'хх = Ay~2(y'x)~1-
Solution in the parametric form:
x = aC\
where A = та64.
23. Уххх = Ау~ \УХ) \УХХ) ¦
Solution in the parametric form:
x = d E~3/2Fexp(^T2) dr + C3, у = ЪЕ'1 exp(=Fr2),
where A =
© 1995 by CRC Press, Inc.

24.
25.
Solution in the parametric form:
x = аСг IE1/2 dr + C3, y = bC2F,
where A = =p8a62.
Solution in the parametric form:
x = ad f F-1'2 ехр(тт2) dr + C3, y = bC2E,
where A = ±a%~2.
> In the solutions of equations 26-33, the following notation is used:
T
E=
1)-
= RE-t.
26.
27.
Solution in the parametric form:
x = 2аС2л/т + 1
where A = — ^-a~664.
Solution in the parametric form:
x = ad I E~1/2<
where A = 2a4b~1.
if xxx ** ** x >¦ ** xx / "
Solution in the parametric form:
C3,
y = ЪС$т,
x =
dr + C3, y = bC2R,
where A =-8a(-b)-5^2.
29.
Solution in the parametric form:
x = aCl f R-^2 dr + C3,
where A = 4a3 (-6)/2.
y =
© 1995 by CRC Press, Inc.

Solution in the parametric form:
x = aCl J t-iI2R-xE~zI2 dr + C3, y =
where A = -\а~%ъ.
3i. yZx = Ay-2(y'xr1(vD3-
Solution in the parametric form:
x = aC-1 f R^E-^F dr + C3, y = Ы
where A = 2a2 b.
32. tC = Ay
Solution in the parametric form:
x = aC{ / т-^R-^E^FdT + C3, y = bCfF2,
where A = a~%7/2.
33. УхХХ = Ay~ (yxx) ¦
Solution in the parametric form:
x = aC^1 / т~2Д-1 dr + C3, у = ЪСгт^Е,
where A = 2ab.
Solution in the parametric form:
x = ±aCl J t(t2 - 1)(t3 - 3t + C2)/2 dr + C3, y = bCf{r2 - if,
where A = ц^^Лта4:Ь~5^2.
35. y'l'xx -
Solution in the parametric form:
x = aC-1 j (t3 - 3t + C2)/2 dT + C3, y = ЬС2т,
where A = 3a~2b~1.
© 1995 by CRC Press, Inc.

36. yZx =
Solution in the parametric form:
x = ±\аСь1т112(т2 - 5) + C3, У = bCf{r3 - Зт + C2),
where A = -^-a6
37. tC
Solution in the parametric form:
l [(t2 - 1)(t3 - 3t + C2)/2(t4 - 6т2 + 4С2т - 3) dr + C3,
x =
where A = т^а
38. yZx
Solution in the parametric form:
x = aCl f (t3 - 3t + C2I/4 dr + C3, y = ±ЪС\&{т/к - 6т2 + 4С2т - 3),
where A = ±72a62
62f—V 3.
a J
39. tC = Ay-5/3«K(ylK.
Solution in the parametric form:
ж = ±aC\ [(t2 - 1)(t3 - 3r + C2I/2[±(t4 - 6т2 + 4С2т - 3)]/2 dr + C3,
here A = т8 • 9-5а660/3.
40. CL =
Solution in the parametric form:
x = аС~7 j (t3 - 3t + C2)/2 dr + C3, y = ±ЪС1{т2 - 1)(т3 - Зт + C2)
Solution in the parametric form:
x = ±aC31 J [±(r2 - l)]/2(r3 - 3r + C2M/4(r4 - 6r2 + 4C2r - 3) dr + C3,
у = ЬС32(тА - 6т2 + 4С2т - ЗJ,
where A = -15 •
© 1995 by CRC Press, Inc.

42- tC = Ay
Solution in the parametric form:
x = aC\7 I (t3 - Зт + С2I/4[±(т4 - 6т2 + 4С2т - 3)]/2 dr + C3,
у = ±ЪС\\тА - 6т2 + 4С2т - 3)~\
where A = ±72а617/3(|-
/3
43. 1С = Ау
Solution in the parametric form:
x = ±aCf f (t3 - Зт + С2)/2(т4 - 6т2 + 4С2т - 3)/3 dr + C3,
С*(т3 -
28 U
у = ЬС*(т3 - Зт + C2)~V " 6т2 + 4С2т - 3J/3,
> In the solutions of equations 44~4^> the following notation is used:
P6(t) = ±(t6 - 15т4 + 20С2т3 - 45т2 + 12С2т + 27 - 8C|).
44. tC
Solution in the parametric form:
x = aC\ I (t3 - Зт + С2I/2[±(т4 - 6т2 + 4С2т - 3)]/2Р6(т) dr + C3,
у = ±6Ci(t3 - Зт + С2K/2(т4 - 6т2 + 4С2т - 3)~\
where A = т^-610/3Bа)/3.
45- 1С 5/27/5
Solution in the parametric form:
x = aCl1 f (t3 - Зт + С2)/2(т4 - 6т2 + 4С2т - 3L/3 dr + C3,
405
л 405 Зг.( , Ь V/2( «
where A = a~6b ±—
8 I 2a/ V 1
2a J V 126
Solution in the parametric form:
x = aC37 I (t3 - Зт + С2M/4[±(т4 - 6т2 + 4С2т - 3)}1/3[Р6(т)}-1/2 dr + C3,
у = ЪС^(тА - 6т2 + 4С2т - 3L/3,
where A = —45 • 2~13а2Ь
© 1995 by CRC Press, Inc.

47. tC = Ау
Solution in the parametric form:
x = ±aCf j (t3 - Зт + С2)/2(т4 - 6т2 + 4С2т - 3M/3Р6(т) <2т + C3,
where A = T28 • ?,'a
48- lC =
Solution in the parametric form:
x = aC\ j (t2 ± 1I/4 dr + C3, y = ЪСЦт3 ± Зт + C2),
, , , 81 5,3 / 36 \ i/3
where A = ±—а~ъЬА —
2 \ a J
49- tC = Ay-s/*y'x(yZxf.
Solution in the parametric form:
x =
[t(t2 ± 1I/2(t3 ± 3t + C2)/2 dr + C3, y = ЪСЦт2 ±
where A = =p-^|g-a46 4/3.
T/5
50. Уххх = Ayyx{yxx) ¦
Solution in the parametric form:
x = aC3 [ (t2 ± 1)/2(t3 ± 3t + C2)/2 dr + C3, y = ЪС1Т{т2 ± 1)/2,
where A = ±56 (
V 36
f(y" )8/5-
Solution in the parametric form:
4 2 3/2a2\3/s
26-3()
x =
1 = T~27 " " V 36
52. y'" = Av~
t J t~1/2(t2 ± 1M/4 dr + C3, y = 6Cf(r3 ± 3r + C2),
Solution in the parametric form:
x = aCl j (r2 ± lI/4(r3 ± 3r + C2)/2 dr + C3, y = 6Cf(r3 ± 3r + C2)~
, 81 5,13/3 / 36 \ 1/3
where A = ±—a613/3 —1 .
2 V a J
© 1995 by CRC Press, Inc.

53. tC = Ау
Solution in the parametric form:
= aC-1 f(±T2 + С2т - 1)(t2 ± 1I/2(t3 ± 3t + C2)/2 dr + C3,
36
54. «"' = *~<~'^-7'
Solution in the parametric form:
x = aCl ( (т2±1)/2(т3±Зт+С2M/6 dr+C3, у ¦-
Solution in the parametric form:
x = aC-1 [ (r2 ± lM/4(±r2 + С2т - l)/2(r3 ± 3r + C2)/3 dr + C3,
r 2
where A = ^Ъа2
о
56. «'" = At/-5/2(w" O/4.
Solution in the parametric form:
x = aC~7 f (т2 ± 1) 3/2(t3±3t
~~ in л !/•>/ i\— 4/ // \5/4
57. yZx = Ay~ ' (ух) (Ухх) ¦
Solution in the parametric form:
x =
i7 f(±T2 + С2т - 1)(t2 ± 1) 3/2(t3 ± 3t + C2J/3 dr + C3,
у = ЬС18(±т2 + С2т - 1J(t2 ± I),
—
where A= -64а69/2(т —
© 1995 by CRC Press, Inc.

> In the solutions of equations 58-69, the following notation is used:
Sx = de2kT + C2e~kr sin(W), w =
S2 = 2кСге2кт + kC2e-kT [л/3 cos(wr) - sinfwr)],
S3 = 4/c2Cie2feT - 2k2C2e-kT [л/3 cos(wr) + sinfwr)],
S^ = Si - 2S!S3, S5 = 5S2S4 + 32k3Si
58. y'l'xx -
Solution in the parametric form:
x = aC3 I S~1/2S2S3 dr + C3, y = bCfSl,
where A = -a%-^l2k3.
59. v'" -
Solution in the parametric form:
x = aC3 I S~3/2S2S±dT + C3, y = bC2S~
where A = 16a-3b9/2k3.
60. y'Zx = Ау(у'хГ9/5.
Solution in the parametric form:
x = aC3 I S3/4 dr + C3, y =
Solution in the parametric form:
x = aCx \ Sx S2S^ dr + C3, у = bC^S^
where A=\- 5-5a66-18/5/c.
62. y'x'xx = Ay{yxx) .
Solution in the parametric form:
= aC~3 f S~3'2 dr + C3, y = bdS~1/2S2,
© 1995 by CRC Press, Inc.

63. tC = Ay
Solution in the parametric form:
x = aC\b I Sl'AS~1/2S^ dr + C3, y =
where A = 7 • 16-4a2
86/c3
64. tC = Ау-™/ЦУхГ9/5.
Solution in the parametric form:
x = aCf J S\/ASf/2 dr + C3, y =
where.4=-160a-V3M
a
65. tC = Ay
Solution in the parametric form:
x =
l f Sl/2S;3/2S5 dr + C3, y =
66. tC = Ay(yx)-9/4(y^)9/r.
Solution in the parametric form:
x =
l J Sf/2St/b dr + C3, y =
1/4
67. y'Zx = A
Solution in the parametric form:
= aCf J S
'2 dr + C3, y =
68. tC = Ay
Solution in the parametric form:
= aC? J S;
x = aC? J S;3/2S~3/5 dr + C3, y =
where A =
208 l
5
© 1995 by CRC Press, Inc.
a^b7'2^I—— )
V b J

69.
Solution in the parametric form:
= aCl1 J S
;3/2S9J5S5 dr + C3,
y =
where A = 208 • 55a~7b13/2k3(—)
V b J
> In the solutions of equations 70-81, the following notation is used:
Ti = cosh(r + C2) cosт, 0i = coshт — sin(r + C2),
T2 = tanh(r + C2) + tan r, 02 = sinh т + cos(t + C2),
T3 = tanh(r + C2) — tanr, #3 = sinhr — cos(t + C2),
T4 = 3T2T3 - 4, 04 = 3(92<93 - 20?.
70.
1°. Solution in the parametric form:
x = ad f T^4
where A = -Заб(—V 3.
V a I
2°. Solution in the parametric form:
x =
. 3 _5 ,/6
where A = —a b[ —
8 \a
1°. Solution in the parametric form:
x = aC\ I TxT~lf
where А = 64-3-7а866/3.
2°. Solution in the parametric form:
/~i2 /"/)l/2/,-l
x = aLjx / v-y 6*2
where A = -256 • 3а8&-16/3.
© 1995 by CRC Press, Inc.
y = bC\TxT2,
y = ЪС$62,
y =
y =

72- tC = Ау(у'хГ1/3(у':х)г/ь.
1°. Solution in the parametric form:
x = aC~2 j T-xTl'2 dr + C3, y =
5 i, i/ b \!/3/2а2\2/5
-b-1() Ы
2°. Solution in the parametric form:
x = aC~2 f в;3/2в1/2 dr + C3, y = ЪСхв
where .4=-a 4 \-) (_)
73. <L = ^у
1°. Solution in the parametric form:
x = aCl1 J TllfiT2T-1/2 dr + C3, y = bCl2lf
5
where A=
ab
2°. Solution in the parametric form:
x = aCl1 j в\/Ав22в-1/2 dr + C3, У = ЪС\2в1,
2 \ з/5
)
74. yZx = Ay
1°. Solution in the parametric form:
x = aCl j T~3/2 dr + C3, y =
2°. Solution in the parametric form:
x = aCl j9~3'2 dr + C3, y = ЬС2в~
© 1995 by CRC Press, Inc.

75- tC = Ay
1°. Solution in the parametric form:
= aCl J Tf
x = aCl J Tf/2T$T3 dr + C3, y =
2°. Solution in the parametric form:
/*
x = аСг I вг в203 dT + C3, у =
where A = 16a~'
1°. Solution in the parametric form:
x = aCl J T-5/%3/2 dr + C3, y
2°. Solution in the parametric form:
x = aC\ j #11/4#~3/2 dT + Сз, У =
where A = |-a-16/3620/3.
О
1°. Solution in the parametric form:
T{T2 ' J
y =
16
2°. Solution in the parametric form:
x = aC2 Г в\/2в~г/2вА dr + C3,
У =
4
© 1995 by CRC Press, Inc.

78- 1С = Ау(у'хГ13/7(у':хO/5.
1°. Solution in the parametric form:
x = aC2 J T^Tl1'6 dT + C3, y =
where A = -—a -r- (-^r) •
4 V 2a / \ 1b J
2°. Solution in the parametric form:
/o2 f /1-3/2-,1 1/6 , , ,-,
5 _2/ 36 \ 6/7/4a2 \ 2/5
-2() ()
79- tC = Ay
1°. Solution in the parametric form:
x = aCl9 f т1*112Т^Т-1/2 dr + C3, y = b
45 ,32l 11/7/2«2 \3/5
where A = —- • 7~3a2b~11/rI—— ) .
Id V 7o /
2°. Solution in the parametric form:
x = aC? J в\1%1%1/2 dr + C3, y =
80. y'" = Ay-5/2(y" I3/10.
1°. Solution in the parametric form:
x = aC-11 f т~11/6Т-1/3 dr + C3, y = bCfT-1/3T2/3,
20 il5/2/2«2 \3/10
where A = —a~ ub/ ' '
3 V b
2°. Solution in the parametric form:
x = aC'11 I (9~3/2<9~1/3 dr + C3, y = bCfe^1^3,
10 lt5/2/ 2a2 \ 3/Ю
where A =—a^572! )
1°. Solution in the parametric form:
ж = aC±9 J Т™/6Т*/3Т4 dr + C3, y = bClsT3T2,
where A = -
2°. Solution in the parametric form:
x = aCl9 I<9~3/2<92/3<94 dr + C3, y = бС^б»^,
© 1995 by CRC Press, Inc.

> In the solutions of equations 82-84, the following notation is used:
Lx = ClTk + С2т~к, JVi = A + k)ClTk + A - к)С2т~к,
L3 = Cisin(fclnr) +C2cos(/clnT), N3 = (d - fcC2)sin(fcbr)
+ (С2 + /сС1)со8(/с1пт).
82. yZx
Solution in the parametric form:
x = JT1/2L^1/2dT + C3, у = т2
1 HA> -1/8,
where к = ^/\1 + 8A\, m = < 2 if A = -1/8,
[ 3 if A < -1/8.
83. tC = Ay(yx)-3(y'^K.
Solution in the parametric form:
/ T^
X = / T^Nm dT + C3, у = TLm,
Г 1 if A < 1,
where к = \J\A- 1|, m = < 2 if A = 1,
I 3 if A > 1.
84. tC = Ay
Solution in the parametric form:
ж = т4 / T^LidT + Сз, y = TzL{,
J
where к = л/1 + 8^4~2.
> /n ifte solutions of equations 85-100, the following notation is used:
7 _ ) C\Jv(t) + C2Yv(t) for the upper sign,
C\Iv{j) + C2Kv(t) for the lower sign,
U1=tZ't + vZ, U2 = UI±t2Z2, U3 = ±^t2Z3 -2UXU2,
where Jv and Yv are Bessel functions, Iv and Kv are modified Bessel functions.
85. tC = AyP(yxK, (Зф-2.
Solution in the parametric form:
x =
where »=
© 1995 by CRC Press, Inc.

86. yZx =
Solution in the parametric form:
= aC1 I t^Ux dr + C3, y = bClTvZ,
2 ,1
where v = —, A = ±—5-a
87. tC = Av-^ivL)', 6^3/2.
Solution in the parametric form:
I* TZv~x
x = ad ITZv~xZdT + C3, y =
i
where » =
88. vZ'm
Solution in the parametric form:
x = aC1 f Zdr + C3, y = ЬС2т~1/3 \tZ2 -Ux f Z dr - C3J7"il,
where v = \, A= ^a6b~3.
89. y'Zx = Ay
Solution in the parametric form:
x = aC1 IrZdT + C3, y = bC2T4/3Z2,
here i/=f, А = -\а3Ь-3'2.
Solution in the parametric form:
x = C1J t~2Z-2U^ dr + C3, y =
where v=^, А = ±±Ъ3/2.
91- yZx = Ay(y'xr3.
Solution in the parametric form:
x = aC1 jZdr + C3, y = ЬС2т~2/3 U2,
where v = \, А = -Щ-а~Ч3.
© 1995 by CRC Press, Inc.

92. УхХХ = Ay 2(y'x) (y'xx) ¦
Solution in the parametric form:
/~i I Г7ТТ rr~l/2 7 , SI 7 /~t2 2/3 Г72
x = aCi / ZUiU2 dr + C3, у = ЬС{т ' Z ,
where v = -|-, А = -^а6Ь~3.
93. v'" -
Solution in the parametric form:
x-djr Z
where ^ = |, ^4 = 46 (т^-) •
94. <L = ^
Solution in the parametric form:
= aC1 I Zbl2U~1/2U2 dr + C3, y =
21 z-l 5/2/ 3 \1/2
l л 21 z-l 5/2/ 3 \
where v=±, A = ^—aAb ' [T^)
95. y'" = Ay {у1)
Solution in the parametric form:
x = ad f tZU~3'2 dr + C3, y =
where ^ =-§-, A = --ifa66.
96. У^'жа; = ^У~2(Уж)~3(Ужа;K-
Solution in the parametric form:
x = d f ZU~3/2U3 dr + d, У =
where ^ = 4-> A = 186.
97. tC = Ay(y4)-3(y;'J3/2.
Solution in the parametric form:
x = aC1J T~2Z-2Ul'2 dr + C3, y =
where v=\, A = -16a~3b2(± —) .
© 1995 by CRC Press, Inc.

98. tC = Ау-Цу'хK(у-хK/2.
Solution in the parametric form:
x = ad / rZti/zU~ ' dr + d, У =
97 / 4 \ 1/2
where v=±, A = ±^a3b~x (±^) .
99. tC = Ay-2(VLK/2-
Solution in the parametric form:
x = d f t~xZ~2 dr + C3, y =
where v = |, A = ±±b2Bb)-1/2.
100. tC = AyVa^
Solution in the parametric form:
x = ad [t-3Z-2U*/2U3 dr + C3, y =
where v=\, A = T^a
> In the solutions of equations 101-138, the following notation is used:
Function p = р(т) is defined implicitly by the above elliptic integral of the first kind.
For the upper sign, the function p coincides with the classical elliptic Weirstrass
function p = р(т + С2, 0, 1). In the solution given below, we can assume p as the
parameter instead of т and use the explicit dependence т = т(р).
101. tC = A(y'xf.
Solution in the parametric form:
x = aC\ Г p~1/2 dr + C3, y =
where A = ±3a26.
102. tC
Solution in the parametric form:
= aCl J T~xl2f dT + C3, y = bCfp,
where A =
© 1995 by CRC Press, Inc.

*'2.
Solution in the parametric form:
юз. tC = АУУ'ямя)*'2.
x = ad J т-^р2 dr + C3, y = bCtf,
where A = -^a3b~3(±Sb)-1/2.
Solution in the parametric form:
„ — ппъ f f-1/2 at _i_ n ¦>/— Ы2т
where A = ±6ab-2(±3b)-1/2.
105. y'l'xx = Ay-b(y'xf.
Solution in the parametric form:
x = aC'1 / T~3/2p-1/2dT + C3, y = ЬС2т-\
where A = ±3a2b.
106. yxxx = Ay2(yx}~ iy'xx) ¦
Solution in the parametric form:
x =
l j'r-3/2(Tf -p)dr + C3, у
where A = =F24a65.
Ю7. yZx 3/25/2
Solution in the parametric form:
-1p2dT + C3, y = bC1(Tf-p),
2 ч1/2
)
where A =
1 / 2 ч1/2
—ab(±—)
2 V a I
108. у'1'хх = Ay-5/4(y'xK(y%xI/2.
Solution in the parametric form:
= aC\ JT3(rf - p)-1'2 dr + C3, y = ЬС\т\
where A = ±^-c
© 1995 by CRC Press, Inc.

109. yZx = Ay
Solution in the parametric form:
= aCl J
y =
where
lot)
no. vZx 1/21/3
Solution in the parametric form:
x = aC-1 [т^2р^2(т/ -p)dr + C3, у = bCf{Tf - pJ,
_¦,¦, /2 / \lb \ i'd / a
where A = =P5a b' [
V a J V 186.
in- tC = АУ
Solution in the parametric form:
x = aU{" I r"'"(r"p =Flj йт + С'з, у =
Леге A = ±3 • 7a26-13/7.
112- tC = Ay\Vxy(yZ)-
Solution in the parametric form:
x — aUj ^ т ^т j + бт р+1)ат-\-<^3, у — о^-^тут р+l),
where A = =p-|
ИЗ- tC = Ay(yxr11/4(yZf/2-
Solution in the parametric form:
C3, у = ЬС3Т-6(т3/ + 3r2pT 1),
3 / 66 \ 3/4/ ч -1/2
(±) F6)
3 / со \^/4/ \-
where A = ±— (±—) (=р66)
2 V a J V /
.„ 15/8/ /\3/ // \l/2
Solution in the parametric form:
where A = ±-^-c
© 1995 by CRC Press, Inc.

Solution in the parametric form:
x =
where A = ^1Ъа
116-
Solution in the parametric form:
у = ВДт3(т2рт IK,
+ 3r2p т l)(r2p T 1I/2 dr + C3,
where A = ±15a
11/2/12Ь
V a
чV3/ a2 x
)
) V186/
Solution in the parametric form:
x = aCf I' т~ъ{т2р--
where A = ±3 • 7a2&-8/7.
us- iC = Ay4y'xVzz/\y'Lf ¦
Solution in the parametric form:
C3, y = ЬС\т-\
x =
where A = Tika
- Ат2р ±6)dr + C3, у = ЬС\2т-\т2р Т 1),
119- tC = Ау(у'хГ2Г/13(у'^M/2.
Solution in the parametric form:
x = aC~
, у = ЬС2т{тгf - Ат2р± 6),
where A = -|i(^
120. tC = Ay
Solution in the parametric form:
= aCf I r2^2(r3f - Ат2р ± 6)/2 dr
y =
where A = ±^
© 1995 by CRC Press, Inc.

121. iC = ^
Solution in the parametric form:
x = aCl9 T-20(T2pTlJdT + C3, у = &CV18t-18(t2PT if,
, 7/20
where A =
122. y'Zx = Ау
Solution in the parametric form:
) у = 6С2т2(т3/ - 4т2р± бJ,
1,1/9/ J-^f \ Iй f О, \
where A = =p20a b ' \ ±
V a J V 186/
13/20
123. <L =
Solution in the parametric form:
= aCl J p~2 dr + C3, y = bClp~2(f ± 2rp2),
where A = -192ab5.
124. yZx = Ay-b/*v'x(y'L
Solution in the parametric form:
x = aC1J fp-2(f±2rp2y
where A = ±4-arb~1^2.
' (y" )8/5-
Solution in the parametric form:
125. tC = Ауух(у>>х)
= aC\z Jp3(f±2rp2)
5
where A = T-
126. tC
Solution in the parametric form:
x
= aC\7 J p~3r1/2 dr + C3, y = bC\%~2{j ± 2rp2),
5 2, z( a \
where A = T-a2b-\-
© 1995 by CRC Press, Inc.

127. tC =
Solution in the parametric form:
x = аС\г J p(f ± 2rp2)/2 dr + C3, y = bCl4p2(f ±
where A = 192a~7b11/2.
128. yZx = Ay
Solution in the parametric form:
x = aC-1 J(f± 2rp2)/2(r/ + 2p) dr + C3, V = bCf(f ±
where A = --^a
129. yZx f
Solution in the parametric form:
x = aCf J p-^2(f ± 2rp2M/4 dr + C3, y = bC\\Tf + 2p),
, , 10 2,_4/2a2\3/5
where A=-a2b\—) .
130. у'ххх = АУ(УХ) ЛУХХ) ¦
Solution in the parametric form:
x = aCl1 J(f± 2rp2)/2(r/ + 2p)/2 dr + C3, y = bC\p{j ± 2rp2)/2,
where A = —10a26~4
131- tC = Ау
Solution in the parametric form:
x = aCf J(f± 2Tp2I/\rf + 2p) dr + C3, y = bC*2(Tf + 2pJ,
where А = -
132. y'xxx = АУ
Solution in the parametric form:
x = aC1 J p(f ± 2тр2I/2 dr + C3, У = bCi(f ± 2тр2K/2,
where A = ±-^-a3b~1^3.
© 1995 by CRC Press, Inc.

133. iC = Ау-*/°(у>хГ7/3.
Solution in the parametric form:
= aCf J(f± 2TP2f\Tf + 2p)~2dT + C3, y = bC32(rf + 2p),
where A = -648a-5623/6f—
V a
134. tC = Ау
Solution in the parametric form:
x =
-1 J(t2P t 1)(/ ± 2rp2I/2(r/ + 2p)~2 dr + C3,
ил l 31. l/3Fb\2/3
where A = a6b~1/6 —
324 V a
135. y'L,
Solution in the parametric form:
X
= aCf J(f± 2тр2) 3/2(т/ + 2pI3/6 dr + C3,
2a2
.here A = -20awb-12(^-)
\ b J
136. y?L =
Solution in the parametric form:
-2p T l)/2(/ ± 2rp2M/4(r/ + 2p)/3 dr + C3,
where A = 20a b
b
137. y'xxx = Ay~1/2(y'x)~4(y'D4/5¦
Solution in the parametric form:
= aCf7 J(t2P T 1)(/ ± 2тр2) 3/2(т/ + 2рL/з
where A = -320a611/2 f—)
V 46 /
2 ,4/5
© 1995 by CRC Press, Inc.

138.
Solution in the parametric form:
x = aC-13J(f±2rp2)-3/2(rf-
r + C3,
r4l/5
4™ \4b) '
> In the solutions of equations 139-Ц0, the following notation is used:
U =
f тк г dr
J z(t)
1п
139.
Solution in the parametric form:
where к = 1/C, A = -2Ъ~&.
140. yxxx = Ay~1(y'x)~fyxx, "f ^
Solution in the parametric form:
x = aC\ I т T-1 z e dr + C3,
where k = , A = a7"^1
7-1
if к = 0;
if /с = -1.
> /n ifte solutions of equations Ц1-110, the following notation is used:
R = 7±Dт3 - 1),
tII h = hh-»ll I5 = 2RI-t2,
where I = J tR~x dr + C<i is the incomplete elliptic integral of the second kind in the
form of Weierstrass.
141- fC = А(у'хГ7.
Solution in the parametric form:
x = aC\
x dr + C3, y = bClr^h,
where A = =f3gT
© 1995 by CRC Press, Inc.

142. yxxx = Ay 4yx(y'xX) ¦
Solution in the parametric form:
x = aC'1 / t~3/2I~1/2 dr + C3, y = bCfr'1,
where A = ±24a46.
143. tC = Ayy'MJ1'*.
Solution in the parametric form:
l1 f ть/21~
x = aUi / т"'1 ' К 'dT + С'з, y =
2,-2/ «2 N
where Л = =f — b =F ^^
144. yZx
Solution in the parametric form:
x = aCl3 [t~2R-3/2 dr + C3, y =
where A = -4a26~3(=F—r
145. tC = Ay(y'x)-7.
Solution in the parametric form:
x = aCl j 1-3/2Ц-г dr + C3, y =
where A = тЗа067.
146. tC = Ay-*(y'xLyL) ¦
Solution in the parametric form:
f T-Wl-^hR-1 dr + C3, y =
where A = ±24а66.
147. tC r/\
Solution in the parametric form:
l f ilR-1 dr + C3, y =
x =
2 о I a
where A = -4a6 (±^-r) .
v бь у
© 1995 by CRC Press, Inc.

148. tC = Ay
Solution in the parametric form:
x = aCl1 j tI-zI-iI2R-x dr + C3, y = bCfr2
149. й;, = Ay-i"(v'Ir'y':If-
Solution in the parametric form:
where A = ^ab [
27 V /
x = aC-1 I Tl'^hir1 dr + C3, у = bCf
126 \ 2/3
1 2,-1/2/ 126
27 V a
150. tC = Ay
Solution in the parametric form:
= aC1 I I-^hhRT1 dr + C3, У = bCl°I-
, . 16 2 1/2/ 36 N 1/3
here A = ^—a2b~1/2[±—
27 V a )
Solution in the parametric form:
Ix I2 I^R'1 dr + C3, у = ЬС3'
where A = =p7 • 2~10a2b~
x =
~ J *
= T7-2-10a2b-^2Fb
152. y'" -
Solution in the parametric form:
/-<13 f r-5/2n-l j , /~i ,„5.-3/2
ж = aCi Ix ' R dr + C3, у = bC°I1 ' ,
7 /a2 \6/7
where A = — a^! — ) .
2 V DO/
153. CL = Ay(y4)-13.
Solution in the parametric form:
x = aC\3 j il'^R-1 dr + C3, y = bCl6I3,
where A = ±3 • 225a~16b13.
© 1995 by CRC Press, Inc.

154. tC = Ау-Цу'хK(у-хK.
Solution in the parametric form:
x = aC'1 I /~1/2/2/3~1/2Д-1 dr + C3, y =
where A =
155. tC = АуЦу'хГ13.
Solution in the parametric form:
x = aCl1 J I^I-WR-1 dT + C3, y =
where A = ±3 • 225a66n.
156. y':L = AV-4y'xy(v'L) ¦
Solution in the parametric form:
Г /1/2/3/2/4Д-1 dr + C3, y =
where A = т192а106.
157. |C = Ay
Solution in the parametric form:
x = aC{ I I'^llRT1 dr + C3, y
where A = ±54a613/6 ^V\
9b
158. |C = AyV2^
Solution in the parametric form:
x = aCl j /1/2/3-1/4Д-1 dr + C3, y =
where A = 8b1/2(—)
> /
159. yxxx = Ay
Solution in the parametric form:
1 1 iq
1 3
where A = т7 • 23a26"9/4
y =
© 1995 by CRC Press, Inc.

160. tC = AvWJ-1'2^I*'7-
Solution in the parametric form:
„ig / т — Ъ/2тАг,— Л i /~i
x = aLx / i1 I3R dr + C3, y =
7 _¦¦ _i / 36 \ i/2 / a2 \6/7
where A =—a 6 (±—
2 V a У V126 У
Solution in the parametric form:
/r
Д dr + C3, у = ЬС\ rR~1dr + C2,
where A = ±6a~1b~1.
162. tC =
Solution in the parametric form:
= aCl j тГ1/2 dr + C3, y = bCfr2,
where A =
163. v'" -
Solution in the parametric form:
x = aC*JT2r1/2R-1dT + C3, y =
where A = —^hra6b~5.
164. y'l'xx -
Solution in the parametric form:
x = aC'1 ГtR~3/2 dr + C3, y = bC\ f tR'1 dr + C2,
where A = 9a~2b~1.
165. v'" -
Solution in the parametric form:
r — ась [ Т-з/2П-г rli-4-Го ii — hC2l-x
Jy — u^^ II П (XT -\- 03, у — yU^J ,
where A = ^¦&а~1Ьъ^2.
© 1995 by CRC Press, Inc.

166. tC =
Solution in the parametric form:
x =
l f Tl-WhR-1 dr + C3, y = ЬС^
where A = т48а67/2.
167. tC 9'ь\
Solution in the parametric form:
x = аС\г J т2/-1/4^-1 dr + C3, y = bCl6I5,
2
where A = —
1125 V a
3/..// \-l
168. yZx = av
Solution in the parametric form:
= aC1J т!3^!-1^-1 dr + C3, y =
where A=^-a~2b2/5.
169. tC = Ау
Solution in the parametric form:
x = aC1 I t2//^-1 dr + C3, y =
where A = —a
Solution in the parametric form:
x = aCf fT^f^hR-1 dr + C3, y =
0, x1/3/ a2 \5/7
Ы (ж) ¦
> In the solutions of equations 171-172, the following notation is used:
С2 + -т2 + 2В1пт if/c = -l.
© 1995 by CRC Press, Inc.

-H-5
. vZ* = ау 4 Ш ¦
Solution in the parametric form
x = aCf J
where k = l—L, А=-
172. y'Zx = Ay-'-Hv'Siy'Lf-
Solution in the parametric form:
where /c = -7 - 2, A = -23-ka1~kB.
173. yZx = Ay^(y'xr(y'Lf/2-
Solution in the parametric form:
x = acl j (V-^V2+±) dT c
J (f/K/VV2 + 4
cl j
J (Tf/K/VV2 + 4
where
2a \
V)
¦y+4/3+5
174. tC = Ay^y'SivZ) -/+2I3+3 , A7# -I-
Solution in the parametric form:
/7+4/3+5
-3/2r7 2G+1) Z-^(]T + Co u-hC1+1U
, л <v_i, _/3_-v / 2a2 \ 7+2/З+3 ,-, rr ( [ dr\ , . . , , .
where A = a' о M I —;—) -?>, t/=exp / , z = z(t) is the solution
V 6 У \J tz J
of the algebraic equation
175. у^'жа; = АУ г(у'х) (vZ) i ^ 7^ 1? 2.
Solution in the parametric form:
x = aCf'3 ITk-1U2^'z'1 dr + C3, y = bC\
6 — 1 . 1 — к _/,o/ 2a2 \6^ Tr //«¦•> / ч . ,
where к = — , A = а о ( j B, U = exp( / ], z = z(t) is the
0 — Zi Zt \ 0 s
solution of the transcendental equation
In \kz - t\ - = — тк + C2.
kz — т к
© 1995 by CRC Press, Inc.

\eT
176. yxxx = Ay 1(y'x) (yxx) ¦
Solution in the parametric form:
x = iCi I e z U dT + G3, у = i^e
±A / — ).
J z J
177. y^xx = A.y~1(y'x) yxx-
Solution in the parametric form:
x = d I eT/2Udr + C3, y = ±dzU,
(P J \
T / — )•
J z J
3.2.5. Some Transformations
Let us consider some transformations of the equation y'xxx = Ахау@{ухУ (ухх) ¦
1. In the special case 7 = 6 = 0, the transformation
1 w
reduces the equation
у?хх=Ахау?>
to an equation of the similar form
2. In the special case ot = 6 = 0, the transformation
f dr _ 1
"" /  7 МО/О > У
reduces the equation
to an equation of the similar form
2/3+7+5
z';tt = az——
3. In the special case /3 = 0, the substitution
u(x) = y'x
reduces the equation
y^xx = Ax^y'x)\
© 1995 by CRC Press, Inc.

to the generalized Emden-Fowler equation
(see Section 2.3 and Section 2.5).
4. In the special case a = 0, the substitution
reduces the equation
in л /3//\7///\<5
У XXX = A%I \Vx) \Vxx)
to the generalized Emden—Fowler equation
(see Section 2.3 and Section 2.5).
3.3. Equations of the Form у?х = f(y)g(y'x)h(y'^x)
3.3.1. Equations Containing Power Functions
This is a special case of equation 3.5.2.17 with f(w) = 1.
2- Уххх = (АУп + вУт)Ух-
This is a special case of equation 3.5.2.1 with f(y) = Ayn + Bym.
3- У'ххх = (АУп + вУт) Ну'хK + Ьу'х].
This is a special case of equation 3.5.2.4 with f(y) = b(Ayn + Bym), g(y) = a(Ayn +
Bym).
4- yZx = У'2 [~ ^g'l Ш*
The substitution w(y) = {y'x) leads to an equation of the form 2.4.2.4:
5- у^ = у~2[^Ш3
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.35:
© 1995 by CRC Press, Inc.

— 7
,M2
The substitution w(y) = (у'х) leads to an equation of the form 2.4.2.31:
7- уг:хх = у[Ю +
y')
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.64:
= у-2
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.6:
9-
The substitution w(y) = {y'x) leads to an equation of the form 2.4.2.26:
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.10:
ххх = у-[тШ + А
y')
The substitution w(y) = {y'x) leads to an equation of the form 2.4.2.12:
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.66:
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.29:
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.14:
© 1995 by CRC Press, Inc.

25 V-Wsr/
„м2
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.8:
ч; =
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.33:
¦u%y = y-2B0w
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.37:
The substitution w(y) = {y'x) leads to an equation of the form 2.4.2.60:
w" = y~2BAw2 - 4rw).
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.62:
w>;y = y~2BAw2 + ±w).
20. yZx=y
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.40:
21- yZx = (y + y)(yx
The substitution w(y) = {y'x) leads to an equation of the form 2.4.2.39:
u%y = BAy4 + 2By3)w-7.
22. yZx = (Ay2 + B)(y'xy9.
The substitution w(y) = {y'x) leads to an equation of the form 2.4.2.16:
«#„ = BAy2 + 2B)w~5.
23. yZx
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.28:
© 1995 by CRC Press, Inc.

24. tC = (A»-V3 +
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.48:
w'Jy = {2Ay~7/3
25. tC = (Ay-4/3 + );)
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.49:
26. tC = (Ay-4/3 + By-7/3)(y/;)-V3
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.24:
27.
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.90:
wvv =
28. yZx = (A
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.89:
29.
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.47:
u%y = BAy2
30. yZx =
The substitution w(y) = (y'x) leads to an equation of the form 2.4.2.46:
ь)'^ = BAy2
+ Byk)y'x(yZ)m-
This is a special case of equation 3.5.4.15 with f(y) = Ayn + Byk.
© 1995 by CRC Press, Inc.

TABLE 3.2
Solvable equations of the form
S
TABLE 3.3
Solvable equations of the form
д
6
arbitrary
0
l
3
2
3
2
2
2
7
1
3
1
0
3
arbitrary
G Ф -1)
-1
Equation
3.3.2.1
3.3.2.9
3.3.2.2
3.3.2.3
3.3.2.7
3.3.2.11
3.3.2.13
6
arbitrary
F^2)
h^
3
2
3
2
2
3
a
0
arbitrary
(/? Ф "I)
-1
l
2
1
0
1
Equation
3.3.2.4
3.3.2.12
3.3.2.14
3.3.2.5
3.3.2.8
3.3.2.6
3.3.2.10
TABLE 3.4
Other solvable equations of the type considered
Form of equation
y':xx = Aeyy'xeM(y'xfWL)S
Ух'хх = Му'хУ1 еМУхх)
y':xx = AyPy'xeMy'L\ РФ-1
Ух'хх = АУ~ХУ'Х ехр(у^)
y':xx = Aeyy'xeM{y'xf+y'L}
Equation
3.3.2.20
3.3.2.15
3.3.2.16
3.3.2.17
3.3.2.18
3.3.2.19
3.3.2. Equations Containing Exponential Functions
Tables 3.2—3.4 represent the equations whose solutions are given in this subsection.
> In the solutions of equations 1-6, the following notation is used:
E= feMT2)dT + C2, G= I t~\1 ± т)к dr + C2,
H= f т'1 ехр(тт) dr + C2.
f т'1 ехр(т
Solution in the parametric form:
x = ad I t^G-1'2 dr + C3, y = ln(bC2S-2T),
where k=^
1-6'
2A - 6)
-2b-1Ba2N.
© 1995 by CRC Press, Inc.

2.
Solution in the parametric form:
x = С
Solution in the parametric form:
x = Ci fE'1 dr + C3, у = т2 +ln(v/2A'1 E'1).
VxL = Ay'x exp [(y'xf] (y'^)S, 6 ф 2.
Solution in the parametric form:
x =
f т-\1 ± t) e-2 [Ы{ЬС{
dr + C3, y =
where fc=7-^, A = ±T-J,
Solution in the parametric form:
= C1F2, A = -V2a~1
6. y™m = Ay
Solution in the parametric form:
x =
= C1H.
> In the solutions of equations 7-8, the following notation is used:
Е=^т{
F = l-
y'i'xx = АеУ(у'хK(у':хK/2.
Solution in the parametric form:
x = aC\
where A = 2г12аЧ.
R =
y = - ln(bC~3E),
Solution in the parametric form:
where A = — 4a~16~3/2.
= bC1F,
© 1995 by CRC Press, Inc.

> In the solutions of equations 9-10, the following notation is used:
C\Jo{t) + C2Yq(t) for the upper sign,
C\Io{t) + C2Kq(t) for the lower sign,
where Jq and Yq are Bessel functions, Iq and Kq are modified Bessel functions.
9- tC = АеУЮ3-
Solution in the parametric form:
x = 2C1(T-1Z-1l2dT + C2, y =
/^Miy'SWLf-
Solution in the parametric form:
x = C1fz'T\\b(±lT)\ dr + C2, y =
Solution in the parametric form:
/ TT \
t + C3, y=U,
where A= ^-a1+1k, U= I —— + C\, / =/(т) is the solution of the transcendental
•J J
equation
7+1
Л/ — Т
A =
12. y'^'xx =
Solution in the parametric form:
U
x =
/3+1/'
\-C\, f = /(t) is the solution of the transcendental
J
w.
equation
Solution in the parametric form:
where W = exp ( /
1
\J т-2Ае
© 1995 by CRC Press, Inc.

Solution in the parametric form:
X = C1 [t-V^t + A
1Wdr + C3,
where W = exp
dr
т + AeT + C2
> In the solutions of equations 15-19, the following notation is used:
V = Сг - ^-(
Л
W=
C2 - /
J
C2- /*ln(Ci-Aln|T|)(iT
J
= Сг-^-{т + 2)e
Л
Ипф -1,
ifn=-l,
"T/2
"+1) dr
2A
Solution in the parametric form:
1 Г 2m+l
X=-Je-V-^
where m = -тгG — 1), A = 2~7A
y=— I e~TV
A
16-
Solution in the parametric form
x= j J exp(-T+-
where m = 0, ^4 = 2A.
y= у
Solution in the parametric form:
x = jW-1'2 dr + C3, у = 2т,
where n = C, A = 2~'d-1X.
Solution in the parametric form:
x= fW~1/2 dr + C3,
where n = —1, A = A.
C2,
© 1995 by CRC Press, Inc.

19. tC = АеУу'х exp [{y'xf + у'Ц.
Solution in the parametric form:
X =
-L J e~T/2M-l
1 dr + C2,
where A = A.
20.
Solution in the parametric form:
X =
y = U,
where
U
-1
dr
ZT"
2-6 1-6
т + In \т\ + С2
\п\т\- — + С2
C2
a 6 = 1,
a 6 = 2.
3.3.3. Other Equations
!• y'vL = Ayyx{cosh\u{y'S]}~2y'L-
The substitution y'x = \/u(y) leads to a second order equation of the form 2.7.4.1:
Kv = Ay[cosb.(uju)]-2u'
2-
tC =
The substitution y'x = ^Ju(y) leads to a second order equation of the form 2.7.4.2:
u'^y = Ay[smh(uu)}-2u'y.
3-
4J] (ylK/2
; cosh[a;(y4J] (yl)
The substitution y^, = ^Ju(y) leads to a second order equation of the form 2.7.4.3:
Uyy = —=ycosh(uu)(u'y) .
v2
4-
5-
The substitution y^, = \/u(y) leads to a second order equation of the form 2.7.4.4:
uvv = —=y8wh{u>u)(uy) .
tC = Acosh(a;y) (
The substitution y^, =
/u(y) leads to a second order equation of the form 2.7.4.5:
© 1995 by CRC Press, Inc.

6. vZx = Asu\\\(ujy) (y'x) (yZ) •
The substitution y'x = \Ju(y) leads to a second order equation of the form 2.7.4.6:
uyy = — sinh(u>y)u(u'y) .
7- tC = A[cosh(a;y)]-2(y4K«J2.
The substitution y^, = y'u(y) leads to a second order equation of the form 2.7.4.7:
uvv = \
8. y'l'xx = A[
The substitution y'x = \/u(y) leads to a second order equation of the form 2.7.4.8:
9- <L
This is a special case of equation 3.5.4.15 with f(y) = A cosh™ {toy) ¦
This is a special case of equation 3.5.4.15 with f(y) = ^4sinhn(tt>y).
This is a special case of equation 3.5.4.15 with f(y) = ^4tanhn(tt>y).
VZX = Acothn(u,y) УЖ.Г-
This is a special case of equation 3.5.4.15 with f(y) = A coth™ {toy)¦
This is a special case of equation 3.5.4.15 with f(y) =
14- V'L = Ax[cos(u>y)]-2yx.
The substitution y'x = \/u(y) leads to a second order equation of the form 2.7.5.1:
u'yy = Ay[cos(uu))-2uy.
The substitution y'x = \/u(y) leads to a second order equation of the form 2.7.5.2:
u'yy = Ay[sm(L0u)]-2uy.
16- tC = A[cos(u,y)]-2(yxK(yZxJ.
The substitution y'x = \/u(y) leads to a second order equation of the form 2.7.5.3:
u'yy = \A[c
© 1995 by CRC Press, Inc.

. tC = A[sin(u>y)}-*(y'xK(yZxJ.
The substitution y'x = ^Ju(y) leads to a second order equation of the form 2.7.5.4:
л I г / / \2т / // \3/2
= луу cos [и? (у ) J (у ) •
The substitution y'x = \/и(у) leads to a second order equation of the form 2.7.5.5:
A ' „,w,,/^/2
# \ 2i /• // \4/2
The substitution yj, = \/u(y) leads to a second order equation of the form 2.7.5.6:
uyy = —=y sm(u)u)(u'yK 2.
20 v'" — л •"<-»=^'х""^-"'^3^<"" ^3/2
The substitution y^, = \/u(y) leads to a second order equation of the form 2.7.5.7:
uyy = — cos(wy)u(^K/2.
The substitution y'x = \/u(y) leads to a second order equation of the form 2.7.4.8:
This is a special case of equation 3.5.4.15 with /(y) = A cos™ (tt>y).
This is a special case of equation 3.5.4.15 with /(y) = ^4sinn(tt>y).
24. y^a; = Atann(u)y) y^iyZY™¦
This is a special case of equation 3.5.4.15 with /(y) = ^4tann(tt>y).
25. yZx =
This is a special case of equation 3.5.4.15 with f(y) = Acotn(u>y).
26. tC = A(arcsin y)"y;(yL)m.
This is a special case of equation 3.5.4.15 with f(y) = ^4(arcsiny)n.
27. tC =
This is a special case of equation 3.5.4.15 with f(y) = ^4(arctany)n.
© 1995 by CRC Press, Inc.

3.4. Some Nonlinear Equations with Arbitrary Parameters
3.4.1. Equations Containing Power Functions
Using the transformation given in 3.5.2.21, we reduce this equation to a nonhomoge-
neous constant-coefficient linear equation.
Solution in the parametric form F^0):
f dr
-J Ит)]3/2
where
2- У'" = axy-5'2 + Ьх3у~7/2.
The transformation x = 1/t, у = w/t2 leads to an equation of the form 3.4.1.1:
</t = -aw-5/2 - bw-У2.
3- VmL = (У + ах2 + ЬХ + С)П.
The substitution z = у + ax2 + bx + с leads to the equation z'^xx = zn whose solvable
cases are outlined in Section 3.2.
4- CL = (ax + b)n(cx
The transfoi
Section 3.2):
CLX I h 1J
The transformation f = , w = —. гтг leads to a simpler equation (see
s cx + d (cx + dJ v
w'^ = A-3Cwm, where A = ad -be.
This is a special case of equation 3.5.2.18 with /(?) = 1.
This is a special case of equation 3.5.1.9 with /(?) = Ь?~п.
This is a special case of equation 3.5.2.2 with /(?) = a?n.
8- y^'L = ax-2n-4yny'x - 2ax-2n-5yn+1.
This is a special case of equation 3.5.2.3 with /(?) = a^n.
The transformation ж = /[</5(т)]~3/2с?т, у = ['/'(т)] leads to a constant coefficient
equation: (z5"^ — \<p'T + bip + a = 0.
© 1995 by CRC Press, Inc.

This is a special case of equation 3.5.2.20 with /(?) = a.
This is a special case of equation 3.5.2.19 with /(?) = a.
12. tC = oy"y4 + bym(y'xK.
This is a special case of equation 3.5.2.4 with f(y) = ayn, g(y) = bym.
This is a special case of equation 3.5.2.9 with f(y) = byn.
m+5
tC = (ax2 + bx + c) 4 (y^)"\
This is a special case of equation 3.5.2.17 with /(?) = ^m.
4 - У)"-
This is a special case of equation 3.5.2.13 with /(?) = a?n.
This is a special case of equation 3.5.2.14 with /(?) = a^n.
^ - 2y)n.
This is a special case of equation 3.5.2.16 with /(?) = a^n.
18- tC = аж2"-7у-"(Жу; - 2yK.
This is a special case of equation 3.5.2.6 with /(?) = a^~n.
19- y^'L = axn-5y-n(xyx - yf.
This is a special case of equation 3.5.2.5 with /(?) = a^~n.
20. tC = ax"j,m(^ - 2у)г.
The transformation t = ж, z = yx~2 leads to the equation
/// _ / i\ij.-n-2m-i-4 m/ /\'
zm — -ч-1-)г z (zt)
which is discussed in Section 3.2.
21- sctC, + 3y^ = axnyn.
The substitution w(x) = xy leads to the equation w'J.'xx = awn which is discussed in
Section 3.2.
© 1995 by CRC Press, Inc.

22. xyZx = -Ых + ax-n-2y2nBxy'x - у).
This is a special case of equation 3.5.3.13 with /(?) = a?2n.
23. xyZx = ~\v'L + ax-n-3y2nBxy'x - уK.
This is a special case of equation 3.5.3.16 with /(?) = a?2n.
24. xyZx + V'L = ax—4xy'x - y)n.
This is a special case of equation 3.5.3.15 with /(?) = a?n.
25. жу^ + A - a)jC = bx2«(xy'x - y)n.
This is a special case of equation 3.5.3.6 with /(?) = 6?n.
26. aj2tC + бжу^ + 6y'x = ax2nyn.
The substitution w(x) = x2y leads to the equation w'^.'xx = awn which is discussed in
Section 3.2.
27. yyixx + \VXVL +
The transformation x = x(t), y= (x't) leads to a nonhomogeneous constant-coefficient
linear equation of the fourth order of the form 4.1.2.2: 2rr"t"t = ax + b.
28. yyZx + \v'xy'L =
The transformation x = x(t), у = (x'tJ leads to the equation of the form 4.2.1.1:
/
29. yyZx + \v'xy'L = kVv y'L
The transformation x = x(t), у = (x't) leads to a constant coefficient linear equation
of the fourth order: 2rr"t"t = ±2/сж"/4 + 2mx"t ± ax't + bx + c, where "+" corresponds
to x't > 0, and "—" corresponds to x't < 0.
30. yyZx + ЪУ'ХУХХ + axnyy'x = bxm.
This is a special case of equation 3.5.3.19 with f(x) = ax", g(x) = bxm.
31- t/tC + 3y'xyZ + a [yyZ + (y'xY] = bxn.
This is a special case of equation 3.5.3.20 with f(x) = bxn.
2 о
32. yyZx ~1~ (ЗУж "I" 2oy)y^,a, + 2а(ух) + а уух = Ьж".
This is a special case of equation 3.5.3.33 with /(ж) = eax, g(x) = bxneax.
33. yyZx + Wx + axny)yZ + axn(y'xJ = 0.
This is a special case of equation 3.5.3.21 with f(x) = axn.
34. (y + a)yZx + by'xyZ + cyny'x = 0.
This is a special case of equation 3.5.3.25 with f(y) = cyn.
© 1995 by CRC Press, Inc.

35. (y + ax + b)y'Zx + Чу'х + a)yZ = cxn.
This is a special case of equation 3.5.3.27 with f(x) = cxn.
36. x(yyZx + Zy'xy'L) + a [yyZ + (y'xJ] = bxn.
This is a special case of equation 3.5.3.28 with f(x) = bxn.
37. x2yyZx + xCxy'x + 2ay)y'x-x + 2ax(y'xJ + o(o - l)yy'x = bxn.
This is a special case of equation 3.5.3.33 with f(x) = xa, g(x) = bxn+a~2.
38. y2y'Zx - ^yy'xy'L + 2(y'xf = <™ny3.
This is a special case of equation 3.5.3.29 with f(x) = axn.
39. y2yZx + 3™УУХУХХ + m{m - l)(|?)s = axky2~m.
This is a special case of equation 3.5.3.30 with f(x) = axk, n = m + 1.
40- 2y'xyZx ~ {y'Lf = КУ'Х? + ay2 + by + с
Differentiating with respect to x and dividing by y'x, we arrive at a constant coefficient
linear equation: 2y'^xx = 2\y'^,x + 2ay + b.
This is a special case of equation 3.5.4.6 with /(?) = a?n.
42- 2y'xyZx Zf ^**^'T
This is a special case of equation 3.5.4.8 with /(?)
43- 2y'xVZx ~
This is a special case of equation 3.5.4.7 with /(?)
44. 2y'xVZx ~
This is a special case of equation 3.5.4.3 with f(x) = axn, g(y) = bym.
45- 2y'xVZx ~
This is a special case of equation 3.5.4.5 with f(y) = ayn, g(y) = by
46- 2y'xVZx ~
This is a special case of equation 3.5.4.4 with f(x) = ax", g(x) = bx
47. 2y'xyZx ~ (yZf = *хп(УхJ + ay2 + 2by + с
This is a special case of equation 3.5.4.1 with f(x) = —Xxn.
© 1995 by CRC Press, Inc.

48- xy'xyZx ~ Зж(у1J + Зу'ху^х = axyn(y'xL + bym(y'xf.
This is a special case of equation 3.5.4.10 with f(y) = ayn, g(y) = bym.
49. yZx = <™-2n4L3
This is a special case of equation 3.5.4.13 with /(?) = a?n.
50. tC = ax-4n-5(xy'x - yTiy'Lf-
This is a special case of equation 3.5.4.12 with /(?) = a?n.
4 - у)"] (y'Lf-
This is a special case of equation 3.5.4.11 with /(?) = 6?n.
52. tC = Mt?)" + by(t?)m + c(t?)h] (у!K + *ШЧу1J-
This is a special case of equation 3.5.4.14 with /(?) = af\ #(?) = &?m, /i(?) = c?fe,
53. жу^ + y^ = axn(xy'x - у)т(у^я)п.
This is a special case of equation 3.5.5.5 with /(?) = a^m, #(?) = ^n.
54. y4'L 4fc
The Legendre transformation x = w't, у = tw't — w leads to the equation w'l[t =
-Atmwl(w't)n(w't'tK~k which is discussed in Section 3.2.
This is a special case of equation 3.5.4.16 with /(?) = a?n,
56. г
This is a special case of equation 3.5.5.7 with /(?) =
57. (y'" J ^ a(x2y" — 2xy' -\- 2y) + by" -\- с
Differentiating with respect to x, we obtain
Equating the second factor to zero and integrating, we find the solution
ax ox _, о „ о ^ч ^ч
У = 720" + ~48~ + 3 + 2 + lX + °'
The integration constants Cj and parameters a, b, and с are related by
36Cf = 2aC0 + 2bC2 + с
This constraint is obtained by substituting the above solution into the original equa-
equation. In addition^to the first factor corresponds the solution у = С^х2 + C\X + Co,
where constants Ci are related by 2aCo + 2bC<i + с = 0.
© 1995 by CRC Press, Inc.

3.4.2. Equations Containing Exponential Functions
!• tC = o(l/ + be' + c)n - bex.
The substitution w = у + bex + с leads to the equation w'xxx = awn whose solvable
cases are outlined in Section 3.2.
2- tC = аехУу'х
This is a special case of equation 3.5.2.4 with f(y) = aeXy', g(y) = 6eM3/.
3- tC = «eAs/i + byn(y'xK.
This is a special case of equation 3.5.2.4 with f(y) = aeXy',
4- <L = аУпУ'х
This is a special case of equation 3.5.2.4 with f(y) = ayn, g(y) = beXy.
5- vZx
This is a special case of equation 3.5.2.9 with f(y) = beXy.
6- tC = 2A2«K + ae*m»(t?r~5-
This is a special case of equation 3.5.2.23 with /(?) = a^m
7- <L = -3y^ + aemxymy'x + aemxym+1 + 1y.
This is a special case of equation 3.5.3.3 with /(?) = a^m.
8- <L 23 ^
This is a special case of equation 3.5.3.2 with f(x) = ae@x.
9- tC 23 х
This is a special case of equation 3.5.3.2 with f(x) = axn.
This is a special case of equation 3.5.3.7 with /(?) = Ь?п.
Solution: y2 = C2x2 + Cxx + Co + 2аХ~3еХх.
This is a special case of equation 3.5.3.19 with f(x) = aeXx, g(x) = belxx.
13. yy'" + 3y' y" -\- aeXxyyf = bxn.
This is a special case of equation 3.5.3.19 with f(x) = aeXx, g(x) = bxn.
© 1995 by CRC Press, Inc.

yvZx ~l~ ^y'xvZ ~l~ ахПУУх = beXx.
This is a special case of equation 3.5.3.19 with f(x) = axn, g(x) = beXx.
. yyZx + zvLv'L + о [yy'L + Ш] = beXx-
This is a special case of equation 3.5.3.20 with f(x) = beXx.
16. yy'Zx ~l~ Cy'x ~\~ 2ay)yZ ~\~ 2a(y'x) -\- a2yyx = beXx.
This is a special case of equation 3.5.3.33 with /(ж) = eax, g(x) =
17- V»^ + Cy'x + aeXxy)yZ + aeXx(y'xJ = 0.
This is a special case of equation 3.5.3.21 with f(x) = aeXx.
18. (y + o)y'" + by' y" -\- ceXyy' = 0.
This is a special case of equation 3.5.3.25 with f(y) = ceXy.
19. (y + ax + b)yZx + Чу'х + a)yZ = keXx.
Solution: (y + ax + bf = C2x2 + Cxx + Co + 2k\~3eXx.
20. yyZx-
Solution: In \y\ = C2x2 + Cxx + Co + a\~3eXx.
21. y2yZx "I" ^rayy'xyZ "I" "m(m — l)(y^,) = aeXxy2~m.
This is a special case of equation 3.5.3.30 with f(x) = aeXx, n = m + 1.
22. x2yyZx + xCxy'x + 2ay)yZ + 2ax(y'xJ + o(o -
= bxa~2eXx
This is a special case of equation 3.5.3.33 with f(x) = xa, g(x) = bxa~2e
23. 2y'xyZx ~ (yZf = keXx(y'xJ + ay2 + 2by + с
This is a special case of equation 3.5.4.1 with f(x) = —keXx.
24. 2y'xyZx ~
This is a special case of equation 3.5.4.3 with f(x) = aeXx, g(y) = 6eM3/.
25. 2y'xVZx ~ 3(yZf = aex*(y'xJ + Ьу™{у'х)\
This is a special case of equation 3.5.4.3 with f(x) = aeXx, g(y) = bym.
26. 2y'xVZx ~
This is a special case of equation 3.5.4.3 with f(x) = ax", g(y) = be1111.
27. 2y'xyZx ~ HvZf = aex*(yxJ
This is a special case of equation 3.5.4.4 with f(x) = aeXx, g(x) = belxx
© 1995 by CRC Press, Inc.

28. 2y'xyZx HVL? (yx? + y\yxf
This is a special case of equation 3.5.4.4 with /(ж) = aeXx, g(x) = bxn
29. 2y'xyZx ~ Hv'Lf = ™пЮ2 + be^y-\y'J>''1'.
This is a special case of equation 3.5.4.4 with /(ж) = ax", g(x) = beXx.
30. 2y'xyZx ~ Hv'Lf =
This is a special case of equation 3.5.4.5 with f(y) = aeXy, g(y) = be11
This is a special case of equation 3.5.4.5 with f(y) = aeXy, g(y) = by1
32. 2y'xyZx
This is a special case of equation 3.5.4.5 with f(y) = ayn, g(y) = beXy.
3.4.3. Equations Containing Hyperbolic Functions
1- tC = acoshn(Xy)yx + bcosh(/iy)(y;K.
This is a special case of equation 3.5.2.4 with f(y) = acoshn(Ay), g(y) = bcosh(/iy).
2- VZX = аУпУ'х + ЬсовЬ(цу)(ухK.
This is a special case of equation 3.5.2.4 with f(y) = ayn, g(y) = bcosh(/iy).
3- tC = asinhn(Xy)yx + bsinh(/iy)(y4K.
This is a special case of equation 3.5.2.4 with f(y) = asinhn(Ay), g(y) = bsinh^/iy).
4- tC = bcosh(Xy)(yxK + a(y'x)-5.
This is a special case of equation 3.5.2.9 with f(y) = bcosh(Xy).
5- VZX = ТЛ2Ш3 +
This is a special case of equation 3.5.2.24 with /(?) = a?2m.
6- VZX = i^(v'xf + a(sinhAy)— 4v'xJm+1-
This is a special case of equation 3.5.2.25 with /(?) = a^2m.
7- VVZX + 3У'ХУХХ = о cosh( Аж).
Solution: у2 = C2x2 + dx + Co + 2aA sinh(Arr).
8- УУХХХ + 3У'ХУХХ = о sinh( Аж).
Solution: у2 = C2x2 + dx + Co + 2aA cosh(Arr).
© 1995 by CRC Press, Inc.

9- УУХ'ХХ + 3УХУХХ = ocosh"(Aa;).
This is a special case of equation 3.5.3.17 with f(x) = acoshn(Ax).
Ю- УУХХХ + 3УХУХХ = a sinh"(Aa;).
This is a special case of equation 3.5.3.17 with f(x) = asinhn(Aa;).
И- УУ'ХХХ + 3У'ХУХХ = ° tanh"(Aa;).
This is a special case of equation 3.5.3.17 with f(x) = atanhn(Aa;).
12- УУ'ХХХ + 3УХУХХ + axnyy'x = b cosh™(Aa;).
This is a special case of equation 3.5.3.19 with f(x) = ax", g(x) = 6coshm(Ax)
This is a special case of equation 3.5.3.19 with f(x) = ax", g(x) = 6sinhm(Aa:).
t/tC + Zy'xy'L + axnyy'x = b tanhm (Xx).
This is a special case of equation 3.5.3.19 with f(x) = ax", g{x) = 6tanhm(Aa:).
15- t/tC + Zy'xy'L + о [yy'L + (y'x)\ = b со3Ь"(ЛЖ).
/„." j-J.,».» ^t..'\^ -„^UDU K^h
This is a special case of equation 3.5.3.20 with f(x) = 6coshn(Ax).
16. УУ'1'ХХ + ЗУ'ХУХХ + а[УУхх + Ш2] = bsinhn(Xx).
This is a special case of equation 3.5.3.20 with f(x) = 6sinhn(Aa;).
17- t/tC + Zy'xy'L + aWL + Ш2] = btanhn(\x).
This is a special case of equation 3.5.3.20 with f(x) = 6tanhn(Aa;).
is- уу'ххх + Cу'х + аУ cosh" x)y'L + а cosh" x(v'xJ = o-
This is a special case of equation 3.5.3.21 with f(x) = a cosh™ x.
19- t/tC + Wx + ay sinh" x)y'^x + a sinh" x(y'xJ = 0.
This is a special case of equation 3.5.3.21 with f(x) = asinh™ x.
20- уу'ххх + Cу'х + avtanh" x)y'L + a tanh" хЮ2 = о-
This is a special case of equation 3.5.3.21 with f(x) = atanh™ x.
21. (y + a)yZx + Ъу'ху'1х + с cosh"(Xy)y'x = 0.
This is a special case of equation 3.5.3.25 with f(y) = ccoshn(Ay).
22. (y + a)y'l'xx + by'xy'lx + с sinhn (Xy)y'x = 0.
This is a special case of equation 3.5.3.25 with f(y) = csinhn(Ay).
© 1995 by CRC Press, Inc.

23. (у + a)yZx + WXV'L + с tanh"(Ay)y; = 0.
This is a special case of equation 3.5.3.25 with f(y) = ctanhn(Ay).
24. y2y'Zx + ZmyvLvL + m{m - l)(y'xf = a coshk (Xx)y2~m.
This is a special case of equation 3.5.3.30 with f(x) = acoshk(Xx), n = m + 1.
25. y2yZx + Z™yy'xy'L + m{m - l)(y'xf = asinhfc(\x)y2-™.
This is a special case of equation 3.5.3.30 with f(x) = asinh (Xx), n = m + 1.
26. y2yZx + *™УУХУХХ + rn{m - l)(y'xf = k
This is a special case of equation 3.5.3.30 with f(x) = atanh (Xx), n = m + 1.
27. 2y'xy'Zx - 3(yZf = acoshn(Xx)(yxJ + Ъу™(у'х)\
This is a special case of equation 3.5.4.3 with f(x) = acoshn(Arr), g(y) = bym.
28. 2y'xy'Zx - 3(yZr = ах"(у'хУ + Ьcosh™(\у)(у'хГ.
This is a special case of equation 3.5.4.3 with f(x) = ax", g(y) = 6coshm(Ay).
29. 2y'xy'Zx - 3(yZf = atanhn(Xx)(y'xJ + Ьtanhm(/xy)(y'xL.
This is a special case of equation 3.5.4.3 with f(x) =atanhn(Aa;), g(y) = 6tanhm(/xy).
30. 2y'xy'Zx - 3(yZr = atanhn(\x)(y'xy + Ъу™(у'х)\
This is a special case of equation 3.5.4.3 with f(x) = atanhn(Aa;), g(y) = bym.
31- 2y'xy'Zx - 3(yZf = ax™(y'xJ + btanhm(Xy)(y'xL.
This is a special case of equation 3.5.4.3 with f(x) = ax", g(y) = 6tanhm(Ay).
32. 2y'xy'Zx - 3(yZf = acoshn(Xx)(y'xJ
This is a special case of equation 3.5.4.4 with f(x) =acoshn(Arr), g(x) = bcoshm(/ix).
33. 2y'xy'Zx - 3(yZJ = acoshn(Xx)(yxJ + Ъх™у-\ух)ъ'2.
This is a special case of equation 3.5.4.4 with f(x) = acoshn(Arr), g(x) = bx
34. 2y'xy'Zx - 3{yZ? = ax"(y'xJ
This is a special case of equation 3.5.4.4 with f(x) = ax", g(x) = 6coshm(Aa:).
35. 2y'xy'Zx - 3(yZf = atanhn(Xx)(y'xJ
This is a special case of equation 3.5.4.4 with f(x) =atanhn(Aa;), g(x) = btanhm(p,x).
36. 2y'xy'Zx - 3{yZ? = atanhn(Xx)(y'xJ + bx™y-\y'xf12.
This is a special case of equation 3.5.4.4 with f(x) = atanhn(Aa;), g(x) = bxm.
© 1995 by CRC Press, Inc.

37. 2y'xyZx
This is a special case of equation 3.5.4.4 with /(ж) = ax", g(x) = 6tanhm(Aa:).
38. 2y'xyZx ~ Hv'Lf = acoshn(Xy)(yxL + bx^ coshm^y)(yxO/2.
This is a special case of equation 3.5.4.5 with f(y) = a coshn(\y), g(y) = b coshm(/лу).
39. 2y'xyZx ~ Hv'Lf = acoshn(Xy)(yxL + bx^ym(yxO/2.
This is a special case of equation 3.5.4.5 with f(y) = acoshn(Ay), g(y) = bym.
40- 2y'xyZx ~ 4V'L? = ayn(y'xL + Ьх-i coshm(Xy)(yxO/2.
This is a special case of equation 3.5.4.5 with f(y) = ayn, g(y) = bcoshm(Xy).
L + Ьх'1
This is a special case of equation 3.5.4.5 with f(y) = atanhn(Ay), g(y) = btanhm(/iy).
42- 2y'xVZx ~ Hv'Lf =
This is a special case of equation 3.5.4.5 with f(y) = atanhn(Ay), g(y) = by
43- 2y'xVZx ~ Hv'Lf = o,yn{y'xf + Ъх-
This is a special case of equation 3.5.4.5 with f(y) = ayn, g(y) = 6tanhm(Ay).
3.4.4. Equations Containing Logarithmic Functions
1- tC = oln"(Ay)y; + Ыпт(цу)(у'хK.
This is a special case of equation 3.5.2.4 with f(y) = alnn(Ay), g(y) = blnm(/iy).
2- VZX
This is a special case of equation 3.5.2.4 with f(y) = alnn(Ay), g(y) = bym.
3- tC = ау™у'х + Ыпт(Ху)(ухK.
This is a special case of equation 3.5.2.4 with f(y) = ayn, g(y) = blnm(Xy).
4- yZx
This is a special case of equation 3.5.2.11 with /(?) = 2aln?.
5- y^ = oy
This is a special case of equation 3.5.2.12 with /(?)
6- х3у'х'хх = °Aпу - inx)(xv'x — у)-
This is a special case of equation 3.5.2.2 with /(?)
© 1995 by CRC Press, Inc.

7- x5yZx = a(lny-2lnx)(xy'x-2y).
This is a special case of equation 3.5.2.3 with /(?) = aln?.
8- tC = -Зу^, + о(ж + In »)"(»; + у) + 2».
This is a special case of equation 3.5.3.3 with /(?) = a In™ ?
9- «y^'L = b(xy'x - У + a In ж)у4'а;.
This is a special case of equation 3.5.3.8 with /(?) = b?.
Ю- VVZL + Zy'xy'L = a \nn(bx).
This is a special case of equation 3.5.3.17 with /(ж) = alnn(bx).
(\x).
This is a special case of equation 3.5.3.19 with /(ж) = ax", g(x) = 61пт(Аж).
12. yy'Zx + 3y'xy'L + <*[yy'L + Ш2] = b \nn(Xx).
This is a special case of equation 3.5.3.20 with /(ж) = 61пп(Аж).
13. (у + a)y'Zx + Wxy'L + с lnn(Xy)y'x = 0.
This is a special case of equation 3.5.3.25 with f(y) = clnn(Ay).
У2УХХХ + 3™УУ'ХУХХ + m(m - l){y'xf = alnk(bx)y2-m.
This is a special case of equation 3.5.3.30 with /(ж) = alnk(bx), n = m + 1.
2y'xyZx ~ 4y'L? = ОУ4(ЬУ; - 2 In у).
This is a special case of equation 3.5.4.6 with /(?) = aln?.
2y'xyZx ~ 4y'L? = alnn(Xx)(y'xJ +Ыпт(цу)(у'х)\
This is a special case of equation 3.5.4.3 with f(x) = а1пп(Аж), g(y) = Ь\пт{ц,у).
2y'xyZx ~ 4y'L? = alnn(Xx)(y'xJ + by™(y'x)\
This is a special case of equation 3.5.4.3 with f(x) = а1пп(Аж), д(у) = bym.
This is a special case of equation 3.5.4.3 with /(ж) = axn, g(y) = blnm(Xy).
2y'xyZx ~ 4y'L? = alnn(Xx)(y'xJ
This is a special case of equation 3.5.4.4 with /(ж) = а1пп(Аж), д(х) = blnm(/ix).
20. 2y'xyZx ~ 4y'L? =
This is a special case of equation 3.5.4.4 with /(ж) = a In™ ж, д(х) = bxm.
© 1995 by CRC Press, Inc.

This is a special case of equation 3.5.4.4 with /(ж) = их", д(ж) = 61пт(Аж).
22. 2t?tC ? 4 17/2
This is a special case of equation 3.5.4.5 with f(y) = alnn(Ay), g(y) = blnm(iiy).
23. 2y'xyZx ~ 3{V'L? = аупШ* + bx'1 lnm(Xy)(y'x)r/2.
This is a special case of equation 3.5.4.5 with f(y) = ayn, g(y) = 61nm(Ay).
24. 2y'xyZx ~ 3(CJ2 = alnn(Xy)(y'x)
This is a special case of equation 3.5.4.5 with f(y) = alnn(Ay), g(y) = bym.
3.4.5. Equations Containing Trigonometric Functions
1- tC = аУпУ'х + Ьсов(\у)(ухK.
This is a special case of equation 3.5.2.4 with f(y) = ayn, g(y) = bcos(Xy).
2- tC = ayny'x + bsin(Xy)(yxK.
This is a special case of equation 3.5.2.4 with f(y) = ayn, g(y) = 6sin(Ay).
3- tC = ayny'x + btan(Xy)(y'xK.
This is a special case of equation 3.5.2.4 with f(y) = ayn, g(y) = 6tan(Ay).
4- tC = acos"(Ay)y; + bym(y'xf.
This is a special case of equation 3.5.2.4 with f(y) = acosn(Xy), g(y) = bym.
5- tC = asinn(Xy)yx + bym(y'xf.
This is a special case of equation 3.5.2.4 with f(y) = asinn(Ay), g(y) = bym.
6- yZx = atann(Xy)y'x + bym(y'xf.
This is a special case of equation 3.5.2.4 with f(y) = atann(Ay), g(y) = bym.
7- tC = ocos"(Ay + e)y^ + bcos(»y + 6)(y'xK.
This is a special case of equation 3.5.2.4 with f(y) =acosn(Xy+e), g(y) = bcos(/iy+6).
8- tC = bcos"(Ay)(y;K + a(y4)-5.
This is a special case of equation 3.5.2.9 with f(y) = bcosn(Xy).
9- y'Zx = btann(Xy)(y'xK + a(y'x)-5.
This is a special case of equation 3.5.2.9 with f(y) = 6tann(Ay).
© 1995 by CRC Press, Inc.

Ю- tC = -1A2«K + o(cosAy)— 4y'xJm+1-
This is a special case of equation 3.5.2.28 with /(?) = a?2™\
11- tC = -iA2«K + o(sinAy)—3«Jm+1.
This is a special case of equation 3.5.2.29 with /(?) = a?2m.
t/tC + 3УХУХХ =
Solution: y2 = C2x2 + Cxx + Co - 2aA sin(Arr).
This is a special case of equation 3.5.3.17 with /(ж) = авт"(Аж).
14- t/tC + Зу^у! = a tan"(Лж).
This is a special case of equation 3.5.3.17 with f(x) = atann(Arr).
15- Vy'vL + 3Vxy'L + asin(Xx)yy'x = bsinn(nx).
This is a special case of equation 3.5.3.19 with /(ж) = asin(Aa;), g(x) = bsmn(p,x).
= Ьхп.
This is a special case of equation 3.5.3.19 with f(x) = asin(Aa;), g(x) = bxn.
17. yy'Z* + 3y'xy'L + axnyy'x = b cosm(Aaj).
This is a special case of equation 3.5.3.19 with /(ж) = ax", g{x) = bcosm(Xx).
18- VVvL + *y'xy'L + axnyy'x = b ъ\тГ(\х).
This is a special case of equation 3.5.3.19 with /(ж) = ax", g(x) = 6sinm(Aa:).
This is a special case of equation 3.5.3.19 with /(ж) = ax", g(x) = ?^апт(Аж).
20. yy'l'xx + 3У'ХУХХ + а[УУхх + Ю2] = bcosn(Xx).
This is a special case of equation 3.5.3.20 with /(ж) = 6совп(Аж).
21- УУХХХ + 3У'ХУХХ + о [УУХХ + Ю2] = Ь tann(Xx).
This is a special case of equation 3.5.3.20 with /(ж) = ?^апп(Аж).
22. yy'Zx + C»; + ay sin" ж)у1 + о sin" Ж(у4J = О.
This is a special case of equation 3.5.3.21 with /(ж) = a sin™ ж.
23. t/tC + Cy'x + ay tan" x)y'^x + a tan" «(y^J = 0.
This is a special case of equation 3.5.3.21 with /(ж) = a tan™ ж.
© 1995 by CRC Press, Inc.

24. (у + a)yZxx + by'xyZ + с cosn(Xy)yx = 0.
This is a special case of equation 3.5.3.25 with f(y) = ccosn(Xy).
25. (y + a)yZxx + by'xyZ + с tann(Xy)y'x = 0.
This is a special case of equation 3.5.3.25 with f(y) = ctann(Ay).
25. y2yZxx + 3гпУУ'хУ'Хх + rn{m — l)(y'xK = a cosk (Xx)y2~m.
This is a special case of equation 3.5.3.30 with f(x) = acosk(Xx), n = m + 1.
26- V2VZX + 3™yy'xy'L + m(m ~ !)Ш3 = atanfc(Aa;)y2-"\
This is a special case of equation 3.5.3.30 with f(x) = atanfe(Aa;), n = m + 1.
27. 2y'xyZx ~ Hy'ZJ = acos(\x + e)(y'xf + bcos^y + 6)(y'x)\
This is a special case of equation 3.5.4.3 with f(x) =acos(Xx + e), g(y) = b cos(/iy+6).
28. 2y'xyZx ~ 4y'L? = acos(Xx)(yxJ + Ьу™{у'х)\
This is a special case of equation 3.5.4.3 with f(x) = acos(Xx), g(y) = bym.
29. 2y'xyZx ~ 3(yZf = <™пШ2 + bcos(Xy)(y'x)\
This is a special case of equation 3.5.4.3 with f(x) = ax", g(y) = bcos(Xy).
30. 2y'xyZxx - 3(yZJ = atann(Xx)(y'xJ + Ьtan™(fiy)(y'xL.
This is a special case of equation 3.5.4.3 with f(x) = atann(Arr), g(y) = btanm(p,y).
- 2y'xVZxx - 3(yZf = аЫп™(Хх)(у'хJ + Ьу™{у'х)\
This is a special case of equation 3.5.4.3 with f(x) = atann(Arr), g(y) = bym.
32. 2y'xVZxx - 3(yZJ = ax"(y'xJ + btan™(Xy)(y'x)\
This is a special case of equation 3.5.4.3 with f(x) = ax", g(y) = bta,nm(Xy).
33. 2y'xyZxx - 3(yZJ = asinn(Xx)(y'xJ
This is a special case of equation 3.5.4.4 with f(x) = asinn(Aa;), g(x) = bsmm(p,x).
34. 2y'xVZxx - 3(yZJ = acos™(Xx)(y'xJ + bxmy-\y'xf/2.
This is a special case of equation 3.5.4.4 with f(x) = acosn(Xx), g(x) = bxm.
35. 2y'xVZxx - 3(yZJ = ax"(y'xJ
This is a special case of equation 3.5.4.4 with f(x) = ax", g(x) = bcosm(Xx).
36. 2y'xVZxx - 3(yZJ = atan"(Xx)(y'xJ
This is a special case of equation 3.5.4.4 with f(x) = atann(Arr), g(x) = bta,nm(iix).
© 1995 by CRC Press, Inc.

37. 2y'xyZx
This is a special case of equation 3.5.4.4 with /(ж) = atann(Arr), g(x) = bx
38. 2y'xyZx
This is a special case of equation 3.5.4.4 with /(ж) = ax", g(x) = 6tanm(Aa:).
39. 2y'xyZx ~ 4V'L? = acos"(\y)(yxL + б* cos™(ny)(yx)r/2.
This is a special case of equation 3.5.4.5 with f(y) = acosn(Xy), g(y) = bcosm(iiy).
40- 2y'xyZx ~ 4V'L? = acos"(\y)(yxL + bx^ym{y'xO/2.
This is a special case of equation 3.5.4.5 with f(y) = acosn(Xy), g(y) = bym.
41- 2y'xyZx ~ Hv'Lf = o,yn{y'xf + Ъх-i cos™(\y)(yxO/2.
This is a special case of equation 3.5.4.5 with f(y) = ayn, g(y) = bcosm(Xy).
42- 2y'xVZx ~ Hv'Lf =
This is a special case of equation 3.5.4.5 with f(y) = atann(Ay), g(y) = 6tanm(/xy).
43- 2y'xVZx ~
This is a special case of equation 3.5.4.5 with f(y) = atann(Ay), g(y) = by
44. 2y'xVZx ~ Hv'Lf = o,yn{y'xf + Ъх-i tan-(Ay)(y4O/2.
This is a special case of equation 3.5.4.5 with f(y) = ayn, g(y) = 6tanm(Ay).
3.5. Nonlinear Equations Containing Arbitrary Functions
3.5.1. Equations of the Form F(x,y)y"'xx + G(x,y) = 0
The substitution w(y) = (y'x) leads to a second order equation: w'yy = ±2f(y)w~1/2.
In particular, with f(y) = ayn the obtained equation is the Emden—Fowler equation
which is discussed in Section 2.3.
2- tC = f(x)y-\
Having integrated the equation, we have
The substitution у = w2 reduces the latter equation to the form
© 1995 by CRC Press, Inc.

The substitution t = In |ж| leads to an autonomous equation of the form 3.5.5.9:
4-
5-
6-
7-
8.
The transformation t = In |ж|, w = yx~x leads to an autonomous equation of the form
3.5.5.9: w'l[t-w't = f(w).
The transformation t = ж, г<7 = yx~2 leads to an autonomous equation of the form
3.5.1.1: <t = -f(w).
VZX = f(y + ax3 + bx2+cx + k).
The substitution w = у + ax3 + bx2 + cx + k leads to an autonomous equation of the
form 3.5.1.1: w'^'xx = f(w) + 6a.
tC = f(y + oeA-)
The substitution w = у + ае^х leads to an autonomous equation of the form 3.5.1.1:
wx'xx = f(W)-
ax2
x + c)yZx = f(x).
The substitution w = у + ax2 + bx + с leads to an autonomous equation of the form
3.5.1.2: wv%'xx = f(x).
9. x(x — aKyZx =
The transformation ? = In
x — a
x
У
, w = —7г leads to an autonomous equation of the
10.
form 3.5.5.9: w'^ - 3w'^ + 2w'( = a~3f(w).
(ax2 + bx + cfyZx = f ! V
The transformation
f У. V
\ ax2 + bx + с '
dx
w =
У
ax2 + bx + с ' w ax2 + bx + с
leads to an autonomous equation of the form 3.5.5.9: w'/i^ + Dac — b )wi = f(w).
11. (ay + be*)yZx + be*y = f(x).
Integrating, we obtain
(ay + bex)y^x - \a(y'xf - be*y'x + be*y = J f(x) dx + С
12- vZx = F(x> у)-
1 W
The transformation x = —, у = —^ leads to an equation of the analogous form:
w
© 1995 by CRC Press, Inc.

3.5.2. Equations of the Form F{x,y,y'x)y'x"xx + G(x,y,y'x) = 0
* ** XXX J V c/ / if X '
Solution:
C3±x= j\c2y + C1+2 jF(y)dy\ dy, where F(y) = f f(y)dy.
2. y"
\2
X \'~ X i
Integrating the latter twice, we obtain a first order autonomous equation:
XI / XI \ ^
The transformation z= —, w = \y' 1 leads to a second order linear equation:
x V x J
11/2
(z-t)f(t)dt\ ,
where z = y/x, zq is an arbitrary number.
3. y;'L = ^-5
\ у
The transformation t = —, z = —5- leads to the equation z'Ht = f(z)z't. The substi-
X X
tution w(z) = (ZjJ next yields a second order linear equation: wzz = 2/(z), whose
solution has the form
Г
w = C2z + C\ + 2 / (z — C)/(C) d?> zo is arbitrary.
Ло
4- yZx
The substitution z(y) = (y'x) leads to a second order linear equation: z'yy = 2g(y)z +
5-
У / У \2
The transformation z= —, w = \y' 1 leads to a second order linear equation:
x V x J
w'z'z = 2f(z)w + 2.
This is a special case of equation 3.5.2.8.
7- y?L = ж/(|")(ж^ - У) +ж-59(|-)(Жу4 - уK.
This is a special case of equation 3.5.3.14 with к = —1.
The transformation t = In ж, z = y/x followed by the substitution w(z) = (z't)
yields a second order linear equation: wzz = 2g(z)w + 2/(z) + 2.
© 1995 by CRC Press, Inc.

8- VZX = ж~5/( Jr)^ - 2y) + x-7g(^)(xy'x - 2yf.
1 у
The transformation t = —, z = —=- followed by the substitution w(z) = (z'tJ leads
xx1
to a second order linear equation: w"z = 2g(z)w + 2f(z).
у, с/ —
The substitution z(y) = (y'xJ leads to Yermakov's equation 2.9.1.12: z'yy = 2f(y)z +
2az~3.
* ^xxx J >¦ if x / "
Solution in the parametric form:
rT dr rT и- Лн- г г ~\1/2
п.
X lc2 4>{t) '
The substitution w(y) = (y'x) leads to an equation of the form 2.9.1.5:
12. v'" V-5/4f( V
1/4
if
The substitution w(y) = (yfx) leads to an equation of the form 2.9.1.6:
<y = y/2jF(^)' where F® = ±r1/2/(±c1/2).
. tC = xf(xy'x - y).
The substitution z = xy'x —y leads to a second order equation of the form 2.9.2.4 with
n = 1: ж.2^ = -4 + ж3/B)-
X
The substitution z = xy'x — у leads to a second order equation: xzxx = z'x + F(z/x),
which is the special case of the equation 2.9.4.12 with n = —1, m = 1, к = —1,
= ?/(?)•
The transformation t = In |ж|, z = xy'x —y leads to a second order autonomous equation:
z"t — 2z't = f(z), which is reduced, with the aid of the substitution w(z) = \z'v the
Abel equation ww'z — w = \f{z) (for some functions /, the solutions of the latter
equation are given in Subsection 1.3.1).
© 1995 by CRC Press, Inc.

The transformation t = —, z = —j" yields z"{t = —f(—z't). The substitution w = —z't
X X
leads to a second order equation of the form 2.9.1.1: w"t = f(w).
»y = w-zF(— W ), where
The substitution w(y) = (y'x) leads to an equation of the form 2.9.1.9:
—
\/ay2
еж4)
2 , . 2 , 4Л-6/4./ Xy'x-2y
+ Ьж^у + еж4) /
{ay2 + bx2y + еж4I/4
The transformation t = —, z = —5- leads to an equation of the form 3.5.2.17:
(az2 + bz +1
СУ'Х ~ У \
у'" = —x~2y' + x~3y + x~3/4y~5/4f
The transformation t = In ж, z = y/x followed by the substitution w(z) = (z't) leads
to a second order equation of the form 2.9.1.6:
«4 = z-^Fiwz-1'2), where F(?) = ±Г1/2/(±\/|)-
20. y'xxx = -x~2y'x + x~3y + x1/2y
= -x~2y' +х~3у + x^y-Wf (XV* V
The transformation t = In ж, z = y/x followed by the substitution w(z) = (z't) leads
to a second order equation of the form 2.9.1.5:
w';z=z-3F(w/z), where F® = ±2Г1/2/(±\/С)-
This is a special case of the equation 3.5.5.9. The transformation
, у=Ыт)}-1 A)
leads to an analogous equation with respect to function ip = (р(т):
Note two important cases of transforming equations of a special form:
© 1995 by CRC Press, Inc.

22 «'" -J-f(JL „' _2-^
22- У^- ^/^^ У, 2
1 у
The transformation ? = —, z = —тг leads to an equation of the form 3.5.5.9: z't"t =
X X
—f(z, —z't), which admits, with the aid of the substitution w(z) = (z't) , lowering of
its order: <г = T^w-^2f(z, Tw1/2).
23. у'ххх = 2X2(y'xy +
This is a special case of equation 3.5.2.32 with ф(у) = e~2Xy.
24. y'xxx = |A2(y;K + (cosh Ay)~3/ ( Vx )y'x.
\ a/cosh Ay /
This is a special case of equation 3.5.2.32 with ф(у) = cosh Ay.
25. y'^ = ±\2(y'xf + (sinh^)-3j( Vx —]yl
V \/sinh Ay /
This is a special case of equation 3.5.2.32 with ф(у) = sinh Ay.
26. «"' = (s:
This is a special case of equation 3.5.2.32 with ф{у) = cothy.
27. yZx = - (cosh y)-2(y'xf + (cothyK f(yx^olhy~)yx.
This is a special case of equation 3.5.2.32 with ф(у) = tanhy.
28- y':L = -|А2(у;K + (cos
/cos Ay /
This is a special case of equation 3.5.2.32 with ф(у) = cos Ay.
29-
sin Ay
This is a special case of equation 3.5.2.32 with ф(у) = sin Ay.
30. |C = (siny)-2(yxK + (tanyKf(yxVt^y~)yx.
This is a special case of equation 3.5.2.32 with ф(у) = cot у.
31- tC = (cosy)-2(y;K + (cotyKf(yxVcbt^)yx.
This is a special case of equation 3.5.2.32 with ф(у) = tan у.
32.
The substitution z = (y'x) leads to a second order equation of the form 2.9.1.14:
yy I
ф
© 1995 by CRC Press, Inc.

33. [ay + f(x)]yZx + 9(У)У'Х + fZJx)y + h(x) = 0.
The equation admits the first integral:
[ay + f{x)}y'lx - \a{y'xJ - fx(x)y'x + fx'x(x)y + J g(y) dy + J h(x) d
34.
x = С
Let F ф (р(х)у'х + ф(х)у + %(ж), i.e., the equation is a nonlinear one. Then, its order
can be lowered by one if the right-hand side of the equation has the following form:
7-1/ / \ ? — 2 p Ж/ \, I [c\ ? ?ttt i / _f2 Pffff , C\ ? ?l ?lll \
F{x, y, yx) = f Ei Ф(и, w)+ [2ffxxxw + (/ fxxxx + 2ffJxxx)u
- Bfx'xxg + !дххх)Е~х - Bffxfx'xx + f2fxxxx - 2kffx"xx)V] dx\, A)
where
/ r . \
V =
u = f-1E-1y + V, w = f-'E-Hfy'x ~ fxV + 9)~ kV;
ф = Ф(и, w), f = f(x), g = g(x) are arbitrary functions, к is an arbitrary constant.
The integral in A) may always be expressed in terms of E and V. The following
cases are possible:
1) For fx'xx ф 0,
f = г2еф(щ w) + r2[Vf'L - (ЛШ + Г3[ff'JL - Vf'J'L + (ЛЛу
+ Г3 {[ffxx - Ю2]д + fU'Jx - fg'L) + W* - fxg -
2) For / = ax2 + bx + с, а ф 0,
F = Г2ЕФ(и, w) + / [ffxg'x - ffx'xg - fg'^ + k(fg'x - fxg - kf)] + k(k2 + Д) V,
where A = 4ac — b2.
3) For / = ax + b, афк, аф—\к,
F = Гк/а~2Чи, w) - Г3 [fg'L - (a + k)fg'x + k(a + k)g] - k(a2 - k2)rk/a~2U,
where
и = !к1а~ху + U, w = fk/ay'x - !к1а~хд +(a-k)U, U= Г fk/a~2g dx.
4) For / = kx + b,
F = Г3 [Ф(и, w) - ?&x + 2kfg'x - 2k2},
where
r
u = y + W, w = fy'x+g, W=jf~1gdx.
© 1995 by CRC Press, Inc.

5) For / = -\кх + Ъ,
, (kx-2bJg''+k(kx-2b)g'+2k2g
F = Ф(и, w) + — ?..
where
u=(kx-2b)~3y-2Q, w = (kx-2b)-2y'x-2(kx-2b)-3g-6kQ, Q= f ¦
J
gdx
(kx - 2bL
6) For / = ax + b, g = с = const, к = О,
F= I ф
( bJ
аУ~С
(ax + bJ \ax
7) For / = kx + b, g = 0,
Ух
In all these cases, the transformation
t = Jf~1dx,
leads to an autonomous equation:
u"{t - 3ku"t + 3k2u't - k3u = Ф(и, u't - ku),
which is reducible, with the aid of the substitution z(u) = u't, to a second order
equation:
z2zlu + z(z'uf - 3kzz'u + 3k2z - k3u = Ф(и, z - ku).
Remark. The original equation can be reduced, with the aid of point transfor-
transformations, to an autonomous form only for function F of the form A).
3.5.3. Equations of the Form
F(x, y, y'Jy'^x + G(x, у, у'х)у^х + H(x, y, y'x) = 0
tC + av'L + by'x + cy = e
The substitution w(x) = ye~^x leads to an autonomous equation of the form 3.5.5.9:
w'xxx + (ЗЛ + aWxx + (ЗЛ2 + 2аЛ + bWx + О3 + аЛ2 + Ь\ + C)W = f(w).
2- tC 23 X
Solution:
: C2x2 + ClX + Co + 4 / (x ~ tff(t) dt,
Jx0
where xo is an arbitrary number.
© 1995 by CRC Press, Inc.

3- tC = -^V'L + 2У + f(exy)(y'x + V).
The transformation z = exy, w = e2x(y'x + y) leads to a second order linear equation:
w"z = 2/B) + 6. Integrating the latter, we find the solution:
/dz , _,
. = ±x + C3,
^/3z2 + C2z + d 2Ф()
where
z = e*y, Ф(г)= f\ f f(z)dz]dz.
4-
The substitution w)(a?) = ху leads to an autonomous equation of the form 3.5.1.1:
W'xxx = f(w)-
5- xv'xxx
The substitution z = xy'x — у leads to a second order equation of the form 2.9.2.7:
Xt'xx = [/W + 1L-
6- xyZx + A - a)y'lx = x2af(xy'x - y).
The substitution z = xy'x — у leads to a second order equation of the form 2.9.2.4:
7- *tC + A - ax)y'Z = e2axf(xy'x - y).
The substitution z = xy'x — у leads to a second order equation of the form 2.9.2.17:
у" nyi — e2ax f(y\
Zxx ~ aZx — e J\Z>-
8- XVXL = f(xVx -V + alnx)y
XX '
The substitution z = xy'x — у leads to a second order equation of the form 2.9.2.22:
xz'x'x=[f(ln(xae*)) + l]zx.
9- x2y'Zx + 6xyZ + 6y'x = f(x2y).
The substitution w(x) = x2y leads to an autonomous equation of the form 3.5.1.1:
w'" = f(w)
XXX J \ /'
The transformation t = ln\x\, z = xy'x — у leads to an autonomous equation of the
form 2.9.6.2: z"t — z't = f(z), which is reduced, with the aid of the substitution w = z't,
to the Abel equation ш^ — w = f(w) (see Subsection 1.3.1).
11- х3Уххх + a^y'L + bxy'x = /(»).
The substitution t = In |ж| leads to an autonomous equation of the form 3.5.5.9:
y<Ht + (a - 3)y& + (b-a + 2)y't = f(y).
© 1995 by CRC Press, Inc.

x3yZx + ax2y':x + Ьху'х = f(xmexy).
The transformation t = In ж, Xw = Xy + mt leads to an autonomous equation of the
form 3.5.5.9:
</t + (a - 3)< + (b - a + 2)w't = f(eXw) + -^-(b - a + 2).
Л
xyZx ^xy:x + f
The transformation ? = ——, z = —\ху'х ~ \v) leads to a second order linear
Y X X
equation 2z"t = 8f(t) + 1 whose solution has the form
ft
z = \t2 + C2t + Ci + 4 / (t- 0/@ rfC, *o is an arbitrary number.
Passing on to variables x, t = уж'2, we obtain an equation with separation of
variables.
- 1)ж2у4'а, + к(к + l)Bfc -
+ f(xky)(xy'x + fey) + x2kg(xky)(xy'x + feyK.
The transformation t = In ж, z = xky followed by the substitution w(z) = (z'tJ yields
a second order linear equation: w"z = 2g{z)w + 2f{z) + 6/c2 + 6/c + 2.
The substitution w(x) = xy'x — у leads to an equation of the form 2.9.1.5: wxx =
x-3f(w/x).
|(); - уK.
v 1
The transformation ? = —=, z = —(%y'x — \уJ leads to a second order linear
Y Ж Ж
equation: 2z? = 16/(i)z + \.
/•Ж
Solution: у = С2Ж + C\X + Co + / (ж — i) f(t) dt, xo is an arbitrary number.
Jx0
18- y(CL + 3ayl + 2o2y4) = f(x).
Having integrated the equation, we obtain
2ayy; - {y'xf = e~2a* [2 J e2axf(x)
19. vvZL + ZyLv'L + f(x)yy'x = g(x).
The substitution w = yy'x leads to a second order linear equation: wxx + f (x)w = g(x)
© 1995 by CRC Press, Inc.

20. yy'Zx + 3У'хУхх + фУхх + (У'х) ] = /(aO-
Solution:
li2 — Cip~ax + Сот + Сл
fx
+ C2x + d+2 (x- t)e~atF(t) dt,
where F(t) = f eatf(t) dt, xq is an arbitrary number.
21. vv'l'xx + Wx + f(*)y]y'L + f(x)(y'xJ =
Solution:
= C3x + C2 + C! I (x - t)e-F{t) dt, where F(t) = f f(t) dt.
J Xq J
22.
Integrating the equation twice, we obtain a first order equation with separation of
variables:
eaxyy'x = C2x
f\x - t)eatf(t) dt.
JIn
23. yy'Zx + Wx + f(x)y]yZ + f(x)(y'x) + 9(х)УУ'х + h(x) = 0.
The substitution w = yy'x leads to a second order linear equation: wxx + f(x)w'x +
g(x)w + h(x) = 0.
24. (y + a)yZxx + by'xyZ = f(x).
Having integrated the equation, we obtain
(y + а)у?х + \{b- l){y'xf = J f(x) dx + С
2
With b ф — 1, the substitution у = w b+l —a leads to the equation
6-3
(for С = 0 and f(x) = Xxn, see Section 2.3).
25. (y + a)y'Zx + by'xVZ + f(y)y'x = 0.
Having integrated the equation, we obtain a second order equation:
(y + a)y':x + \(b - l)(y'xf + J f(y) dy = C,
which is reduced, with the aid of the substitution w(y) = (y'x) , to a first order linear
equation:
(y + a)w'y + (b - l)w + 2 / f(y) dy = 1С.
© 1995 by CRC Press, Inc.

26. (у + a)y'Zx + Wxy'L
Having integrated the equation, we obtain
(y + a)y'lx + ±(b-l)(y'xJ+ f(y)dy= g(x) dx + С
27. (у + ax + b)yZx + 3(У'Х + a)yZ = f(x).
Solution:
(y + ax + bf = C2x2 + ClX + C0+ I {x- tJf(t) dt,
•I Xo
where xq is an arbitrary number.
28.
Solution:
y2 = C3x2~a + C2x + d + 2 / (x - t)t~aF(t) dt,
J Xq
where F(t) = /i°~1/(i) dt, xq is an arbitrary number.
29. y2y'Zx ~ ЗУУХУХХ + 2(y^,) = f(x)y3.
Solution:
In \y\ = C2x2 + ClX + C0 + ± f\x - tff{t) dt,
Jxo
where Xq is an arbitrary number.
30. y2y'" +3(n-
Solution:
y" = C2x2 + ClX + C0 + ^- Г(х - tJf(t) dt,
1 Jxo
where xq is an arbitrary number.
У2УХХХ = Sy2yZ + 2y3 + y2f(e*y)(y'x + у) + д(ё°у)(у'х + уK.
The substitution z{x) = exy, followed by lowering of the equation order and the
substitution w(z) = (z'xJ, leads to a second order linear equation: w"z = 2z~2g(z)w +
2/B) + 6.
32. y(fyZx + ifLy'L + ifLy'x)=9(x), f = f(x).
Having integrated the equation, we obtain
Vyy'L + fxvv'x - fiv'xf = zjg(x) dx + a
33. fyyZx + C/t? + 2/^y)y^ + 2/^(y;J + fx'xyy'x =g, f = f(x), g = g(x).
on twice, we obtain a first order linear eqi
f(x)yy'x = C2x + d + I (x - t)g(t) dt.
Integrating the equation twice, we obtain a first order linear equation with separation
of variables:
© 1995 by CRC Press, Inc.

3.5.4. Equations of the Form
4-
1- 2ухуххх - (уххJ + f(x)(v'xJ = av2 + 2Ьу + c-
Differentiating both sides of the equation with respect to x and dividing by y'x, we
arrive at a fourth order linear equation: yxxxx + fyxx + \fxv'x = ay + b.
2. 2y'xyZx ~ (Ухх) ~ ^УХУХХ + F(x)(y'x) = еХх(аУ2 + 2by + c).
Multiplying both sides by e~Xx, we arrive at an equation of the form 3.5.4.17 with
f(x) = e~Xx, g(x) = e~XxF(x).
dx
3. 1y'y'" —3(y" ) =/(ж)(у') +9(у)(у')
Solution:
J и (у) J
гиЦх) +О>
where и = u(y) and w = w(x) are the general solutions of the second order linear
equations
= 0 and 4<X + f(x)w = 0.
V
The substitution w(x) = —-= leads to a nonhomogeneous second-order linear equa-
Vx
tion: ^wxx + f(x)w + g(x) = 0.
5- 2y'xyZx ?
Taking у as the independent variable, we obtain an equation of the form 3.5.4.4 for
x = x(y):
2х'ух;'уу - 3D'3/J = -f(y){x'vf-g(y)x-\x'v)bl2.
6- 2y;y^-3(ylJ = y4/D-
у
The substitution w(x) = —т=- leads to the second order autonomous equation of the
\/Ух
form 2.9.1.1: w'^x = F(w), where F(w) = -\wbf(w-2).
'• ЛУХУХХХ Л\УХХ) — x I
The substitution w(x) = —-== leads to the second order equation of the form 2.9.1.6:
/У'х
/2), where F(?) = -\{
2,,/
8.
у
The substitution w(x) = —-=¦ leads to the second order equation of the form 2.9.1.5:
/
Ух
1), where F(Q = -^
© 1995 by CRC Press, Inc.

9- 2xy'xyZx ~ *WL? + ny'xy':x + F(x)(y'xJ = x^iay2 + 2by + c).
Multiplying both sides by ж™, we arrive at an equation of the form 3.5.4.17 with
f(x) =xn, g(x) = xn-1F(x).
ю- xy'xyZx - 3x(vLJ + 3yxvL = xf(y)(y'xL + g(y)(y'xf.
Taking у as the independent variable, we obtain an equation of the form 3.5.3.23 for
x = x(y):
xx'yyy + Зх'уХуу = -g(y) - f(y)xx'y.
ii- tC = [*3f(*yx -v) + a*] (y'Lf-
The Legendre transformation x = w't, у = tw't — w leads to an equation of the form
3.5.2.9: </t = -f(w){w'tK-a{w't)-5.
ж4
The Legendre transformation x = w't, у = tw't — w leads to an equation of the form
3.5.2.12: w'Ht = w-s^Fiw'tW-1/4), where F(?) = -C/(C)-
/// _
Уххх —
The Legendre transformation x = w't, у = tw't — w leads to an equation of the form
3.5.2.11: w'l[t = w-^Fiw'tW-1/2), where F{?) = -C/(C)-
- vZx = ЫЮ + У9Ю + Hy'Mv'Lf + 4>(y'x)(y'L?-
The Legendre transformation x = w't, у = tw't — w leads to a linear equation:
tg(t)]w't + g(t)w - h(t).
* ** XXX J V c/ / if X \c/XX /
Solution with m/1:
С3±х =
Solution with m = 1:
dy, F(y) = Jf(y)dy.
C3±x= {C2 e^y>dy + CA dy, F(y)= f(y)dy.
16. yx'xx = xf(xy'x ~ у)(ухх) + хд(ху'х ~ у)(ухх) ¦
The Legendre transformation x = w't, у = tw't — w leads to the equation
Lowering its order with the substitution z(w) = w"t (z'w = w"lt/w't), we have the
Bernoulli equation z'w = —f(w)z — g(w)z3~k.
2 / / # \ 2 о
Differentiating the both sides of the equation with respect to x and dividing by y'x,
we arrive at a fourth order linear equation:
© 1995 by CRC Press, Inc.

3.5.5. Equations of the Form
'x,y"x)y'"xx +
F(x, y, y'x,y"x)y'"
Уххх = f(Vxx)-
Solution in the parametric form:
2^T2
_ Г dri _ Г dn Г1 т;
x=JCl Ты"' y ~ Jc2 ты" Jc3ты ¦
* Уxxx J V i/ / с/xU \У хх / "
/а;У Vi/азаз /
\{y')
Integrating the equation and substituting w(y) = \{y'x) , we arrive at a first order
equation:
J~^k = Jf(y)dy + C, where ? = w'y.
Solving this equation for w'y, we obtain an equation with separation of variables.
3. y'Zx = f(y)g(y'x)Hy'L)-
The substitution w(y) = \{y'x) leads to a second order equation:
wvv = f(y)(P(w)h(w'y), where <p(w) = ±-
whose solvable cases for some functions /, g, and h are outlined in Section 2.7.
4- Ух'хх = Xf(XV'x - У)9(Ухх)-
The Legendre transformation x = w't, у = tw't — w, where w = w(t), leads to an
equation of the form 3.5.5.2: w't"t = — f (w)w'tg( —— J (w't't) .
5- xVzx + v'L =
The substitution w(x) = xy'x — у leads to an equation of the form 2.9.4.2: wxx =
f(w)g(w'x).
6- yZx = f(xM2
The substitution w(x) = x2yxx — 2xy'x + 2y leads to a first order equation with sepa-
separation of variables: w'x = x2f(x)g(w).
T /// УхУхх , Г „ (y'xf 1 .( Ух
y> y"
The transformation t = —^-, w = —^- leads to a first order equation with separation
У У
of variables: w't = f(t)g(w).
The substitution u(x) = y'x leads to a second order equation: uxx = F(x,u,u'x).
© 1995 by CRC Press, Inc.

9-
Autonomous equation.
The substitution w(y) = (y'x) leads to a second order equation:
г/
и'" — iiF I ——
Уххх — Уг \ 1
У У
Ух „„ _ Ухх
The transformation t = ——, w = —— leads to a first order equation: (w — t2)w'f =
У У
-tw + F(t,w).
tC = *Ч*, жу^ - у, xyZJ.
The substitution z = xy'x — у leads to a second order equation: zxx = x~1z'x +
xF(x,z,z'x).
Homogeneous equation in the extended sense.
The transformation t = lnx, z = xky leads to an autonomous equation:
4't = 3(/c + 1L - C/c2 + 6/c + 2)z't + k{k + l)(/c + 2J
, 4 - kz, 4 - B/c + l)z't + k(k
The substitution w(z) = (z't) leads to a second order equation:
w"zz = ±3(jfc + 1)«;-1/2< - 6/c2 - 12/c - 4 ± 2jfc(jfc + l)(jfc + 2)гги~1/2
±2w~1/2F(z, ±w1/2-kz, ±w'zTBk + l)w1
У У
Homogeneous equation in the extended sense.
ITfl
The transformation t = xkym, z = —— leads to a second order equation.
У
У У
This is a special case of equation 3.5.5.13.
xxi x xi"
The transformation z = ——, w = — leads to a first order equation:
У У
(w + z — z )w'z = 2w — zw + F(z,w).
The Legendre transformation x = w't, у = tw't — w leads to the equation
<t = -F
(w't,tw't-w,t, -Jr)«K.
© 1995 by CRC Press, Inc.

. tC = e-a'F(ea'y, e™y'x, e^y'^).
The substitution z = eaxy leads to an autonomous equation of the form 3.5.5.9:
zx ~ a z = * \zt zx~ azt zxx ~ ^azx + a
17 v'" — vF\ eaxv
*-' • Уххх — У1 \ e У
III —„.тг1„аХ„. УХ УХХ
*xxx
Exponential homogeneous equation.
y>
The transformation z = eaxy, w = —^- leads to a second order equation:
У
z (w+a) w'!,z+z (w+a)(w'z) +z(w+a)D:W+a)w'z+w =F(z, w, z(w+a)w'z+w ).
y'Zx = x~3F(xmey, xy'x, x2y'D-
Exponential homogeneous equation.
The transformation z = xmey, w = xy'x + m leads to a second order equation:
z2w2wzz + z2w(w'z) + zw2w'z — 3zww'z + 2w — 2m = F(z, w — m, zww'z — w + m).
© 1995 by CRC Press, Inc.

Chapter 4
Fourth Order
Differential Equations
4.1. Linear Equations
4.1.1. Preliminary Comments
1. A nonhomogeneous linear equation of the fourth order has the form
hy'xxxx + hy'xxx + hyxx + hy'x + hy = 9{x), h = fk(x). A)
Let уо=Уо(х) be a nontrivial particular solution of the corresponding homogeneous equation
(with g = 0). Then, the substitution
У = Уо{х) / z(x)dx B)
J
leads to a linear equation of the third order:
/4Уо2/"+D/4^+/зУоJ"+F/4^/+3/з^+/2уоJ/+D/4^"+3/з^/+2/2^+/1уоJ = <?) C)
where prime denotes differentiation with respect to x.
2. Let уг = Уг{х) and 2/2 = yi[x) be two nontrivial linearly-independent particular
solutions of equation A) with g = 0. Then, the substitution
y2w dx — 2/2 / yiw dx D)
yields a second order linear equation:
UAlW" + (З/4А2 + /3Д1V + [/4CA3 + 2e) + 2/3A2 + f2A1]w = g, E)
where
Ai = 2/i2/2 - 2/12/2, A2 = 2/i'2/2 - 2/12/2, ^3 = 2/1/2 - 2/12/2", e = 2/"У2 - У1У2 ¦
4.1.2. Equations Containing Power Functions
1°. Solution with a = 0:
y = Ci+ C2x + C3x2 + С4Ж3.
© 1995 by CRC Press, Inc.

2°. Solution with a = 4A;4 > 0:
у = С г cosh kx cos kx + C<i cosh kx sin kx + C3 sinh /еж cos kx + C4 sinh kx sin /еж.
3°. Solution with a = -kA < 0:
y = C\ cos /еж + Сг sin /еж + C3 cosh /еж + C4 sinh /еж.
2- V™ + ЛУ = ож3 + Ьж2 + еж + s, Л# 0.
Solution: у = — (ах3 + Ьх2 + сх + s) + w(x), where w(x) is the general solution of
Л
the equation 4.1.2.1: wxxxx + Xw = 0.
3- tC» = axv + b-
This is a special case of equation 5.1.2.4 with n = 4.
4-
For m = -2, -4, -6, -8, and -9, see equations 4.1.2.34, 4.1.2.42, 4.1.2.47, 4.1.2.48,
and 4.1.2.53, respectively.
The transformation ж = i, у = wt~3 leads to an equation of the similar form:
w"" = at
"" = at~m-
utttt
w.
This is a special case of equation 4.1.2.24.
6- УГ^
This is a special case of equation 4.1.2.13 with n = 1.
7- tC» + 4ожУ4 + Bа - о2ж4)у = 0.
This is a special case of equation 4.1.2.13 with n = 2.
8- V™ + ОЖBЬ - За - a2x2)y'x + ЬBа -b + a2x2)y = 0.
The substitution w = y!xx — axy'x + by leads to a second order linear equation of the
form 2.1.2.28: wxx + axw'x + Ba - b + a2x2)w = 0.
9- I/™ + axmy'x - 3axm~1y = 0.
Particular solution: yo = ж3.
The substitution z = xy'x — 3y leads to a third order equation of the form 3.1.2.7:
¦у'" Л. птт? П
Integrating yields a third order equation: y'xxx + axmy = C.
П- tfZm. + axmy'x + a(m + 3)xm~1y = 0.
The transformation ж = t~x, у = wt~3 leads to an equation of the form 4.1.2.10:
w"tlt + btnw't + bntn~1w = 0, where b = —a, n = —m — 6.
© 1995 by CRC Press, Inc.

yZxx + Ьхту'х ~ а(а3 + Ьхт)у = 0.
This is a special case of equation 4.1.5.1 with / = bxm.
Ухххх ~l~ 2a,nxn 1y'x + a\n(n — \)xn 2 — ax2n]y = 0.
The substitution w = yxx+axny leads to a second order equation of the form 2.1.2.7:
w" - axnw = 0.
)у4 + y = 0.
Particular solution: yo = е~Ьж.
Ухххх + (ажп+1 + Ьж")У; - ахпу = 0.
Particular solution: yo = ах + Ь.
16- 1С» + 2оу1 + о2У = 0.
1°. Solution with a = к2 > 0:
у = (Ci + С2ж) сов(/сж) + (Сз + С4ж) sin(fca;).
2°. Solution with a = -к2 < 0:
у = (Ci + Сгж) ехр(/сж) + (Сз + С4Ж) ехр(—кх).
17- УГ^ + (а + Ъ)у'1х + aby = 0.
The case of a = b is given in 4.1.2.16. Let афЬ.
1°. Solution with a = a2 > 0, 6 = /32 > 0:
у = C\ cos(aa;) + Ci sin(aa;) + C3 cos(/3a?) + C4 sin(/?a;).
2°. Solution with а = а2 > 0, 6 = -(З2 < 0:
у = C\ cos(aa;) + C2 sin(aa;) + C3 ехр(/3ж) + C4 exp(—/Зж).
3°. Solution with a = -a2 < 0, b = /32 > 0:
у = C\ ехр(аж) + C2 exp(—ax) + C3 сов(/3ж) + C^ sin(Cx).
4°. Solution with a = -a2 < 0, b = -f32 < 0:
у = C\ ехр(аж) + Ci exp(—ax) + C3 ехр(/3ж) + С± exp(—fix).
© 1995 by CRC Press, Inc.

tC, - 2a2v'L + «4У - Max - Ь)Ь?а - a?y) = 0.
This equation is met with in the turbulence theory. Assuming
z(x) = Ухх - а2У> (!)
yields a second order linear equation of the form 2.1.2.12:
zxx - a2z - X(ax - b)z = 0. B)
Given the boundary conditions
y@) = y'x@) = 0, y(l)=y'x(l)=0, C)
we obtain
2ay = eax I e-axzdx-e~ax f eaxzdx
Jo Jo
The latter is the solution of equation A) that satisfies the first pair of the boundary
conditions C). In order to satisfy the second pair of the boundary conditions, the
solution z(x) of equation B) must meet the requirements
/ e~axzdx = f eaxzdx = 0.
JO JO
tC» + axnv'L + Haxn - b)y = 0.
1°. Particular solutions with b > 0: y\ = cos(xVb), y2 = sin(xVb).
2°. Particular solutions with b < 0: y\ = exp(—ж\/—b), yi = ехр(ж\/—b).
The substitution w = yxx + by leads to a second order linear equation: wxx +
(axn - b)w = 0.
20- tC» + axn+1vL - ±a,xny'x + baxn~^y = 0.
Particular solutions: y\ = x2, yi = x3.
The substitution w = x2yxx — ^xy'x + 6y leads to a second order linear equation
of the form 2.1.2.7: w" + axn+1w = 0.
21. y'n'J.xx + Waxny'^x + Wanxn-Iy'x + [3an(n - 1)ж"-2 + 9a2x2n]y = 0.
This is a special case of equation 4.1.5.26 with / = axn.
22. у'^хх + (axn + b)y'ix + abxny = 0.
1°. Particular solutions with b > 0: y\ = cos(a?v6), yi = sin(a;v6).
2°. Particular solutions with b < 0: У\= ехр(—ж\/—b), Vi = ехр(ж\/—Ь).
The substitution w = yxx + by leads to a second order linear equation of the form
2.1.2.7: w'x'x + axnw = 0.
23. y'Zxx + av'L + bxny'x +
Intergating yields a third order equation: y'xxx + ayxx + bxny = s J xm dx + C.
© 1995 by CRC Press, Inc.

24. С + о3у1 + а2у'х + аоу = 0.
For ao = 0, the substitution w(x) = у'х leads to a third order equation. Let ao ф О
and -P(A) = A4 + азА3 + агА2 + aiA + ao be the characteristic polynomial.
1°. Let P be factorizable, so that
P(A) = (A - Ai)(A - A2)(A - A3)(A - A4),
where Ai, A2, A3, and A4 are real numbers. The following cases are possible:
a) Aj are all different, then
у = deXlX + C2eX2X + C3eXsX + CAeXiX;
b) Ai = A2; A3 and A4 are different and not equal to Ai, then
у = (Ci + C2x)eXlX + C3ex*x +
c) Ai = A2 = A3 ф А4, then
y=(C1+ C2x + C3x2)eXlX
d) Ai = A2 = A3 = A4, then
y=(C1+ C2x + C3x2 + O±x3)eXlX.
2°. Let
P(A) = (A - Ai)(A - A2)(A2 + 26iA + b
where Ai and A2 are real numbers, and b\ — bo < 0. If
a) Ai ф \2, then
у = CieXlX + C2eX2X + e~blX[C3 cos(iix) +
b) Ai = A2, then
у = (d + C2x)eXlX + e~blX[C3 cos(iix) +
3°. Let us assume that
P(A) = (A2 + 26A + 60)(A2
where b\ - b0 < 0 and /32 - ft < 0. If
ц, = ф0 - b\;
ц, = фо - b\.
ft A
a) (h - AJ + F0 - ftJ ф 0, then
у = e~blX[Ci cos(iix) + C2 sm(iix)] + e~plX[C3 cos(i^x) + C4 sin(ra)],
where ц, = ^Jb0 -b\,v= ^/ft - /32;
b) 61 = /3i and 60 = ft, then
У =
C2x)
(C3 + C±x) sin(iix)], jjl = Jbo — b\.
© 1995 by CRC Press, Inc.

25. у"" + 4аа;|/"' + 6а2ж2у" + 4а3ж3у' + о4ж4у = 0.
*xxxx ' * »ж
Solution:
4
%=\
A4
where Aj are the roots of the biquadratic equation A4 — 6aA2 + 3a2 = 0.
26. vZxx ~l~ (ож "I" tyyZx ~l~ [^(° + с)ж + c]vZ ~l~ b2cxy'x — Ь2су = 0.
Particular solutions: y! = x, y<i= e~bx.
Particular solutions: y^ = ехр(А^ж) (к = 1, 2, 3), where Afe are the roots of the
cubic equation A3 — b = 0.
28- vZxx + axn+3yZx ~ 3axn+2yZ + baxn+^y'x - 6axny = 0.
Particular solutions: y\ = x, y2 = x2, уз = x3.
The substitution w = хгу'^'хх — Ъх2у'хх + &xy'x — 6y leads to a first order linear
equation: w'x + axn+3w = 0.
29. yZxx + axnyZx + bxm+1yZ ~ 1bxmy'x + 2bxm~1y = 0.
Particular solutions: y\ = x, У2 = x .
The substitution w = x2yxx — 2xy'x + 2y leads to a second order linear equation:
xw%x + (axn+1 - 2)w'x + bxm+2w = 0.
30- vZxx + axnyZx + bxmyZ + acxny'x + c(bxm - c)y = 0.
1°. Particular solutions with с > 0: y\ = cos(xt/c), y2 = sin (ж-у/с)-
2°. Particular solutions with с < 0: y\ = exp(—Жу/—с), уг = exp(a?v/—с).
The substitution w = ухх + су leads to a second order linear equation: wxx +
axnw'x + (bxm - c)w = 0.
31- vZxx + axnyZx + (bxm + c)yZ + acxny'x + bcxmy = 0.
1°. Particular solutions with с > 0: y\ = cos(xt/c), y<i = sin^v^c).
2°. Particular solutions with с < 0: y\ = exp(—X\f^c), y<i = ехр(жу/—с).
The substitution w = yxx + cy leads to a second order linear equation: wxx +
axnw'x + bxmw = 0.
32. xyZxx + ±vZx + а*У = 0-
The substitution w(x) = xy leads to a constant coefficient equation of the form 4.1.2.1:
wxxxx + aw = 0.
33. xyZxx ~ ^^vZx "I" axV = 0' m = 1, 2, 3, ...
Solution:
where w = г<;(ж) is the general solution of the constant coefficient equation 4.1.2.1
wxxxx + aw = 0.
© 1995 by CRC Press, Inc.

34. x2yxxxx = ay.
This is a special case of equation 5.1.2.23 with n = 2.
35. x2y'xxxx ~ 2(ож2 + Q)yxx + a(ax2 + 4)y = 0.
Particular solutions: y\ = x~1^2Ii/2(x^/a), y2 = x~1/2K1/2{x\/a), where I1/2
and K1/2 are modified Bessel functions.
36. x2y'"' -
The equation of transverse vibrations of a pointed bar.
Solution:
у = -^= [d Л B^A^) + C2Y1 BV\x) + C3I1
\JX
where J\ and Y\ are Bessel functions, 1г and Кг are modified Bessel functions.
37. x yxxxx "I" 2(o + 2)жу^,а, + (о + l)(o + 2)y^fa, — by = 0.
Solution:
where ? = 2b^/x, Ja and Уа are Bessel functions, Ja and АГа are modified Bessel
functions.
The substitution w(x) = x2y leads to a constant coefficient equation of the form
4.1.2.1: w'"' T + aw = 0.
39. x2y'^xx + 8xy'Zx + 12yl = ax3y + b.
The substitution w(x) = x у leads to an equation of the form 4.1.2.3: w'^xx = axw+b.
40. *?y'Zxx + ™y'xxx + (bxn+1 + c)yZx + (a - 4)bxny'x + b(c - 2o + в)хп~1у = 0.
The substitution w(x) = x2yxx + (a — ^)xy'x + (c — 2a + 6)y leads to a first order
equation of the form 2.1.2.7: wxx + bxn~1w = 0.
41 x3v"" + 2x2vf" — xv" +vf— a4x3v — 0
Ч:±. а. ухххх т^ -^л yxxx •Lyxx \ Ух и а, у — и.
Solution:
у = Ci J0(ax) + C2Y0(ax) + C3I0(ax) + C4K0(ax)],
where Jq and Yq are Bessel functions, Iq and Kq are modified Bessel functions.
42. x4yxxxx
Solution:
у = ClXkl
where k1>2 = -j ± J-j + Va + T, /c3L = f ±y|- Va + T.
© 1995 by CRC Press, Inc.

43. x^y'l'L* + A3x3yZx + ^V'L + Aixy'x + Aoy = 0.
The Euler equation.
The substitution t = \n\x\ leads to a constant coefficient equation of the form
4.1.2.24:
yZt + (A3 - 6J/^ + A1 - 3A3 + A2)y't't + BA3 -A2 + Ax- 6)y't + Aoy = 0.
44. х4у^хх2 4
where n is a positive integer.
Solution:
where \v are four different roots of the equation A4 + a = 0, and Pv is some definite
polynomial of the degree < An. For a = 0, we have the Euler equation 4.1.2.43.
45. x4yxxxx + 2B - n)x3yxxx + A - n)B - n)x2yxx - a4x2ny = 0.
Solution:
У = Vx [Ci Jl/n(C) + C2^1/n@ + СзЛ/п@ + C4-Ki/n(?)]>
where ? = xn^2, Jv and 1^, are Bessel functions, Iv and Kv are modified Bessel
n
functions.
+ жA6ж2 + 1 — о2 — Ь2)у^, + (8ж2 + a2b2)y = 0.
Solution with ab Ф 0:
у = СМх)Мх)
where JM and Y^ are Bessel functions, 2/л = a + b, 2v = a — b.
47. x*y'Zxx = ay.
This is a special case of equation 5.1.2.24 with n = 2.
48- a^tCr» = «»•
The transformation x = t~1, y = wt~3 yields a constant coefficient equation: w""tt = aw.
49. x8yxxxx + 4xryxxx = ay.
The substitution w(x) = xy leads to an equation of the form 4.1.2.48: xsw'x/'x'xx = aw.
50. (ax + bL(cx + dLyxxxx = fey.
The transformation
ax + b у
? = In ¦
еж + d ' (ex + dK
leads to a constant coefficient equation.
© 1995 by CRC Press, Inc.

51. (ax2 + bx + cLyx.
The transformation
<¦/¦
dx у
w =
ax2 + bx + с ' (ax2 + bx + cK/2
leads to a constant coefficient equation:
4«? ~ TD4? + (^6D2 ~ k)w = °> where D = b2- Aac.
52. (ax + bJ(cx + d)etC» = ky.
The transformation
ax + b у
с = W =
4 cx + a"
W =
cx + a" (cx + dK
leads to an equation of the form 4.1.2.34:
?,2w'^? = kA~4w, where A = ad — be.
The transformation x = t , у = wt leads to an equation of the form 4.1.2.3:
w""tt = atw + b.
54. (ax + b)9y'^xx = (ex + d)y.
The transformation
ex + d у
с = w =
ax + b'
w =
ax + b' (ax + bK
leads to an equation of the form 4.1.2.3:
4 where A = ad - be.
4.1.3. Equations Containing Exponential, Hyperbolic, and Logarithmic
Functions
!• y'l'L* + а3У'х + beax(a2 - beax)y = 0.
The substitution w = y'J.x + ay'x + beaxy leads to a second order linear equation of the
form 2.1.3.10: w%x - aw'x + (a2 - beax)w = 0.
2- tC» + аеХхУ'х ~ (abeA- + Ь4)у = О.
Particular solution: yo = e .
3- У'1'Lv + 2a\eXxy'x + a(X2eXx - ae2Xx)y = 0.
The substitution w = yxx + aeXxy leads to a second order linear equation of the form
2.1.3.1: w'^x - aeXxw = 0.
'Х + a.beXxy = 0.
Particular solution: yo = е~Ьх¦
© 1995 by CRC Press, Inc.

5- С + (ах + Ь)еХху'х - аеХху = 0.
Particular solution: yo = ах + Ь.
6- y'Zxx + ™Xxy'L ~ ЧаеХх + Ъ)у = 0.
1°. Particular solutions with b > 0: y\ = exp(—xVb), yi = ехр(жл/&).
2°. Particular solutions with b < 0: yi = cos(a?\/—b), уг = sm(xy/—b).
The substitution ад = yxx — by leads to a second order linear equation of the form
2.1.3.10: wlx + {aeXx + b)w = 0.
7- tC» + (° + beXx)v'L + abeXxy = 0.
1°. Particular solutions with a > 0: yi = сов(жу/й), 2/2 = sin^ya).
2°. Particular solutions with a < 0: г/i = exp(—Жу/—а), У2 = ехр(жу/—«)•
The substitution ад = yxx + ay leads to a second order linear equation of the form
2.1.3.1: w'^ + beXxw = 0.
8- t/™ + Юое^»^ + 10оЛеАа;у; + (ЗоЛ2еАа; + 9а2е2Хх)у = 0.
This is a special case of equation 4.1.5.26 with /(ж) = аеЛа:.
Particular solution: yo = e ax.
1П 11"" — npXxii'" -\- hit' nhpXxii
Particular solutions: y^, = e'3''21 (fc = 1, 2, 3), where fa are the roots of the cubic
equation /33 — b = 0.
^L + oceAa:y4 + cibe^ - c)y = 0.
1°. Particular solutions with с > 0: yi = сов(жу/с), У2 = sin(a;^
2°. Particular solutions with с < 0: yi = exp(—ж^/—с), У2 = ехр(ж^/—с).
The substitution ад = у^ + су leads to a second order linear equation:
wxx + aeXxw'x + c(be^x - c)w = 0.
1°. Particular solutions with с > 0: y\ = cos(xy/c), yi = sin(xy/c).
2°. Particular solutions with с < 0: y\ = exp(—Жу/—с), уг = ехр(жу/—с).
The substitution ад = ухх + су leads to a second order linear equation: wxx +
aeXxw'x + be^xw = 0.
13- Ухххх + ax3eXxy'Zx ~ 3ax2eXxy'Z + 6axeXxy'x - 6aeXxy = 0.
Particular solutions: уг = ж, у^ = ж2, уз = ж3.
The substitution ад = х3ух'хх — Зх2ухх + &ху'х — 6у leads to a first order linear
equation: w'x + ax3eXxw = 0.
© 1995 by CRC Press, Inc.

14. (ое- + Ъ)у'^хх = ае*у.
Particular solution: yo = aex + b.
15. (ox™ + Ьех + c)v'Zxx = Ье*у, т = 1, 2, 3.
Particular solution: yo = &xm + ^e21 + c-
16. (axme* + b)y'^xx = by, m = 0, 1, 2, 3.
Particular solution: yo = «a:m + be~x.
") y^ + o[b ехр(Лж") - a]y = 0.
This is a special case of equation 4.1.5.5 with /(ж) = 6ехр(Ажп).
- tC» + [« + b ехр(ЛЖ")]у^ + ab ехр(ЛЖ") у = 0.
This is a special case of equation 4.1.5.6 with /(ж) = 6ехр(Ажп).
tC» + bsinh"(Aa;) y^ + olbsinh^CAa;) - o]y = 0.
This is a special case of equation 4.1.5.5 with /(ж) = 6sinhn(Aa;).
20- tC» + [° + b sinh"(Aa;)]yl + ab sinh"(Aa;) у = 0.
This is a special case of equation 4.1.5.6 with /(ж) = 6вт11п(Аж).
21- tC» + Ь со3Ь"(АЖ) y'lx + а[Ь со3Ь"(АЖ) - о]» = 0.
This is a special case of equation 4.1.5.5 with /(ж) = 6совЬп(Аж).
22. y'Zv* + [а + Ь совЬп(\х)]Ух-х + ab cosh^A*) у = 0.
This is a special case of equation 4.1.5.6 with /(ж) = 6совЬп(Аж).
23. 3?y'Zxx + 2axy'x - o[l + ож2 1п2(Ьж)]у = 0.
The substitution w = y'J.x + а1пFж) у leads to a second order linear equation: w'J.x
а1пFж)«; = О.
24. у'^хх + а 1гГ(Аж)(x3yZx ~ 3x2y'L + 6xy'x - 6y) = 0.
This is a special case of equation 4.1.5.15 with /(ж) = а1пп(Аж).
4.1.4. Equation Containing Trigonometric Functions
!• Ухххх + 2abcos(bx) y'x - a[b2 sin(bx) + a sin2(bx)]y = 0.
The substitution w = y'xx + авт(бж) у leads to a second order linear equation of the
form 2.1.6.3: wxx — asin(bx)w = 0.
2- tC» + ° sin"(Aaj) y'x + b[a 3тгг(Аж) - Ь3]у = 0.
Particular solution: yo = e~bx.
© 1995 by CRC Press, Inc.

3- tC» + [asinn(Xx) + Ь3]у'х + absinn(Xx) у = 0.
Particular solution: yo = е~Ьх¦
Xx) y'x + b[a tan"(Aaj) - bs]y = 0.
Particular solution: yo = е~Ьж.
5- tC» + [° tan"(A*) + b3]y4 + ob tan"(А*) у = 0.
Particular solution: yo = е~Ьж.
6- v'l'L* + a ыпп(\х) y'lx + b[a sin"(Aaj) - b]y = 0.
The substitution w = y'^x + by leads to a second order linear equation: wxx
[asinn(Xx)-b}w = O.
7- У'ЛХХ + [a + b sinn(Xx)]y':x + ab s\nn(\x) у = 0.
The substitution w = yxx + ay leads to a second order linear equation: wxx
bsinn(Xx)w = 0.
8- VZn + b tan"(AaJ) yl + o[btan"(A*) - a]y = 0.
This is a special case of equation 4.1.5.5 with /(ж) = 6tann(Arr).
tC» + [° + b tan"(Aa;)]yL + ab t^n(Xx) у = 0.
This is a special case of equation 4.1.5.6 with /(ж) = 6tann(Arr).
^ _ 6y) = 0
This is a special case of equation 4.1.5.15 with f(x) = asinn(Aa;)
АЖ) <L + Ы - «Ь зт"(АЖ) у.
Particular solutions: y^ = e@kX (k = 1, 2, 3), where /?fc are the roots of the cubic
equation /33 — b = 0.
12. tC,,, = а Ыпп(Хх) yZx + by'x - ab tan^A*) y.
Particular solutions: yk = e/3kX (k = 1, 2, 3), where Ck are the roots of the cubic
equation C3 — b = 0.
13- a^tC. + о sin"(Aa;) (ж2»^, - 4xy'x + 6y) = 0.
The substitution w = x2y'^.x — ^xy'x + 6y leads to a second order linear equation:
wxx + asinn(Aa;) w = 0.
© 1995 by CRC Press, Inc.

14. sin4 ж у™т + 2 sin3 ж cos x y'i'xx + sin2 x (sin2 x - 3) y^
+ sin ж cos ж B sin2 x + 3)y'x -\- (o4 sin4 ж — 3)y = 0.
TTie equation of a loaded rigid spherical shell.
If a4 = 1 — A2 then the equation can be written as
LL(y) — A2y = 0, where L = -—5- + cot ж— cot2 ж.
dxA ax
This equation falls into two second order equations:
L(y) + Ay = 0, L(y) - Ay = 0,
which differ only in the sign of parameter A. The transformation ? = sin2 x,w = y/ sin ж
reduces the latter equations to the hypergeometric equations 2.1.2.158:
?(? - 1)«? + (|C - 2)«4 + |A T A)w = 0.
15. (o cos ж + Ь)ухххх = о cos ж у.
Particular solution: yo = a cos ж + 6.
16. (аж™ + bcosx)y'x"x'xx = bcosxy, m = 1, 2, 3.
Particular solution: yo = ажт + 6 cos ж.
17. (о sin ж + Ь)ухххх = о sin ж у.
Particular solution: yo = a sin ж + 6.
18. (axm + bsinx)yxxxx = b sin ж у, m = 1, 2, 3.
Particular solution: yo = axm + b sin ж.
4.1.5. Equations containing arbitrary functions
Notation: f, g, and h are arbitrary functions of x; a, b, and с are parameters.
* *? XXXX ' v if X >¦ «^ ' / «/
Particular solution: yo = eax.
2. y'xxxx + (/ + a3)v'x + °/У = 0-
Particular solution: yo = e~ax.
Particular solution: yo = ж .
The substitution z = жу^, — 3y leads to a third order equation: z'^'xx + xfz = 0.
4- <!. + (ax + b)fy'x - afy = 0.
Particular solution: yo = ax + b.
© 1995 by CRC Press, Inc.

1°. Particular solutions with a > 0: yi = cos(xy/a}, У2 = sin^ya).
2°. Particular solutions with a < 0: yi = exp(—Жу/—а), У2 = exp(a?v/—«)•
The substitution w = yxx+ay leads to a second order equation: wxx + (f — a)w = 0.
6- tC» + (/ + «)tC + o/i/ = o-
1°. Particular solutions with a > 0: yi = cos(x\/a), yi = sm(x\/a).
2°. Particular solutions with a < 0: yi = exp(—ж-\/~а), У2 = exp(xy/—a).
The substitution ад = yxx + ay leads to a second order equation: wxx + /ад = 0.
7. у"" + f(x)(x2y" — 4жу' + 6y) ^ 0.
Particular solutions: y\ = x2, уг = ж3.
The substitution ад = x2yxx — ^xy'x + 6y leads to a second order linear equation:
wxx + x2fw = 0.
Particular solution: yo = o,x2 + bx + c.
Particular solution: yo = x .
Ю- tC,,, + /y^'L - ^V'L ~ a2fy'x + о4У = 0.
Particular solutions: yx = e~ax, yi = eax.
ii- y'l'L* + fy*L + gy'L + o/i/i + «(» - °)y = °-
1°. Particular solutions with a > 0: yi = cos(a;-\/u)) У2 = sin(xy/a).
2°. Particular solutions with a < 0: yi = exp(—ж-\/~а), У2 = exp(xy/—a).
The substitution ад = yxx + ay leads to a second order linear equation: wxx
fw'x + (g- a)w = 0.
12. tC,, + fy'Zx + (9 + o)y'L + afy'x + а9У = 0.
1°. Particular solutions with a > 0: У\= сов(жу/й), Уг = sa\[x\fa).
2°. Particular solutions with a < 0: yi = exp(—ж-\/~а), У2 = ехр(жу/—«)•
The substitution ад = y^,/a,+ay leads to a second order equation: wxx+fwx+gw =
уГ.. + /tC + fl»»» + ^Ы - л» = о.
Particular solution: yo = x.
yx + y) = 0.
Particular solutions: yi = x, уг = x .
The substitution z = x2yxx — 2жу^, + 2y leads to a second order equation: xzxx
(xf - 2)z'x + x3gz = 0.
© 1995 by CRC Press, Inc.

Particular solutions: y\ = x, У2 = x2, уз = ж3.
The substitution w = x3yx'xx — Ъх2ухх + 6xy'x — 6y leads to a first order linear
equation: w'x + ж3/ад = 0.
16. yZxx = fvZx + ay'x - afy = 0.
Particular solutions: y^ = eXkX (k = 1, 2, 3), where Afe are the roots of the cubic
equation A3 — a = 0.
17- tC» = (/ - a)tC + (o/ - b)t/™ + (V - С)У4 + с/У = 0.
Particular solutions: y^ = eXkX (k = 1, 2, 3), where A^ are the roots of the cubic
equation A3 + a A2 + ЬХ + с = 0.
18- t/™ + (/ + «OtC» + Ы + 9 + axgWL + a2xgy'x - a2gy = 0.
Particular solutions: уг = x, j/2 = e~ax.
19. y'l'L* + (/s + o)y4'L + (/2 + o/s)»S- + (Л + °Л)У4 + о/1У = 0,
where fk = fk{x) (k = l, 2, 3).
Particular solution: yo = e~ax.
20. x^M + 4tC + ожу = /(ж).
The substitution ад (ж) = жу leads to a nonhomogeneous constant-coefficient linear
equation: wxxxx + aw = f(x).
21- xyZxx + xfy'x - [(x + l)f + x + 4]y = 0.
Particular solution: yg = xex.
22. x2y'Zxx + axyZx + (x2f + Ъ)у^х + (о - 4)xfy'x + (Ь-2а + 6)fy = 0.
The substitution ад = x2yxx + (a — ^)xy'x + (b — 2a + 6)y leads to a second order
equation: wxx + /ад = 0.
23. xS'l'L* + <*x3yZx + xfvL + (« - 3)/У = 0-
Particular solution: yo = x .
24. y'Zxx + fvL + fLv = 9-
Integrating yields y'xxx + fy = f gdx + C.
25. уГ^+2/Х + (/1-/2)У = 0.
The substitution ад = yxx + fy leads to a second order equation: wxx — /ад = 0.
26. yZ'L. + lo/tC, + io/?t? + (з/^ + 9/2)y = o.
Solution:
У = C\w\ + C2W\W2 + C3W1WI + C4VJ2,
where W\ and W2 are nontrivial linearly-independent solutions of the second order
equation wxx + /ад = 0.
© 1995 by CRC Press, Inc.

27. y'Zxx + (/ + g)v'L + zfLvL + (fL + fg)v = o.
The substitution w = yxx + fy leads to a second order equation: wxx + gw = 0.
28. yZxx+fyZx + ^fL + f + 9)y:xHfL + ffL+f+f9 +
+ 3B fxg + 5fg'x + 6f2g + g'^ + 3g2)y = 0.
Solution:
у = C\w\ + C2W1W2 + C3W1W2 + C4W2,
where Wi and W2 form a fundamental set of solutions of the second order equation
w'xx + fw'x +gw = 0.
29.
The equation of transverse vibrations of a bar.
tion: y = d+C2x+ f ^-^- (C3 + CAt)
30. y'Zxx + fv'x + (/tan x-l)y = 0.
Particular solution: yo = cos ж.
31- Ухххх +/»i-(l + / cot x)y = 0.
Particular solution: yo = sinx.
32. y'Zxx = f{*)V-
The transformation x = t~x, у = wt~3 leads to an equation of the similar form:
33. у"" =*<аХ + Ь\ У
Ухххх
8
сх + d ' (сх + d)
ах ~\~ b xi
The transformation ? = , w = — -^- leads to a simpler equation:
cx + d (ex + dK
where A = ad — be.
34. y'Zxx + f(x)v'x + 9(x)y + h(x) = 0.
The transformation x = t~x,y = wt~3 leads to an equation of the similar form:
© 1995 by CRC Press, Inc.

4.1.6. Asymptotic Solutions
This subsection presents asymptotic solutions, as e —> 0 (e > 0), of some fourth-order
linear ordinary differential equations containing arbitrary functions (sufficiently smooth),
with the independent variable being a real number.
1. Consider the equation
e4y'::xx-f(x)y = 0 A)
on a closed interval a<x<b. With the condition / > 0, the leading terms of the asymptotic
expansions of the fundamental system of solutions, as e —>• 0, are given by the formulae
|||| y2 =
Уз = [/(Ж)]-3/8с[1 J[f{)f'4}
2. Now consider the "biquadratic" equation
e%Zxx ~ ^29{x)y'L ~ №V = 0- B)
Introduce the notation
D(x) = [g(x)}2 + f(x).
In the region where the conditions f(x) ф 0 and D(x) ф 0 are satisfied, the leading terms
of the asymptotic expansions of the fundamental system of solutions of equation B) are
described by the formulae
J \h{x)dx-± J
/D(x)
where
dx); к = 1, 2, 3, 4.
= \g(x) + JD(x), X2(x) = -\ g(x) + JD(x),
- y/D(x), А4(ж) = -у <?(ж) - y/D(x).
4.2. Nonlinear Equations
4.2.1. Equation Containing Power Functions
i- tC, = ау~5/3-
Multiply both sides of the equation by y5/3 and differentiate the resulting expression
with respect to x. We have
Integrating the latter equation three times, we obtain a chain of equalities:
xx + 2y'xy':xx - (y'Zf = 2C2, A)
Х-У'ХУ1Х=Ж2Х + СЪ B)
'1х - 2{y'xf = C2x2 + ClX + Co, C)
© 1995 by CRC Press, Inc.

where Co, C\, and C2 are arbitrary constants. By eliminating the highest derivatives
from (l)-C) with the help of the original equation, we obtain a first order equation:
BPy'x - Wxyf = 9(C? - 4C0C2)y2 - 2P3
where P = C2x2 + C\X + Co. The substitution у = (P/wK^2 leads to an equation
with separation of variables whereof integration finally yields
Vxxxx —
By integrating, we obtain (m ф —1)
where С is an arbitrary constant. The substitution w(y) = (y'x) leads to a second
order equation:
ЗА
wyy =
2m+ 2
¦ут+1+С)т-ъ/3.
The value С = 0 corresponds to the Emden—Fowler equation whose integrable cases
are specified in Section 2.3 for some values of m (to those cases correspond three-
parameter families of particular solutions of the original equation).
3- Ухххх =
The transformation x = t~1, у = t~3w(t) leads to an equation of the form 4.2.1.2:
W'xxxx = Aw™-
3m,+5
4- VXLX = Ax —ym.
This is a special case of equation 4.2.3.3 with f(w) = Awm.
5- V'ZXX = («У + Ьхк)т, к = 0, 1, 2, 3.
The substitution aw = ay + bxk leads to an equation of the form 4.2.1.2: wxxxx =
amwm.
6. ж3"г+1 (ах + ЪУу'1ххх = cVm-
This is a special case of equation 4.2.3.5 with f(w) = cwm.
Зтгг+5
Ухххх
7- V'ZXX = (ах2 + Ъх + с) 2 у-
This is a special case of equation 4.2.3.6 with f(w) = wr'
/""
8- У'х
The transformation ? = ex^a, w(?) = ?3^2у leads to an equation of the form 4.2.1.1:
© 1995 by CRC Press, Inc.

-rii"" A- Aii'" — 4т-5/3?/-5/3
^Ухххх Т ^Уххх — AAj У
The substitution w(x) = xy leads to an equation of the form 4.2.1.1: wxxxx = Aw~5^3.
KVxxxx + 2Уххх = "(xy'x - У)т-
The substitution w(x) = xy'x — у leads to a third order equation: w'xxx = awm (Sec-
(Section 3.2 presents its solutions for m = — у, — ¦§-, —2, —-j, — -g-, —-i-, 0, and 1).
11. x2y"" -
The substitution w(x) = x2y leads to an equation of the form 4.2.1.1: wxxxx = (
= оУ~Ъ
The substitution t = In |ж| leads to an equation of the form 4.2.1.1: yxxxx =
* ** **XXXX **X **X
Having integrated this equation, we obtain the third order equation y'xxx = Cya whose
solvable cases are specified in Section 3.2.
14- yy'Zxx + ±У'хУххх + ЧУххJ = «*"•
This is a special case of equation 4.2.3.22 with f(x) = axn.
* ** ** xxxx ' Уx Уxxx ' V ** xx / {/ ¦
The substitution w = y2 leads to an equation of the form 4.2.1.1: wxxxx = 2aw~5'3.
The transformation x = x(t), у = (x't) leads to a constant-coefficient fifth-order linear
equation: 2xf = ax + b.
17- У3Ухххх = ^У'хУххх + ЗуЧУххJ - Ну'хL-
This is a special case of equation 4.2.3.27 with / = 0.
Solution in the parametric form:
* & XX^XXXX V УXXX/
Solution:
y = J Co + Cia; + (C2 + С3ж) i-а if а ф 1,
1
3-2a
if a = 1.
© 1995 by CRC Press, Inc.

i -v) + Pv'x + 7-
Differentiating with respect to x yields
yx1x(yx5)-ax-0)=O.
By equating the expression in the parentheses to zero and integrating it, we find the
solution:
у = a-||- + /3-|p + С4ж4 + С3Ж3 + C2x2 + dx + Co.
The constants Cfc and parameters a, [3, and 7 are related by the constraint
48C2C4 - 18Cf = -aC0 + /3Ci + 7
obtained by means of substituting the solution into the original equation. In addition,
there exists the solution
y = C1x + C0, where aC0 - f3C1 - 7 = 0.
20. у"" = ауку' Ы" Y.
&XXXX €t €t a; \ &XXX '
This is a special case of equation 4.2.3.29 with f(y) = Ayk, g(w) = ws. For к = — 1
and s = 1, see equation 4.2.1.13.
The first integral has the form:
-С
= с
= с
11 /С
о
if*;
ф-1, s
ф-1, s
= -1,3
^1;
= 1;
= L
(!)
B)
C)
For С = 0, equality A) is changing to the equation
УXXX [ jfc + 1
which is discussed in Section 3.2 (the solutions given there generate 3-parametric
families of particular solutions of the original equation for к = A — s)C — 1, where
Ц = --I, -f, -2, -f, --I, -|, 0, and 1).
4.2.2. Equations Containing Exponential, Hyperbolic, Logarithmic, and
Trigonometric Functions
This is a special case of equation 4.2.3.1 with f(y) = aeXy.
2- V™ = °(У + be-)/3 - be-.
The substitution w = y+bex leads to an equation of the form 4.2.1.1:
© 1995 by CRC Press, Inc.

3. tC» = «(У + be')m ~ Ье^.
The substitution w = y + bex leads to an equation of the form 4.2.1.2: wxxxx = awm.
4- tC, ~ 4AtC + 6A2yL - 4A3< + Л4у = оexp(f ^
The substitution w(x) = ye Xx leads to an equation of the form 4.2.1.1: wxxxx =
aw~5/3.
5. y'xxxx ~ ^У'ххх ~l~ ®^2УХХ — 4A3y^, + A4y = aexA~rn^xym.
The substitution w(x) = ye~Xx leads to an equation of the form 4.2.1.2: wxxxx = awm.
6. yy"" +4y/y/// + 3(y" Г = аеХх.
if & XXXX & X & XXX >¦ &XX ' -^-w
Solution: y2 = C3x3 + C2x2 + Cxx + Co + 2a\~AeXx.
7- УУ'1'L* + ±y'xvZx + 3(tO2 = а со3Ь(ЛЖ).
Solution: y2 = C3x3 + C2x2 + dx + Co + 2aA cosh(Arr).
'ХУХХХ
8- УУ'ЛХХ + ±У'ХУХ
This is a special case of equation 4.2.3.22 with f(x) = atanhm(Aa;).
9- tC^
This is a special case of equation 4.2.3.1 with f(y) = alnm(by).
This is a special case of equation 4.2.3.2 with f(w) = alnw.
nil"" — пт
This is a special case of equation 4.2.3.3 with f(w) = 2alnw.
yy'Zxx + ±У'ХУХХХ + ЧУХХJ = о lnm(Aa;).
This is a special case of equation 4.2.3.22 with /(ж) = а1пт(Аж).
This is a special case of equation 4.2.3.1 with f(y) = acosm(Xy).
This is a special case of equation 4.2.3.1 with f(y) = atanm(Ay).
J2 = а со3(АЖ).
Solution: y2 = C3x3 + C2x2 + dx + Co + 2aA cos(Arr).
16- yy'Zxx + ±У'ХУХХХ + Hy'LJ = о tan-САж).
This is a special case of equation 4.2.3.22 with f(x) = atanm(Aa;).
© 1995 by CRC Press, Inc.

4.2.3. Equations Containing Arbitrary Functions
By integrating, we obtain
ZVxVxxx - {y'Lf = 2 / f(v) dy + 2G.
The substitution w(y) = \y'x\3'2 leads to a second order equation:
2- tC, = ж/(у
The transformation x = t~1, у = wt~3 leads to an equation of the form 4.2.3.1:
<ш = /И-
3. У'1'L* = x
The transformation x = e*, у = x3/2w leads to an equation of the form 4.2.3.14:
//// 5 // 9 _\_ f f \
The substitution w = у + ax3 + fix2 + jx + 6 leads to an equation of the form 4.2.3.1:
<xxx = /W-
CLX I h 1J
The transformation ? = In , w = —r- leads to an autonomous equation of the
form 4.2.3.34.
6- VZXX = (<™2 + Ьх + с)
(ож2 + ож +
1°. The transformation
ж2 + bx + с ' (ax2 + bx + cK/2
leads to an autonomous equation of the form 4.2.3.14 for w = w(^):
w^xx-^Aw^ + ^A2w = f(w), where A = b2 - 4ac.
Therefore, having integrated the latter equation, we obtain
ЧЧ« - i«?J - f AKJ = --kAW +1 /Иdw + c-
The substitution z(w) = \w'A leads to a second order equation:
f(w) dw + C] z-b'3.
2°. The first integral of the original equation has the form
(Py'x - \P'xy)y'^xx - \P{y'Lf + \P'ML + ^ayt/xx - MVxf = JfH dw + C,
where P = ax2 + bx + c, w = yP~3^2.
© 1995 by CRC Press, Inc.

7- tC» = f(v + «О - ae*.
The substitution w = у + аеж leads to an autonomous equation of the form 4.2.3.1:
<xxx = /W-
8- Ухххх = f(V + о cosh ж) - о cosh ж.
The substitution w = y+a cosh x leads to an autonomous equation of the form 4.2.3.1:
<xxx = /W-
9. y'x'Jxx
The substitution w = y + asinhx leads to an autonomous equation of the form 4.2.3.1:
<xxx = /W-
10- У'х'ххх = f(y + a cos x) - a cos ж.
The substitution ад = y + acosx leads to an autonomous equation of the form 4.2.3.1:
<xxx = /W-
И- Ухххх = f(V + a sin Ж) - О Sin Ж.
The substitution ад = у + a sin ж leads to an autonomous equation of the form 4.2.3.1:
<xxx = /W-
By integrating, we find
For <7(ж) = 0, the order of this equation can lowered by one with the help of the
substitution w(y) = y'x.
13. y'Zxx = *-Af(*y'x - v)-
The transformation t = In |ж|, ад = ху'х —у leads to a third order autonomous equation
of the form 3.5.5.9: ад™ - 5ад^ + 6ад? = /(ад).
Having integrated this equation, we obtain
2Ух1/ххх - (v'Lf + <У'х? = 2 / f(v) dy + 2C
where С is an arbitrary constant. The substitution w(y) = |y^|3'2 leads to a second
order equation:
= -f aw-1/3 + I-
+ I- У f(y) dy + с] ад/3.
© 1995 by CRC Press, Inc.

'x - v)v'L-
The substitution t = \n\x\, w = xy'x — у leads to a third order equation:
Integrating it, we obtain a second order automous equation:
w"t -5w't + 6w = / f(w) dw + С
The substitution z(w) = \w't leads to the Abel equation of the second kind:
zz'w-z = -± [-6w + J f(w) d
(see Section 1.3).
dw
The substitution w = x2yxx — 2xy'x+2y leads to a second order equation: xwxx — 2wx =
xm+3f(w).
For m = —4, the substitution z(w) = \xw'x leads to the Abel equation of the
second kind: zz'w — z = -^f(w) (see Subsection 1.3.1).
tC» + atC + by'L + cy'x = eA-/(ye-A-).
The substitution w(x) = ye~Xx leads to an autonomous equation:
<Lx + DA + a)w':xx + FA2 + 3aA + b)<x
+ DA3 + 3aA2 + 26A + c)< + (A4 + a\3 + b\2 + cX)w = f(w),
which can be reduced to a third order equation by means of the substitution z(w) = w'x.
For a = —4A and с = 8A3 — 2bX, the above equation coincides, to a precision of the
notation, with the equation 4.2.3.14 and can be reduced to a second order equation.
The substitution w(x) = xy leads to an equation of the form 4.2.3.1: wxxxx = f(w).
19- ^y'Zxx 2
The substitution w(x) = x2y leads to an autonomous equation of the form 4.2.3.1:
<xxx = /W-
20. ^y'Zxx + a3x3yZx + a2x2y'x-x + alXy'x =
The substitution t = In |ж| leads to an autonomous equation:
y'Zt + («з - 6)Уш + (И " 3a3 + a2)y'trt + Bа3 - а2 + d - 6)y't = f(y),
the order of which can lowered with the help of the substitution w(y) = y't. For аз = 6
and ai = й2 — 6, the latter equation coincides, to a precision of the notation, with the
equation 4.2.3.14 and can be reduced to a second order equation.
© 1995 by CRC Press, Inc.

The transformation i = In ж, ад = ya?fe leads to an autonomous equation of the form
4.2.3.34.
Solution: y2 = C3x3 + C2x2 + Cxx + Co + | f (x - tff(t) dt.
JIn
22.
23. yy'Zxx + ay'xy'Zx + (о -
Having integrated this equation, we find
VV'L + \^Ш2 = Сгх + С0+ Г(х- t)f{t) dt.
xn
The substitution ад = (yy'x)xx leads to a first order linear equation: «4 + /ад + g = 0.
Solution:
y2 = C2x2 + dx + Co + I (x - tJw(t) dt,
Jx0
where ад (ж) = e~F^[Cs — f eF^g(x)dx], F(x) = f f(x)dx; xq is any number.
25. yy'Zxx + (±y'x+fy)yZx+4yLJ + (zfy'x+gy)yZ+9(yxJ+hyyx+s = o,
where / = /(ж), g = g(x), h = h(x), s = s(x).
The substitution w = yy'x leads to a nonhomogeneous third-order linear equation:
w'xxx + fwxx + 9w'x + hw + s = 0.
26. (y + ax + b)yZxx + Цу'х + a)y'Zx + 3«J2 = f(x).
Solution: (y + ax + bJ = C3x3 + C2x2 + Cxx + Co + | / (x - tff{t) dt.
¦J Xq
* ** **XXXX *7X**XXX ' ^& XX' О I |_c? ifXX ^**X' J v I
У \ У
у1 у" (у1 \2
The transformation f = ——, w = —^ I —— leads to a second order linear
У У \ У J
У У \ У J
equation for w2: {w2)'L = 24?2 + 2/(?). Integrating yields
ад =
+ d + 2C4 + 2 / (C - t)f(t) dt.
Taking into account that ?'x = w, y'x = ^y, y'? = t;y/w, we find the solution in the
parametric form:
J w \J w
where
ад = ±WC2? + d + 2C4 + 2 J (C - t)f(t)
dt.
© 1995 by CRC Press, Inc.

28. y'Lv'L'L* - 3(tCJ2 = f(xy'x - y){y'Lf-
The Legendre transformation x = u't, у = tu't — и leads to an equation of the form
4.2.3.1: uZt = -/(«)•
29. y'Zxx
By integrating, we obtain a third order autonomous equation:
/ ~Jw) = I
the order of which can be lowered by means of the substitution z(y) = y'x.
30.
L - у
The substitution w(x) = xy'x — у leads to a third order equation of the form 3.5.2.11:
where F{Q = Г5/@-
„ = fi^Vxx - ^ХУх + 2уЫх2Уххх)-
The substitution ад (ж) = х2ухх — 2ху'х + 2у leads to a second order equation of the
form 2.9.4.2: w'^ = f(w)g(w'x).
32. y'xxxx = f(x)9(x3y'xxx — 3x2yxx -\- 6xy'x — 6y).
The substitution w(x) = x3y'xxx — Ъх2у'хх + 6xy'x — 6y leads to a first order equation
with separation of variables: w'x = x3f(x)g(w).
33. y'Zxx = /(«, y'x, y'L, у'ххх)-
The substitution w(x) = y'x leads to a third order equation: w'xxx = f(x, ад, w'x, wxx).
* ** xxxx J \c/' Уx ' ** xx ' & xxx / "
Autonomous equation.
The substitution w(y) = (y'x) leads to a third order equation:
ww'yyy + \w'yWyy
(v' v" v'"
or aJfff n»-?\ x "xx "xxx
35. yxxxx - yt{—' —, —
yl y" ( y> \2
The transformation ^ = ——, ад = ——— I —— leads to a second order equation:
У У \ У J
/ V1I> <r2ii" <rzii'"
ofi ..//// _ ...,-4 f / „km ХУх Х У xx X У xxx
ЛЬ- Ухххх~УХ I\X V , —jj~, " , "
The homogeneous equation in the extended sense.
I'll
The transformation t = xkym, z = —— leads to a third order equation.
У
© 1995 by CRC Press, Inc.

о7 //// _ -4 / ХУ А ^ А У
/ 111
The transformation z = ——, w = — leads to a second order equation.
У У
//// —„-4*(„т ay I 2 // 3 /// \
Ухххх — Л /\л е » "ЬУх! Л Ужа;' Л Уххх)'
The transformation z = жтеаз/, w = ху'х leads to a third order equation.
г'"
39.
"a; "a;a; "xxx
У У У
The transformation z = еахут, w = у'х/у leads to a third order equation.
© 1995 by CRC Press, Inc.

Chapter 5
Higher Orders
Differential Equations
5.1. Linear Equations
5.1.1. Preliminary Comments
In this Chapter, we denote higher derivatives by yx that stands for .
dx11
1. The general solution of a nonhomogeneous linear equation of the nth order
fn(x)yxn) + fn-iyxn-1] + ¦¦¦ + fi(x)y'x + fo(x)y = 0 A)
has the form
У = Ciyi(x) + C2y2(x) -\ h Cnyn(x), B)
where уг{х), у2(х), ..., yn{x) make up a fundamental set of solutions (y^ are linearly-
independent solutions; j/fe ^0); C\, C2, ..., Cn are arbitrary constants.
2. Let yo = yo{x) be a nontrivial particular solution of equation A). Then, the sub-
substitution у = yo(x) J z(x) dx leads to a linear equation of the (n — 1) th order for function
z(x).
Given m linearly-independent solutions уг{х), y2(x), ..., ym(x) of equation A), its
order can be lowered down to (n — m) by the following technique. The substitution у =
Ут(х) J z(x) dx leads to an (n — 1) th order equation for z(x), with the following linearly-
independent solutions known:
2/m-l
^m—1 —
f—V z = f—V
Ут
Furthermore, the substitution z = zm-i(x) f w(x) dx yields an (n — 2) th order equation,
etc. Thus, the above procedure applied m times results in an (n — m) th order homogeneous
linear equation.
3. A nonhomogeneous linear equation of the nth order has the form
fn(x)yin) + fn-W^ + ¦¦¦ + h(x)y'x + fo(x)y = g(x). C)
The general solution of equation C) is the sum of its particular solution and the general
solution of the corresponding homogeneous equation A).
© 1995 by CRC Press, Inc.

Let {2/1 (ж), ..., yn{x) } be a fundamental set of solutions of the homogeneous differ-
differential equation A), and W(x) is the Wronskian:
2/2, ••., yn) =
2/i О)
2/n
D)
where y(™\x) = 4"lr> m = 1, 2, ..., n - 1; к = 1, 2, ..., n. Denote by W^(rr)
determinant D) wherein the J^th column is replaced by the column 0, 0, ..., 0, g (from top
to bottom). Then, the general solution of the nonhomogeneous linear equation C) can be
written as
У =
v=l
^yv{x) I —
Wv{x)dx
[x)W(x)
5.1.2. Equations Containing Power Functions
+ay = 0.
1°. Fora = 0,
2°. For a =
у = d + C2x + C3xz + dx6 + С5Ж4 + C6x
lev
у = C\ cos kx + C<i sin kx + cos -— (C3 cosh ^ + C4 sin
sm— (C5 cosh ? + C6 sinh ?),
3°. For a = -k6 <0,
kx
kx
у = C\ cosh kx + C<i sinh kx + cosh -— (C3 cos ^ + C4 sin
Z
. . kx ,
+ smh— (
о B«) In
2. yx = ay.
Solution:
У =
where ipk = «жcos
n
arbitrary constants.
C2e~ax
n-l
fe=l
= axsin ; C\,
n
Bfesin^fe),
(fc = 1, 2, ..., n - 1) are
© 1995 by CRC Press, Inc.

3. yin) + an^yn~x +••• + alV'x + aoy = 0.
The homogeneous constant-coefficient linear equation.
To solve this equation determine the n roots of the characteristic polynomial
P(A) = A" + an-iA" + • • • + aiA + a0.
The general solution is determined by these characteristic roots. Several cases are
possible:
1. The roots are all real and different. Denote them by Ai, A2, ..., An. Then, the
general solution of the original equation is
у = C\ exp(Aia?) + C2 ехр(А2ж) -\ + Cn exp(A2n).
2. There are m < n equal real roots, Ai = A2 = ¦ ¦ ¦ = Am, while the other roots are
real and different. In this case, the general solution is
у = exp(Aia;)(Ci + C2x + ¦ ¦ ¦ + V1)
+ Cm+1 exp(Am+irr) + Cm+2 exp(Am+2rr) -\ \-Cn exp(A2n).
3. There are m equal pairs Bm < n) of complex conjugate roots, A = a ± г/3, while
the other roots are real and different. Then, the general solution has the form
у = ехр(аж) cos(/3rr)(^i + A2x -\ h Amxm~1)
+ exp(arr) sin(/3rr)(Bi + B2x + ¦ ¦ ¦ + Bmxm-1)
+ C2m+i exp(A2m+ia;) + C2m+2 ехр(А2т+2ж) Н YCn exp(A2n),
where A\, ..., Am, Вг, ..., Bm are arbitrary constants.
4. In the general case, there are r different roots Ai, A2, ..., Ar of multiplicities
Wi, m2, ..., mr, respectively. Hence, the characteristic polynomial can be factorized:
P(A) = (A - A!)mi(A - A2)m2... (A - \r)m\
where m\ + m2 + ¦ ¦ ¦ + mr = n. Then, the general solution of the original equation is
given by the formula
r
y = Y, exp(Afea;)(CM + Ck>1x + ¦¦¦ + Ск.т*-^*-1),
fe=i
where Ck,i are arbitrary constants.
If -P(A) has complex conjugate roots, in the above solution the real and imaginary
parts should be taken, in view of the formula: exp(a ± г/3) = ea(cos/3 ± г sin/?).
4. Ух = axy + b, о > 0.
Solution:
" /-oo r ^n+1 I
У = V Cvev I exp evxt ——— dt,
^ Jo L a(n + l)\
( 2nvi \ -A b ,2
where ev = exp — , > Lv = —, г = —1.
V n+ 1 / -^ a
© 1995 by CRC Press, Inc.

5. yxn) + axvy'x + auxv~xy = 0.
This equation can be reduced to an (n — 1) th order equation: y% + axvy = C,
where С is an arbitrary costant.
ft »,(") j.ni.Hl,/ _ „(r, _ iWfc?/ — П
The substitution z = xy'x — (n — l)y leads to an (n — 1) th order equation: z% +
axk+1z = 0.
7. yin) + axk+1y'x + a(k + n)xky = 0.
The transformation x = t~x, у = wt1~n leads to an equation of the form 5.1.2.5:
wt(n) + btvw't + bv^-^w = 0, where b = a(-l)n+1, v = l-k-2n.
8. yxn) + axky(xm) - (abmxk + bn)y = 0.
Particular solution: yo = еЬх ¦
9. yin) + (axk - bn-m)yxm) - abmxky = 0.
Particular solution: yo = ebx ¦
10. yin) + (axm+1 + bxm)y'x - axmy = 0.
Particular solution: yo = ax + b.
11. yin) + ay{x~^ + bxmy'x + abxmy = 0.
Particular solution: yo = e~ax.
12. xy(xn) - nmyxn~x) + axy = 0, n = 2, 3, 4, ..., m = 1, 2, 3, ...
Solution:
у = x
where w is the general solution of the constant coefficient equation u4 + ^w = 0.
13. xyx + nyx = axy + b.
The substitution w = xy leads to a constant coefficient equation: «4 =
aw
14. xyx + nyx = ax у + b.
The substitution w = xy leads to an equation of the form 5.1.2.4: «4 = olxw + b.
15. xy(xn) + (n-m- l)yi"~1) + axky'x - amxk~xy = 0.
Particular solution: yo = xm.
16. xyx + axkyx — (axk + amxk~1 + x + n)y = 0.
Particular solution: yo = xex.
© 1995 by CRC Press, Inc.

n— 1
17. хУх = /_j [(o^4.^+i — Au)x + Au+i]y^>,
where An = 1, ^4o = 0; a and Aj, are arbitrary numbers (v = 1, 2, ..., n — 1).
Denote /(A) = X)"=o-^н-i^"- L6* a^ ^e roots ^i> ^2, •••, An_i of the equation
/(A) = 0 be different, and /(a) 7^ 0. Then, the solution is as follows:
18. ^ (avx + bv)y^ = 0.
The Laplace equation.
Particular solutions:
У/с =
where P(i) = 5Z™_0 aut", Q{t) = X)"=o ^*"! ak an<i /^fc are found from the condition
= 0.
In many cases, the path of integration should be chosen on the complex plane.
19. aryi y + 2ггжу^ y + n(n — l)yi y = ax*y + b.
The substitution w = ж2у leads to a constant coefficient equation: wj." = aw + 6.
20. x^y^. ' + 2nxy±. ' + n(n - l)yi ' = axdy + b.
The substitution w = x2y leads to a equation of the form 5.1.2.4: «4 = axw + b.
21. x{x + m)yin) + x(axk - ж - n)yim) - a(x + m)xky = 0.
Particular solution: yo = xex.
22. x2nyin) =
ay.
The transformation x = t , у = wt n leads to a constant coefficient equation:
w(n) = (_]
23. ж"у?2гг) = ay.
Solution:
fe=i
where Jn and ATn are modified Bessel functions; /?i, /З2, • • •, 0n are the roots of the
equation Cn = i/a.
© 1995 by CRC Press, Inc.

24. x3nyi2n) = ay.
The transformation x = i, у = wt1~2n leads to an equation of the form 5.1.2.23:
tnwfn) = aw.
25. xn+1^y{2n+1) = ay.
Solution:
2n
У = xBn+i)/4 ^2 Cfe[J_n_1/2B/3feV^) + iJn+1/2B[3kVx~)],
k=0
where /?o, Pi, • • •, p2n are the roots of the equation /32n+1 = —ai; i2 = —1.
26. x3n+3/2yi2n+1) = ay.
The transformation x = i, у = wt~2n leads to an equation of the form 5.1.2.25:
.„_]_! /2 Bn+l)
tn+ I w\ ' = —aw.
27. anxnyin) + an-1xn-1yin~1) -\ \- alXy'x + aoy = 0.
The Euler equation.
If all the roots A^ (k = 1, 2, ..., n) of the algebraic equation
n
a»4x - 1) • • • (A - rv + 1) = -a0
are different, the general solution of the original differential equation has the form
У
= d\x\Xl + C2\x\X2 + ¦ ¦ ¦ + C
ЖГ".
In the general case, the substitution t = In |ж| leads to a constant coefficient
equation of the form 5.1.2.3:
™ d
\^ avD(D — 1)... (D — v + l)v = —any, where D = ——.
t—1 ax
v=l
28. x2n+1yin) =ay + bxn.
The transformation x = t~1, у = wt1~n leads to an equation of the form 5.1.2.4:
»
29. x^+iy™ + nx^y^ = axy.
The substitution w = xy leads to an equation of the form 5.1.2.22: a?2n«4 = aw-
30. я!2"+1|/?") + пх^у^ = ay.
The substitution w = xy leads to an equation of the form 5.1.2.28: x2n+1w^ = aw.
о-. n Bn) , „ n —1 Bn-l)
31. a;"yi ' + 2ггж" 1yi ^ = ay.
The substitution w = xy leads to an equation of the form 5.1.2.23: xnwx = aw.
© 1995 by CRC Press, Inc.

32. x3ny{2n) + 2nx3n-1y(x2n-1) = ay.
The substitution w = xy leads to an equation of the form 5.1.2.24: x3nwx n' = aw.
гг. „4.1 Bti+l) , /„ , 4\ n Bn) /—
33. xn+1y^ T ' + (In + l)xny^ ' = ал/ху.
The substitution w = xy leads to an equation of the form 5.1.2.25: rrn+1/2«4 n =aw.
34. x3n+3/2yi2n+1) + (In + I)x3n+1^2y{2n) = ay.
The substitution w = rry leads to an equation of the form 5.1.2.26: x ' Wx n =
aw.
35. (ax + Ь)'2п+1у(^) = (ex + d)y.
The transformation
ex + d у
с = w =
ax + b '
w =
ax + b ' (ax + 6)™-1
leads to an equation of the form 5.1.2.4: wj1 = A~n^w, where A = be — ad.
36. (ax + b)n(cx + d)nyin) = ky.
1°. The transformation
ax + b
l
w=
err + d ' (err + d)n~
leads to a constant coefficient equation.
2°. The transformation
ax + b у
C= —7, w =
ex + d ' (ex + d)n~1
leads to the Euler equation 5.1.2.27: ?n«A = kA~nw, where A = ad — be.
37. (ax2 + bx + c)nyin) = ky.
The transformation
dx
2 , , , ,
arr2 + 6rr + с
leads to a constant coefficient equation.
38. (ax + b)n(cx + dKnyi2n) = ky.
The transformation
ax + b
/2т n 1
w; = y(arr2 + 6rr + c) 2
ex + d ' (ex + dJn~1
leads to an equation of the form 5.1.2.23: ?nwi = kA~2nw, where A = ad — be.
© 1995 by CRC Press, Inc.

39. (ax + b)n+1/2(cx + dKn+3/2yx2n+1) = ky.
The transformation
ax + b у
?= —г, w =
cx + d' (ex + dJn
leads to an equation of the form 5.1.2.25: ?n+i/2w&n+i) = kA-2n-iw^ where д =
ad — be.
ЛС\ p t (тЛи —I— P о (тЛи —I— ... —I— Pi (тЛи —I— (ti i т —I— hi }ii tnfi til ^^ П
^t\j* ± п—lv^^/c/aJ i^ fi—2 V. ^ / Ух i^ i^ ± 1 V. / Ухх '^ vu'lt*y i^ *-/Ух "tI*ly — ^?
where Pv are polynomials of the degree < v, m is a positive integer, a\ ф 0.
A particular solution of this equation is the polynomial of degree m which can be
written as
m 1 ч fe
2/0 = E(-—) [xmIx-m~1(Pn-1Dn + ¦¦¦ + PXD2 + b1D)}kxm,
k=0 X
d xv+1
where D = ——, Ixv = with v Ф —1.
dx v + 1
41. [anxn + Ргг_1(ж)]у1гг) H h [oioj + P0(x)]y'x + aoy = 0,
where Pv are polynomials of the degree < v.
Assume that for some integer m > 0,
C>!a, = 0, where C»m =
v=0
m!
and m is the least of the numbers satisfying this condition. Then, there exists a
solution in the form of a polynomial of degree m such that no polynomial of a smaller
degree satisfies the original equation.
5.1.3. Equations Containing Exponential Functions
1. yin) + (ax + b)eXxy'x - aeXxy = 0.
Particular solution: yo = ax + b.
2. yin) + (aeXx - bn-m)yim) - abmeXxy = 0.
Particular solution: г/о = с •
3. yin) + ayin-x) + beXxy'x + abeXxy = 0.
Particular solution: г/о = e~ax.
4. yin) + aeXxyim) - (abmeXx + bn)y = 0.
Particular solution: г/о = еЬх ¦
© 1995 by CRC Press, Inc.

n— 1
5. у™ = J2 (Ak+ieXx + ЬАк+1 - Ak)yxk\
k=0
where An = 1, Aq = 0; 6 and A^ are arbitrary numbers (k = 1, 2, ..., n — 1).
Particular solutions: ym = e^mX, where /лт are the roots of the polynomial equation
6. xyin) + axeXxyim) - [a(x + m)eXx + x + n]y = 0.
Particular solution: yo = xex.
7. xyin) + (n-m- l)yi"~1) + axeXxy'x - ameXxy = 0.
Particular solution: yo = xm.
8. ж(ж + m)yx + x(aeXx — ж — n)yx — a(x + m)eXxy = 0.
Particular solution: yo = xex.
9. (axm + bex + c)yxn) = bexy, ra = 0, 1, 2, ..., n - 1.
Particular solution: yo = axm + bex + с
10. (axmex + b)yxn) = (-l)nby, ra = 0, 1, 2, ..., n - 1.
Particular solution: yo = axm + be~x.
(П 1 ч
aex + У~] bkxk I yxn^ = aexy.
k=O '
n-1
Particular solution: yo = aex + / b^x .
k=o
12. yxn) + a coshfc ж yxm) - (abm coshfc ж + bn)y = 0.
Particular solution: yo = e .
13. yxn) + (a coshfc ж - bn-m)y(xm) - abm coshfc ж у = 0.
Particular solution: yo = ebx.
14. yx + (ax + b) coshm(Xx)y'x — a coshm(\x)y = 0.
Particular solution: yo = ax + b.
15. жуа," + ax coshfc ж yx — \a(x + ra) coshfc ж + ж + n]y = 0.
Particular solution: yo = xex.
16. yin) + a sinhfc ж y?m) - (abm sinhfc ж + b")y = 0.
Particular solution: yo = ebx.
© 1995 by CRC Press, Inc.

17. yin) + (a sinhfc ж - bn-m)yim) - abm sinhfc ж у = 0.
Particular solution: yo = ebx.
18. yin) + (ax + b) sinhm(Xx)yfx - asinhm(Xx)y = 0.
Particular solution: yo = ax + b.
19. xyin) + ax sinhfc ж yxm) - [a(x + m) sinhfc ж + ж + п]у = 0.
Particular solution: yo = жеж.
20. yin) + a tanhfc ж y(xm) - (abm tanhfc ж + bn)y = 0.
Particular solution: yo = ebx.
21. yin) + (a tanhfc ж - bn-m)yxm) - abm tanhfc ж у = 0.
Particular solution: yo = e .
22. yin) + (ax + b) tanh™^^ - otanh"г(Aж)y = 0.
Particular solution: yo = ax + b.
23. xyxn) + ax tanhfc ж yxm) — [a(x + m) tanhfc ж + ж + п]у = 0.
Particular solution: yo = xex.
24. yin) + a cothfc ж y?m) - (obm cothfc ж + b")y = 0.
Particular solution: yo = ebx.
25. yxn) + (a cothfc ж - Ьгг~"г)у?т) - obm cothfc ж у = 0.
Particular solution: yo = e .
26. yi") + (ax + b) coth"г(Aж) y^ - а соШ^Аж) у = 0.
Particular solution: yo = ax + b.
27. xyx + ax cothfc ж yx — \a(x + m) cothfc ж + ж + п]у = 0.
Particular solution: yo = xex.
5.1.4. Equations Containing Trigonometric Functions
y x yxm^ (
Particular solution: yo = ebx.
2. yi"} + (o sinfc ж - b"-m)yim) - abm sinfc ж у = 0.
Particular solution: yo = еЬж.
© 1995 by CRC Press, Inc.

3. yin) + ayjf-V + bsinm(Xx)yx + absinm(Xx)y = 0.
Particular solution: yo = e~ax.
4. yin) + (ax + b) sinm(Xx)yx - a sinm(Xx)y = 0.
Particular solution: yo = ax + b.
5. yin) + a cosfc ж yim) - (abm cosfc ж + bn)y = 0.
Particular solution: yo = еЬх ¦
6. yin) + (о cosfc ж - b"-m)yim) - abm cosfc ж у = 0.
Particular solution: yo = eba:-
7. yi"} + ау^п~г) + bcosm(Xx)y'x + abcosm(Xx)y = 0.
Particular solution: yo = e~ax-
8. yi"} + (ax + b) cosm(Xx)y'x - a cosm(Xx)y = 0.
Particular solution: yo = ax + b.
9. yin) + a tanfc ж yim) - (abm tanfc ж + b")y = 0.
Particular solution: yo = ebx.
10. yi™} + (o tanfc ж - Ьгг-"г)у?"г) - abm tanfc ж у = 0.
Particular solution: yo = еЬж.
11. yin) + аухп-г) + btanm(Xx)y'x + abtan™^)y = 0.
Particular solution: yo = e~ax.
12. yin) + (ax + b) tanm(Xx)y'x - о tan™^)y = 0.
Particular solution: yo = ax + b.
13. xyx + o,x sink(Xx)yxm — [a(x + m) з1п'в(Лж) + ж + гг]у = 0.
Particular solution: yo = xex.
14. xyin) + ax cosk(Xx)yim) - [a(x + m) сов'г(Лж) + ж + п]у = 0.
Particular solution: yo = xex.
15. xyx + о,х tanfc(Xx)yx — [а(х + га) tanfc(Лж) + ж + п]у = 0.
Particular solution: yo = хех.
16. (ахт+ bsinx)yxn) = bsin(x + ^Ау, га = 0, 1, 2, ..., п - 1.
Particular solution: yo = ахт + b sin x.
© 1995 by CRC Press, Inc.

n—1
7ГП
( \ (
17. I osina; + 2_, bkXk )y^ = osinla; -\ ]y.
n-1
Particular solution: уп = a sin ж + ^^ h ™k
k=o
18. (axm+ bcosx)yxn) =bcos(x +^^)y, т. = 0, 1, 2, ..., n - 1.
Particular solution: yo = o-xm + b cos ж.
П—1
/ ^-^ h\ ( \ ( 7Ггг \
19. la cos x + y, bkX I yxn> = о cos I x -\ 1 у.
у —^ J V 2 ^
n-1
Particular solution: yo = a sin ж + ^J b^xk.
k=o
5.1.5. Equations Containing Arbitrary Functions
1. yin) = f(x).
Solution:
n1 rx i _ f\n-
( J
where жо may be chosen arbitrarily.
2. yin) + xf(x)y'x - mf(x)y = 0.
If rn = 0, 1, 2, ..., or n — 1, the equation has a particular solution yo = xm, and the
substitution z = xy'x — my leads to an (n — 1) th order equation:
ax
In particular, for m = n — 1, we have z^ + xf(x)z = 0.
3. yin) + (ax + b)f(x)y'x - af(x)y = 0.
Particular solution: yo = ax + b.
4. yin) + f(x)(x2y'L ~ 2xy'x + 2y) = 0.
Particular solutions: yi = x, У2 = % ¦
The substitution z = x2yxx — 2xy'x + 2y leads to a linear equation of the (n — 2) th
order.
5. yin) + f(x)yim) - [о" + amf(x)]y = 0.
Particular solution: yo = eax-
© 1995 by CRC Press, Inc.

6. yin) +(f- an~m)yim) - amfy = 0, / = f(x).
Particular solution: yo = eax.
7. yx -\- ayx -\- fy -\- dfy ^ 0> f ^ f'ух).
Particular solution: yo = e~ax-
8. yin) + f(x)yin-1) + g(x)yin-2) + h(x) = 0.
The substitution w(x) = y^ leads to a linear equation of the second order: wxx +
f{x)w'x + g(x)w + h(x) = 0.
9. yi + an-iyxn H h oiy^ + °оУ = f(x).
The nonhomogeneous constant-coefficient linear equation.
The general solution of this equation is the sum of the general solution of the
corresponding homogeneous equation (see 5.1.2.3) and any particular solution of the
nonhomogeneous equation.
If all the roots of the polynomial
P(A) = A™ + an-iA™ H h aiA + a0
are different, the original equation has the general solution
у = у CveKx + V JL , / f{x)e~Kx dx
> CveKx+} \
^-j ^-j Р((А„) J
(with complex roots, the real part should be taken).
In Table 5.1 are listed the forms of particular solutions corresponding to some
special forms of functions on the right-hand side of the linear nonhomogeneous equa-
equation.
10. xnyin) + Ъп-гхп-ху{*~Х) + ••• + blXy'x + boy = f(x).
The substitution x = aef (а ф 0) leads to an equation of the form 5.1.5.9.
n— 1
11. »(,») + f(x) J2 (-l)fcfe!C^_1a;"-fc-1y?"-fc-1) = 0,
fc=0
where Cl, = ——. —- are binomial coefficients.
m fc!(m-fc)!
Particular solutions: ym = xm, where m = 1, 2, ..., n — 1.
n-l
The substitution z = ^2(-l)kk\C*_1xn~k~1yin~k~1) leads to a first order linear
fe=o
equation: z'x + xn~1f(x)z = 0. Having solved this equation, we obtain an (n — 1) th
order equation of the form 5.1.5.10 for function y(x).
n—l
12- !/<,"> =X>h+1/-ah)t/<,h\
fc=O
where / = f(x); an = 1, ao = 0; a^ are arbitrary numbers (k = 1, 2, ..., n — 1).
Particular solutions: y^ = eXkX (k = 1, 2, ..., n — 1), where Afe are the roots of the
polynomial equation X)fe=o ak+i^k = 0.
© 1995 by CRC Press, Inc.

TABLE 5.1
The forms of particular solutions of the nonhomogeneous constant-coefficient
linear equation yxn) + orl_iy?"~1) -\ \- a^y'x + aoy = f(x)
which correspond to some special forms of function /(ж).
The form
of function f(x)
Pm{x)
Pm(x)eax
(a is a real number)
Pm(x) cos fix
+ Qn{x) sinfix
[Pm(x)cos/3x
+ Qn{x) sin fix]eax
Roots of the characteristic equation
A™ + an-x\n~x Л V aiA + a0 = 0
Zero is not a root of the
characteristic equation (i.e., a§ ф 0)
Zero is a root of the
characteristic equation (multiplicity r)
a is not a root of the
characteristic equation
a is a root of the
characteristic equation (multiplicity r)
iC is not a root of the
characteristic equation
iC is a root of the
characteristic equation (multiplicity r)
a + г/3 is not a root of the
characteristic equation
a + г/3 is a root of the
characteristic equation (multiplicity r)
The form of a particular
solution у = y(x)
Pm(x)
xrPm(x)
Pm{x)eax
xrPm{x)eax
Pv (x) cos fix
+ Qv{x) sin fix
xr[Pv{x) cos fix
+ Qv (x) sin fix]
[Рг, (ж) cos fix
+ Qv{x) sin fix]eax
x1'[Pv(x) cos fix
+ Qv{x) sin fix]eax
Notation: Pm and Qn are polynomials of the degrees m and n with given coefficients;
Pm, Pv, and Qv are polynomials of the degrees m and v whose coefficients are de-
determined as a result of substituting the particular solution into the basic equation;
v = max(m, n); i = — 1.
13. xy(xn) + xfy(xm) - [(ж + га)/ + ж + п]у = 0, / = /(ж).
Particular solution: yo = хех.
14. ж(ж + т)ух + ж(/ — ж — п)ух — (ж + га) /у = 0, / = /(ж).
Particular solution: yo = хех.
15. хкух + xf(x)y'x — га/(ж)у = 0, га = 0, 1, 2, ..., п — 1.
Particular solution: yn = жт.
16. ж^у!^ + (п - га - 1)ж"-1у?гг-1) + xfy'x - mfy = 0,
Particular solution: yo = жт.
/ = fix)-
© 1995 by CRC Press, Inc.

(n)
im)
17. xny(xn) + xmfyim) - (n\Cl + m\C™f)y = 0,
where / = /(ж), С™ = ——у1 -у are binomial coefficients, Г (a) is the gamma-
it. i. I ct Tt ~т~ J-J
function.
Particular solution: yo = xa.
18.
n—l
fc=O
where / = /(ж); an = 1, ao = 0; m and a^ are arbitrary numbers (k = 1, 2, ..., n — 1).
Particular solutions: y^ = eAfea: (fc = 1, 2, ..., n — 1), where Afe are the roots of the
polynomial equation X)fe=o afe+i^fe = 0-
19.
fc=2
Particular solution:
The substitution
; - у).
= x.
= xy'x — у leads to an (n — 1) th order equation.
20.
fc=3
; + 2y).
Particular solutions: yi = x, У2 = x2.
The substitution w(x) = x2y'J.x — 2xy'x + 2y leads to an (n — 2) th order equation.
21.
fc=4
Particular solutions: y\ = x, yi = x2, уз = ж3.
The substitution ад (ж) = х3ух'хх — ix2yxx + &xy'x — 6y leads to an (n — 3) th order
equation.
22.
fc=m+l
where ClL =
fc=O
are binomial coefficients.
Particular solutions: ys = xs, where s = 1, 2, ..., m.
The substitution z = Y;T=o(~1)kk]Cmxm~kyim~k) leads to an (n ~ m)th order
equation:
n
E •
k=m+l
where D =
23.
fc=0
where fk = fk(x) (k = 1, 2, ..., n), /n+i = /0 = 0.
Particular solution: yo = eax.
© 1995 by CRC Press, Inc.

24. ? Ли + (к- m)/h+1]»W = 0,
fc=O
where /fe = fk(x) (к = 1, 2, ..., n), /n+i = /0 = 0.
Particular solution: yo = &"*•
^O. J yx — Jx у U5 J J \<ly I.
Particular solution: yo = /(ж).
The first integral has the form
In .
E(—l)k fx n~ yx = I q(x)dx + C.
\ / JX УХ J »V )
k=0 J
27. sinxyxn) +sinxf(x)yxm) - [sin (ж + -^) + /(ж) sin (ж -\ — )\y = 0.
Particular solution: yn = sin ж.
28. соэжу^ + cosxf(x)yxm^ — соэ(ж -\ j + /(ж) соэ(ж Н j у = 0.
Particular solution: уо = совж.
29. yin) = f(x)y.
The transformation x = t~x, у = wt1~n leads to an equation of the similar form:
30.
The transformation ? = , w = — —— leads to the equation wj1 =
cx + d (crr + d)™ ?
A-nf(?)w, where A = ad - be.
31. yin) + f(x)y'x + g(x)y + h(x) = 0.
The transformation x = t~1, у = wt1~n leads to an equation of the similar form:
w;
32. yi"+2) + /(ж)[ж2у1 - 2пжу; + n(n + l)y] = 0.
The substitution w(x) = x2y'^.x — 2nxy'x + n(n + l)y leads to an nth order equation:
w{xn) + x2f(x)w = 0.
© 1995 by CRC Press, Inc.

33. y^+yi /()у д(
X -1 ^ X
The transformation t = ж, w = yx2~n leads to an nth order equation: w]. =
(-l)n[f(t)w
34. x*yin+2)+vi+0v
+ f(x)[x2y'^x + (a - 2n)xy'x + (f3 - an + n2 + n)y] = 0.
The substitution
w(x) = x2yxx + (a - 2n)xy'x + (C - an + n2 + n)y
leads to an nth order equation: wx + f(x)w = 0.
5.1.6. Asymptotic Solutions
This subsection presents asymptotic solutions, as e —>• 0 (e > 0), of some fifth-order
linear ordinary differential equations containing arbitrary functions (sufficiently smooth),
with the independent variable being a real number.
1. Consider an equation of the form
eny(^ ~ f(x)y = 0. A)
on a closed interval a < x < b. Assume that / ф 0. Then, the leading terms of the asymptotic
expansions of the fundamental system of solutions, as e —>• +0, has the form
"* + ^ expj ^f- J [f(x)} ^ <fc j [1 + O(e)},
where u>\, u>2, ¦ ¦ ¦, ton are the roots of the equation uin = 1:
/ 2ттт \ . . / 2wm
tt>m = cos J+isinl , m = 1, 2, ..., n.
V n / V n /
2. Consider an equation of the form
e«y(") + e—V^i^yi") + • • • + еЛ(ж)у; + fo(x)y = 0. B)
on a closed interval a < x < b. Let Am = Ат(ж) (т = 1,2, ...,n) be the roots of the
characteristic equation
P(x,X) = Xn + /n_i(rr)An-1 + • • • + Л(ж)А + fo(x) = 0.
Let all the roots of the characteristic equation be different on the interval a < x < b, i.e.,
the conditions Ат(ж) ф Xk(x) if m ф к are satisfied, which is equivalent to the fulfillment
of the conditions P\(x, Am) ф 0. Then, the leading terms of the asymptotic expansions of
the fundamental system of solutions, as e —>• +0, are given by the formulae
P ( \ ( W
exp<^ — / Am(rr) dx - — /
nx Px(x,Xm(x))
where
ж, A) = -It- = пХп~г + (n - lJf-'A11 + ... + 2А/2(ж) + Д(:
РЛЛ(Ж, A) = ^- = n(n - 1)A" + (n - l)(n - 2)fn-1Xn~3 + ... + 6Xf3(x
© 1995 by CRC Press, Inc.

5.2. Nonlinear Equations
5.2.1. Equations Containing Power Functions
This is a special case of equation 5.2.6.1 with f(x) = axn.
2. yi6) V5
Multiplying both sides by y7/5 and differentiating with respect to x, we obtain the
equation 5yyx + 7y'xyx = 0. Having integrated the latter three times, we arrive at
a chain of equalities:
5yyx6) + 2y'Jx5) - 2y'lxyZxx + Wxxf = 2C2, A)
х - 3yxyxxxx + УХХУХХХ = 2С2ж + Ci, B)
х - 8j4Cx + l(y;'J2 = с2Ж2 + с1Ж + Co, (з)
where Co, C\, and C2 are arbitrary constants. Eliminating the highest derivatives
from (l)-C), with the aid of the original equation, we can obtain a third order
equation which can be reduced to a second order equation (see equation 5.2.1.4 with
n = 3).
3. yyLe) + 6y'xVib)
This is a special case of equation 5.2.6.4 with f(x) = axn.
l+2n
4. yx2n^ = Ay i-2« .
2n+l
Multiply both sides by у 2«-i and differentiate with respect to x. As a result we
obtain
Bn-l)yyx2n+1) + Bn+l)y'xyx2n)=0.
Three integrals containing arbitrary constants Co, C\, and C2 are presented in 5.2.6.22
wherein we should let / = 0. Eliminating the highest derivatives from those integrals
and the original equation, we may always obtain a Bn — 3) th order equation. With
the aid of the transformation
J P'
l-2n
t= —, w = yP 2 , where P = C2xz + dx + Co,
the latter equation can be reduced to the autonomous form 5.2.6.40. Therefore, the
substitution z(w) = w't finally leads to a Bn — 4) th order equation with respect to
z = z(w).
5. yi2n) = Ayk, к ф -1.
Having integrated, we arrive at
where С is an arbitrary number. Further, the order of the obtained autonomous
equation can be lowered by the substitution w(y) = y'x.
© 1995 by CRC Press, Inc.

6. yin) = ах~пут.
This is a special case of equation 5.2.6.10 with f(y) = aym.
7. yin) = axkym.
1°. The transformation x = i, у = t1~nw(t) leads to an equation of the similar
form: Wt(n) = [-l)nAt-k-(n-l)m-n-lwm^
2°. The transformation ? = xn+kym~1, z = xy'x/y leads to an (n—1) th order equation.
8. УУп = axn + b.
This is a special case of equation 5.2.6.16 with f(x) = axn + b.
n ».(") — „m—nm—n—l(n1. i и„п—1\т
This is a special case of equation 5.2.6.11 with f(w) = (aw + 6)m.
m—2nm—2n—1 / 2n—1 \ m
10. y^fn) = x 2 (ay + bx2
This is a special case of equation 5.2.6.12 with f(w) = (aw + b)m.
11. yin) = (ay + Ьхк)т; к = 1, 2, ..., n - 1.
The substitution aw = ay + 6a?fe leads to an autonomous equation: wxn = amwm (see
5.2.1.4, 5.2.1.5, and 5.2.6.40).
m.—nm—n—1
12. y^n) = (ax2 + bx + c) * ym.
This is a special case of equation 5.2.6.21 with f(w) = wm.
13. yin) = (ax + b)~n(cx + d)m-nm~xym.
This is a special case of equation 5.2.6.20 with f(w) = wm.
t л Bn+l) / Bn)
14. ууп = ay'xy^ .
The equation admits two different (with аф —1) first integrals:
Bn) _ ri a
Ух — <-аУ >
n-l
уУх2п) + (a + i)Y (-l)myim)yi2n~m) + i(-l)"(« + 1) \yin)}2 = C2,
m—1
where C\ and Ci are arbitrary constants. Eliminating the highest derivative from the
integrals, we arrive at a Bn — 1) th order autonomous equation:
n-l
У^ (_l)m (m) Bn-m) _^_ W_^n Гу(п)] 2 = Q ya+1 + Co,
m—1
where C\ = , Co = • The order of the obtained equation next can be
a+l a+l
lowered by the standard substitution w(y) = y'x.
© 1995 by CRC Press, Inc.

л- (п— 2) (тг) / (тг—1)\2
15. yi ;yi = а(ух ') .
Solution:
y = I Co + ClX + ¦ ¦ ¦ + Cn-3xn~3 + (Cn_2 + Cn-lX)n-2+-^ if а ф 1,
I Co + dx + ¦ ¦ ¦ + Cn-3xn-3 + Cn-2 exp(Cn_irr) if a = 1.
16. yi") = ax™-^1-™^)™.
This is a special case of equation 5.2.6.15 with f(w) = awm.
л- ("•+!) к I I (n)\m
17. yi T = ayky'x{yyx >) .
This is a special case of equation 5.2.6.17 with f(y) = y~k, g(w) = awm.
18. yln)=ax
The substitution w(x) = xy'x — у leads to an (n — 1) th order equation:
dxn-2
/^л чгп-з / (тг 1)\
I Jy ф *Jx ^^~ \JbrJLr С/ I С/ J * * * \ У «с / ~~^ ¦
Homogeneous equation in the extended sense.
Til
The transformation ? = xxylx, w = ——, where
У
A = n + mi — тез — 2m.4 — • • • — (n — l)mn+i, /x = m.2 + тез + ¦ ¦ ¦ + топ+1 — 1,
leads to an (n — 1) th order equation.
20. xy(xn) + пухп~1] = axmym.
This is a special case of equation 5.2.6.23 with f(w) = awm.
in~2) = ax2
у
m
21. x2yxn) + 2nxyxn 1} + n(n - l)y
This is a special case of equation 5.2.6.24 with f(w) = awm.
22. Bn - I)yyi2"+1) + Bn + l)y'xy{xn) = axm.
This is a special case of equation 5.2.6.22 with /(ж) = axm.
23.
The transformation x = x(t), у = (x't) leads to a constant coefficient linear equation:
2x{n+1) = ax + b.
n—l
24. 2 ? (-lry^yi2™-™) + (-irfy^]2 + X(y'xf = ay2 +by + c.
m=l
Differentiating both sides with respect to x and dividing by y'x, we arrive at a constant
coefficient equation: 2yx — 2Xyxx + 2ay + 6 = 0.
© 1995 by CRC Press, Inc.

n— 1
25. 2
Differentiating both sides of the equation with respect to x, we have
У:х[2УЧп-1)-ах-0\=О. A)
Equating the second factor to zero, we find
1 ax2n 1 px2n~x 2^'
+
Integration constants Cfe and parameters a, [3, and 7 are related by the equality
n-l
2 Y, (-l)mm\ Bn - m)! CmC2n_m + (-l)"(n!JC2 = /3d - aC0 + 7,
m=2
which is obtained as a result of substituting the above solution into the original
equation.
In addition, there is the solution corresponding to equating the first factor in A)
to zero:
C0, where /3Ci - aC0 + 7 = 0.
n— 1
26. 2 ^(-1)-^-)^"—) + (_!)-[y(")]2 + s(y;'j2 =
where n is an integer greater than or equal to 3.
With s = 0 see 5.2.1.25. Let s ф 0. Differentiating the equation with respect to x, we
have
Equating the second factor to zero and integrating, we obtain
axA px3 _, 2 _, _, /•/•/• , , ,
у = —г— + ^7— + C2x2 + Cxx + Co + wdxdxdx,
48s 12s JJJ
where w = w(x) is the general solution of a constant coefficient equation of the form
5.1.2.2: wxn~ +sw = 0. The constants of integration are related by an equality which
is found as a result of substituting the obtained solution into the original equation.
In addition, there is the solution у = C\X + Co, where the constants of integration
are related by /3Ci — aCo + 7 = 0.
m — 1 v
2 2
27. Х!М2Е (-^vviv)V^m-v) + (-l)m[ylm)}2 = «У2 + 2/3y + 7-
m=l ^ v=\ '
Differentiating with respect to x, we arrive at a constant coefficient linear equation:
En Bm) , , /э n
m=i amyx + ay + P = 0.
© 1995 by CRC Press, Inc.

5.2.2. Equations Containing Exponential Functions
1. yyi5) + by'xy'
'xy'Zxx
This is a special case of equation 5.2.6.1 with f(x) = aeXx.
2. yyle) + 6y'xyib) 2 ^
This is a special case of equation 5.2.6.4 with f(x) = ae .
3. yx2n) = ае^У.
This is a special case of equation 5.2.6.6 with f(y) = aeXy.
4. yxn) = ax-neXy.
This is a special case of equation 5.2.6.10 with f(y) = aeXy.
5. yxn) = axkeay.
This is a special case of equation 5.2.6.31 with f(w) = aw, m = к + п.
6. yxn) = Aeaxym.
This is a special case of equation 5.2.6.11 with m = m\ and mi = m.3 = • • • = mn = 0.
7. yy(xn+1) = aex* + b.
This is a special case of equation 5.2.6.16 with f(x) = aeXx + b.
8. yxn) =aebyecxm, m = 1, 2, ..., n - 1.
The substitution 6w = by+cxm leads to an autonomous equation: «4 = aebw, which,
for even n, admits lowering of its order by two (see 5.2.2.3).
9. Bn - l)yyx2n+1) + {In + l)y'xyx2n) = aeXx.
This is a special case of equation 5.2.6.22 with f(x) = aeXx.
tn (n+1) \,i / I (n)\m
10. yx ^ = аехУу'х(ух ') .
This is a special case of equation 5.2.6.17 with f(y) = e~Xy, g(w) = awm.
11. yx = Aeaxymi (y'x) 2...(yx ')
The substitution w(x) = ye@x, where /3 = , leads to an
mi + mi Л + mn — 1
autonomous equation of the form 5.2.6.40.
12. yxn) = Ае<*Ух™1(ухГ*(Ух-хГ° ... (yi-1^".
The transformation z = xaeay', w = xy'x, where a = n + mi — mi — 2тз — 3m.4 • —
(n — l)mn, leads to an (n — 1) th order equation.
© 1995 by CRC Press, Inc.

5.2.3. Equations Containing Hyperbolic Functions
vZ* = acoshm(Xx).
This is a special case of equation 5.2.6.1 with /(ж) = acoshm(Aa;).
2. yyi5) + by'xy'^xx + lOy'^vZ, = о sinhm(Xx).
This is a special case of equation 5.2.6.1 with /(ж) = asinhm(Aa;).
3. yyi5) + by'xy'^xx + lOy'LvZL = «tanbm(Xx).
This is a special case of equation 5.2.6.1 with /(ж) = atanhm(Aa;).
4. yyi5) + by'xy'^xx + lOy'LvZL = acothm(Xx).
This is a special case of equation 5.2.6.1 with /(ж) = асоШт(Аж).
5. yyi6) + 6y'xyi5) + ^yZvZxx + 10(vZJ2 = acosh™(Aa;).
This is a special case of equation 5.2.6.4 with /(ж) = асовЬт(Аж).
6F) I л / E) 1 -i r" // //// i -t r\ / fff \ 2 • i 771. / \ \
УУх + ®УхУх + 15yxxyxxxx + Ю(уххх) = о sinh (Аж).
This is a special case of equation 5.2.6.4 with /(ж) = авшЬт(Аж).
7. yyi + ®у'хУх + 15yZy'xxxx "I" Ю(у^4а;) = °tanhrTl(Aa;).
This is a special case of equation 5.2.6.4 with /(ж) = о^аппт(Аж).
8. уухв) + 6у'хУх5) + ibyZv'xxxx + 10(vZxJ = ocothm(Aa;).
This is a special case of equation 5.2.6.4 with /(ж) = асоШт(Аж).
9. yi^^
This is a special case of equation 5.2.6.6 with f(y) = acoshm(Xy).
10. yi^^
This is a special case of equation 5.2.6.6 with f(y) = asinhm(Ay).
11. yi^^
This is a special case of equation 5.2.6.6 with f(y) = atanhm(Ay).
12. yi^^
This is a special case of equation 5.2.6.6 with f(y) = acothm(Ay).
13. yin) = ax-ncoshm(Xy).
This is a special case of equation 5.2.6.10 with f(y) = acoshm(Xy).
© 1995 by CRC Press, Inc.

14. yxn) =ax-nsinhm(Xy).
This is a special case of equation 5.2.6.10 with /(y) = asinhm(Ay).
15. yin) = ax-ntanhm(Xy).
This is a special case of equation 5.2.6.10 with /(y) = atanhm(Ay).
16. yin) = ax-ncothm(Xy).
This is a special case of equation 5.2.6.10 with /(y) = acothm(Ay).
17. yyi2n+1) = acoshm(Xx).
This is a special case of equation 5.2.6.16 with f(x) = acoshm(Aa;).
18. yyi2n+1) = asinh™(Aa;).
This is a special case of equation 5.2.6.16 with f(x) = asinhm(Aa;).
19. yyi2n+1) = atanhm(Xx).
This is a special case of equation 5.2.6.16 with f(x) = atanhm(Aa;).
20. yyi2n+1) = acothm(Xx).
This is a special case of equation 5.2.6.16 with f(x) = acothm(Aa;).
21. Bn - l)yyBn+1) + Bn + l)y'xyBn) = acoshm(Xx).
This is a special case of equation 5.2.6.22 with f(x) = acoshm(Aa;).
22. Bn - l)yyl2n+1) + Bn + l)y'xyBn) = asinhm(Xx).
This is a special case of equation 5.2.6.22 with f(x) = asinhm(Aa;).
23. Bn - l)yyi2n+1) + Bn + l)y'xyi2n) = аЫпЬт(Хх).
This is a special case of equation 5.2.6.22 with f(x) = atanhm(Aa;).
24. Bn - l)yyi2n+1) + Bn + l)y'xyi2n) = acothm(Xx).
This is a special case of equation 5.2.6.22 with f(x) = acothm(Aa;).
„_ (n+l) , k/\ \ I I (n)\'m
25. yx T = о cosh (Xy)y'x(yx >) .
This is a special case of equation 5.2.6.17 with f(y) = cosh" (Ay), g(w) = awm.
26. yxn+1) = osinhfc(Ay) y'x(yin))m.
This is a special case of equation 5.2.6.17 with f(y) = sinh~ (Ay), g(w) = awm.
27. yi"+1) (у)ух(ухТ
This is a special case of equation 5.2.6.17 with /(y) = tanh~ (Ay), g(w) = awm.
28. yx T = acothB(Ay) y'x(yx ') ¦
This is a special case of equation 5.2.6.17 with /(y) = coth~ (Ay), g(w) = awm.
© 1995 by CRC Press, Inc.

5.2.4. Equations Containing Logarithmic Functions
This is a special case of equation 5.2.6.1 with f(x) = a\nm(bx).
2. wwi + 6wlwi5 -
This is a special case of equation 5.2.6.4 with f(x) = alnmFa?).
3. yi2n) =a\nm(by).
This is a special case of equation 5.2.6.6 with f(y) = a\nm{by).
4. yyi2n+1) =a\nm(bx).
This is a special case of equation 5.2.6.16 with f(x) = a\nm{bx).
5. Ух = y(cix + rainy + b).
This is a special case of equation 5.2.6.30 with /(w) = In w + b.
Ух — "^ \^^У I ifb 1П Ж | ** } ¦
This is a special case of equation 5.2.6.31 with /(w) = In w + b.
7. yi™^ еж"™ In™(by).
This is a special case of equation 5.2.6.10 with f(y) = a\nm{by).
8. yxn) = ax-™-1 [In у + A - n) In ж].
This is a special case of equation 5.2.6.11 with /(w) =
9. yin) = ax-n-k(ln у + к In ж).
This is a special case of equation 5.2.6.13 with /(w) =
10. yx = ayx~n (rainy + fe 1пж).
This is a special case of equation 5.2.6.14 with /(w) = a In w.
2тг+1
11. у(,2гг) = ax 2 [2 In у + A - 2n) In ж].
This is a special case of equation 5.2.6.12 with /(w) = 2a In w.
тг+1
12. yxn) = (ax2 + c) 2 [2 In у + A - n) 1п(ож2 + с)].
This is a special case of equation 5.2.6.21 with 6 = 0, /(w) = 2Inw.
13. yin) =beax(\ny-cxx).
This is a special case of equation 5.2.6.29 with /(w) = 6In w.
© 1995 by CRC Press, Inc.

14. {In - l)yyi2n+1) + Bn + I)y4yi2"} =
This is a special case of equation 5.2.6.22 with /(ж) = а1птFж).
., _ («•+!) , fc/. \ / / (п)\ггг
15. j/i т = oln {by)y'x{yx >) .
This is a special case of equation 5.2.6.17 with f{y) = ln~ {by), g{w) = avf
16. j/i ^ ' = aymyyxlnyx '.
This is a special case of equation 5.2.6.17 with f{y) = y~m, g{w) = a In ад.
5.2.5. Equations Containing Trigonometric Functions
1. yyib) + 5y'xyZxx + lOt&tC, = о cos-СЛж).
This is a special case of equation 5.2.6.1 with /(ж) = асовт(Аж).
2. yylb) + 5y'xyZxx + lOt&tC, = asinm(AaJ).
This is a special case of equation 5.2.6.1 with f{x) = asinm(Aa;).
3. yyl5) + by'xy'Zxx + lOt&tC, = о tanm(Aa;).
This is a special case of equation 5.2.6.1 with f{x) = atanm(Aa;)
4. yyi5) + by'xy'^xx + lOt&.tC, = о cotm(Aa;).
This is a special case of equation 5.2.6.1 with /(ж) = acotm{Xx).
5. УУх + ®УхУх + ^-"*УХХУХХХХ "Ь Ю(у^д,а,) = ocosrTl(Aa;).
This is a special case of equation 5.2.6.4 with /(ж) = асовт(Аж).
л F) i л / E) . i ~. // //// i tr\/ III \2 . m / \ \
6. yyx + oyxyx + *-5yxxyxxxx + Ю^у^^) = о sin (Аж).
This is a special case of equation 5.2.6.4 with /(ж) = авшт(Аж).
7. yyi6) + 6yxyi5) + 1Ьу'ххУхххх + 10{Ух'ххJ = atanm{Xx).
This is a special case of equation 5.2.6.4 with /(ж) = о^апт(Аж).
8. уУхв) + 6у'хУх5) + ibyZy^L* + ЩУ'ХХХJ =
This is a special case of equation 5.2.6.4 with f{x) = acotm{Xx).
9. y(x2n) = a cos™(Ay).
This is a special case of equation 5.2.6.6 with f{y) = acosm{Xy).
10. yi2n) =asinm{Xy).
This is a special case of equation 5.2.6.6 with f{y) = asinm{Xy).
© 1995 by CRC Press, Inc.

11. yx2n) = atanm(Xy).
This is a special case of equation 5.2.6.6 with f(y) = atanm(Ay).
12. yBn) = a cot™(Ay).
This is a special case of equation 5.2.6.6 with f(y) = acotm(Xy).
13. yin) = ax-ncosm(\y).
This is a special case of equation 5.2.6.10 with f(y) = acosm(Xy).
14. yin) = еж"™ sin™ (Ay).
This is a special case of equation 5.2.6.10 with f(y) = asmm(Xy).
15. yin) = ax-ntanm(Xy).
This is a special case of equation 5.2.6.10 with f(y) = atanm(Ay).
16. yin) = ax-ncotm(Xy).
This is a special case of equation 5.2.6.10 with f(y) = acotm(Xy).
17. yyi2n+1) = a cosm (\x).
This is a special case of equation 5.2.6.16 with f(x) = acosm(Xx).
18. yyi2n+1) =asinm(Xx).
This is a special case of equation 5.2.6.16 with f(x) = asinm(Aa;).
19. yyi2n+1) = atanm(Xx).
This is a special case of equation 5.2.6.16 with f(x) = atanm(Aa;).
20. yyi2n+1) = acotm(Xx).
This is a special case of equation 5.2.6.16 with f(x) = acotm(Xx).
21. Bn - l)yyi2n+1) + Bn + l)y'xy{2n) = acosm{Xx).
This is a special case of equation 5.2.6.22 with f(x) = acosm(Xx).
22. Bn - l)yyi2n+1) + Bn + l)y'xyi2n) = asin^A*).
This is a special case of equation 5.2.6.22 with f(x) = asinm(Aa;).
23. Bn - l)yyBn+1) + Bn + l)y'xyBn) = atanm(Xx).
This is a special case of equation 5.2.6.22 with f(x) = atanm(Aa;).
24. Bn - l)yyi2n+1) + {In + l)y'xyBn) = асо^(Аж).
This is a special case of equation 5.2.6.22 with f(x) = acotm(Xx).
© 1995 by CRC Press, Inc.

25. ух т = acosk(Xy)y'x(yx >) .
This is a special case of equation 5.2.6.17 with f(y) = cos~k(Xy), g(w) = awm.
„_ (n+l) . k/\ \ i I (n)\Tn
26. yx ^ = оsinfe(Ay)»i(»i ;) .
This is a special case of equation 5.2.6.17 with f(y) = sin~fe(Ay), g(w) = awm.
„— (n+l) , h/\ \ I I (n)\Tn
27. yl ^ ' = atank(Xy)y'x(y^ ') .
This is a special case of equation 5.2.6.17 with f(y) = tan~k(Xy), g(w) = awm.
28. yyx T = acotfe(Ay) y'x{yl ') .
This is a special case of equation 5.2.6.17 with f(y) = cot~k(Xy), g(w) = awm.
5.2.6. Equations Containing Arbitrary Functions
i. yyi5) + ьу'хУ'^хх + ioy'LvZL = /(^)-
Solution:
у2 = С4ж4 + C3x3 + C2x2 + ClX + C0 + -}- Г(х - tL/(t) dt,
where xo is an arbitrary number.
2. yyi5) + ay'xVZxx + Ca - 5)y^y^ = f(x).
Integrating the equation three times, we obtain
yy'L + ^-Ю2 = c2x2 + clX + c0 + \ f\x - tfm dt,
where xq is an arbitrary number.
3. (a + y)yi5) + by'xy'Zxx + cy'^vZL = /(*)•
Integrating, we obtain
(« + vML* + (ь- i)y'xyZx + ^0--ь + c)(y':xf = J f(x) dx + a
4. уУхв) + 6y'xyx5) + \by'lxy'Zxx + 10(y^LJ = /(^).
Solution: y2 = C5x5 + С4Ж4 + С3Ж3 + C2x2 + Cxx + Co + -?- f (x - tff{t) dt.
5. yi6) = (ож2 + bx + c)-r/2f((ax2 + bx + c)~5/2)
This is a special case of equation 5.2.6.21 with n = 6.
© 1995 by CRC Press, Inc.

6. yx = f(y).
The first integral of the equation is
? (-1)ЫтЫ2п"т) + | (-1)" [y^]2 + / f(y) dy = С
m=l J
Next, the order of the obtained equation can be lowered by the substitution w(y) = y'x.
1 • Ух —
Having set u(x) = yx , we obtain u'x = f(u). Further, find и from the relation
.- , + C\. Then, the (n — l)-fold integration yields y.
fyu) _
The solution in the parametric form is written as
u du fu dui fUl du2 fUn-з dun-2 fUn~2 un-\ dun-\
J J J
fu dui fUl du2 fUn-з dun-2 fU
J f()' Jc2 /Ю Jc3 f(U2) '"Ус„_1 f{Un-2) JCn
Jc2 /Ю Jc3 f(U2) Ус„_1 f{Un-2) JCn f(Un-l
8. yin)(in2))
Setting u(x) = yxn , we obtain the equation uxx = f(u) whose solution has the form
du г f 11/2
+С h () ±\C+2 f()d\
/du г f 1
—г-г+Съ, where ip(u) = ±\C1+2 f(u)du\
(p(u) L J J
Expressing и in terms of x and integrating the resulting relation (n — 2) times, we
find y.
The solution in the parametric form is written as
ptX 7 ftX 7 Рил 7 fUrn Q 7 fUrn Q 7
аи / dui / au2 / dun-2 / «-n-2 dun-2
x = , у = / / ¦ ¦ ¦ / / ¦
9. yi"} = /(у + аж™), m = 0, 1, 2, ..., n - 1.
The substitution w = у + axm lead to an autonomous equation: wx = f(w), which,
for even n, admits lowering of its order by two (see 5.2.6.6).
10. yin) = x-nf(y).
The substitution t = In |ж| leads to an autonomous equation of the form 5.2.6.40.
П»/") — „—n—1 f (tr-l— п„Л
• yx — ^ J \^ У )•
The transformation x = i, у = t1~nw leads to an autonomous equation: W)J =
(—l)nf(w), whose order, for even n, can be lowered by two (see 5.2.6.6).
2n+l / l — 2n
12. yx2n> = X 2 flX 2 у
2n-l
The transformation x = e , у = x 2 w(i) leads to an autonomous equation of the
form 5.2.6.25, whose order can be lowered by two.
© 1995 by CRC Press, Inc.

13. yin) = x-n-kf(yxk).
This is a special case of equation 5.2.6.44.
The transformation t = In ж, w = xky leads to an autonomous equation of the
form 5.2.6.40.
14. yin) = yx-nf(xkym).
ITfl
The transformation t = xkym, w = —— leads to an (n — 1) th order equation.
15.
У
XII* X и"
The transformation z = ——, w = — leads to an (n — 2) th order equation.
У У
16. yyi2n+1) = f(x).
Having integrated the equation, we obtain
2 J2 (-1)ЫтЫ2"~т) + (-i)n [y(:]]2 = 2 / f{X) dx+c,
m=0
where the notation y^ = у is used.
Having integrated the equation, we obtain
dw f dy (n)
= / + C> Where w = Vx ¦
f dw f
J g(w) J
Next, the order of this equation can be lowered by the substitution z{y) = y'x.
18. yin) = fix, y).
The transformation x = t~1, у = t1~nw(t) leads to an equation of the similar form:
¦!¦»• Ух — J[X, yx i Ух )¦
The substitution w(x) = yx leads to a second order equation: w'^x = f(x, w, w'x).
20. (ax + b)n(cx + d)yx^ = f
icx + d)n~1
ax ~\~ b xi
The transformation ? = In , w = -. ^—т- leads to an autonomous equation
ex + d (ex + d)n i
of the form 5.2.6.40.
© 1995 by CRC Press, Inc.

__ l-\-n / 1—n
21. yf> = (ax2 + Ьх + с) 2~
1°. The transformation
dx , , . , . ,±?L
leads to an autonomous equation with respect to w = w(t), which admits lowering of
its order by the substitution z(w) = w't.
2°. Let n = 2m be an even integer (m = 1, 2, 3, ...). In this case, transformation A)
yields an equation of the form 5.2.6.25, whose order can be lowered by two.
2m-l
Setting P = ax +bx + c, у = wP 2 and multiplying both sides of the original
l+2m / 1 — 2m \
equation by w'x = P 2 (Py'x-\ P^yJ, we obtain
Integrating both sides of this equality with respect to x (the left-hand side is integrated
by parts), we have
m-2
fe=0
where
(-1) V?fc)y?2m-1-fc) + (-I)™1 ^<ro-1)y<m+1) dx = J f(w) dw + C, B)
(remind that n = 2m). It can be shown that the integrand on the left-hand side of B)
is the total differential. Finally, we arrive at the first integral
m-2
V^ i ntln.lttil , I i_ ... , ^ \n/..l«J , nU(U _
fe=0
. 2\ (m-2) (m) . О171 г (m_i)-,2~l
+ a(l - mz)yx 'yx ' H — [yx >\ V
f
= / f(w) dw + C.
22. Bn - l)yyi2n+1) + (In + l)y'xyi2n) = f(x).
Having integrated the equation, we have
n-l
Bn - l)yyi2n) + 2Y{-l)i+1yxi)yBn-i) + (-l)n+1 [y^]2 = I f(x) dx + 2C2.
The second integration yields
i=0
The third integration leads to a Bn — 2) th order equation:
]Г(г + l)Bn - г - 1)(-1Уу^ух2п-2-г) + ji-ir-W [
i=0
j\x - t)f{t) dt.
= C2x2 + dx + C0 + ^ l\x - tff{t) dt.
1 J
© 1995 by CRC Press, Inc.

23. xyx > 4- nyx ' = f(xy).
The substitution w(x) = xy leads to the autonomous equation «4 = fiw) (see 5.2.6.6
and 5.2.6.40).
24. x yx 4- 2nxyx 4- nyn — i-)yx = fyxy).
The substitution wix) = x2y leads to the autonomous equation «4 = fiw) (see
5.2.6.6 and 5.2.6.40).
n
25. Y^ amyx2m) = f(y).
m=l
The first integral has the form
П sTn—1 -. N
E aA E (-^ух^ух2™'^ + -i-ir[yim)]2 \ +
where С is an arbitrary constant. Further, the order of the obtained equation next
be lowered by the substitution u>(y) = y'x.
26. У^ arr,xmy(m) = f(y).
/ j ill"" &X *f \i? /
The substitution t = In |ж| leads to an autonomous equation of the form 5.2.6.40.
"*' У / j атУх — J\x)-
m=O
Having integrated the equation, we obtain
n s m—1 -\ /.
am< 2 ^^i—iyyi yx m + (—l)m[yi ] f = 2 / f(x)dx + C,
where yx stands for y.
n
ОЙ V^ n »,("») Bn+l-m) _ f(\
m=O
The first integral has the form
n — 1 ^
2 Y, АтУхт)У(хП~т) + An[yxn)]2 = 2 / fix)dx + C,
m=0 ^
where
m
-f*-m — / ^ J-J ttfc — C^m C^m—1 ~I~ ^m — 2 ^m—3 ~I~ ¦
fe=0
If the condition An = 2^^L0(—l)n~1+mAm is satisfied, the obtained equation can
be integrated two times more (see, in particular, equation 5.2.6.22).
© 1995 by CRC Press, Inc.

29. yin) = eaxf(ye-ax).
The substitution w(x) = ye~ax leads to an autonomous equation of the form 5.2.6.40.
30. yin) = yf(eaxym).
The transformation z = eaxym, w(z) = y'x/y leads to an (n — 1) th order equation.
o-i («) _ -nf(may\
OJ-' Ух — ••' J \* e )¦
The transformation z = xmeay, w(z) = xy'x leads to an (n — 1) th order equation.
32. yin) = f(y + aeXx) - a\neXx.
The substitution w(x) = y + aeXx leads to an autonomous equation: wx = f(w) (see
5.2.6.6 and 5.2.6.40).
33. yx = f(y + о cosh ж) — о cosh ж.
The substitution w(x) = у + a cosh ж leads to an autonomous equation: wx = f(w)
(see 5.2.6.6).
34. yx = f(y + о sinh ж) — о sinh ж.
The substitution w(x) = у + a sinh ж leads to an autonomous equation of the form
5.2.6.6: wx2n) = /(го).
35. yx = f(y + о cosh ж) — о sinh ж.
The substitution w(x) = у + a cosh ж leads to an autonomous equation of the form
5.2.6.40: wx2n+1) = f(w).
36. yx = f(y + о sinh ж) — о cosh ж.
The substitution w(x) = у + a sinh ж leads to an autonomous equation of the form
5.2.6.40: wi2n+1) = f(w).
37. yxn> = f(y + о cos ж) — ocosl ж -\ 1.
The substitution w(x) = у + a cos ж leads to an autonomous equation: wx = f(w)
(see 5.2.6.6 and 5.2.6.40).
38.
The substitution w(x) = у + a sin ж leads to an autonomous equation: wx = f(w)
(see 5.2.6.6 and 5.2.6.40).
39. F(x, y'x, Vx-X, ..., yin))=0.
The substitution w(x) = y'x leads to an (n — 1) th order equation:
F(x, w,w'x, ..., wxn-1])=0.
© 1995 by CRC Press, Inc.

40. F(y, y'x, Ух-х, ..., yin))=0.
Autonomous equation.
The substitution w(y) = y'x leads to an (n — 1) th order equation. The derivatives
of the original equation and the transformed one are related by the formulae
yxx = ww'y, yxxx = w2w'yy + w(w'yf, ..., yxn)= w{yxn~1])'y.
41. xy'x — у = F(x, yxxi y'x'xxi • • • i Ух j-
The substitution w(x) = xy'x — у leads to an (n — 1) th order equation:
wL d "•"' - лп-2
42. x2yxx — 2xy'x +2y = F(x, yxxx, ..., yxn)) = 0.
The substitution w(x) = x2yxx — 2xy'x + 2y leads to an (n — 2) th order equation:
w' dn~3
w = F[x, -f-, ..., l-f-l =0.
V x1 dxn~6 \r//
43. ^(-l)fcfe! Ctxm-kyxm~k) = F(x, yxm+1\ ..., t/<,">),
fc=0
yn\
where ClL = —-r-, гтг are binomial coefficients.
к! (m — к)!
The substitution w(x) = Yuk=o(~^-)k^- C^nxm~kyxm leads to an (n — m) th order
equation; the derivatives on the right-hand side are calculated in consecutive manner
using the formula yx = x~mw'x.
44. F(xky, хк+гу'х, ..., xk+nyin)) = 0.
Homogeneous equation in the extended sense.
The transformation t = In ж, w = xky leads to an autonomous equation of the
form 5.2.6.40.
45. F(-^=-, ^^, ..., ^Л?—) =о.
V v v у J
Homogeneous equation in the extended sense.
xy' x y"
The transformation z = ——, w = — leads to an (n — 2) th order equation.
У У
46. Ffo- S&.,^&.,...,^L)= 0.
V у у у J
Homogeneous equation in the extended sense.
I'll
—
У
I'll
The transformation t = xkym, z = —— leads to an (n — 1) th order equation.
У
© 1995 by CRC Press, Inc.

47 P(paxii paxii' po"cii>> paxii^n^\ — П
Exponential homogeneous equation.
The substitution г<7(ж) = еаху leads to an autonomous equation of the form
5.2.6.40.
v' y" y(n
48. F[ eraum, -^-, -^2., ..., ^ ) = o.
У У У
Exponential homogeneous equation.
y>
The transformation z = eaxym, w = —^- leads to an (n — 1) th order equation.
У
4Q jrirprripay „„./ t-2..// TniSn^\ — П
¦*». г yi, e . «bf/^, a yxxi . . . , a yx j — u.
Exponential homogeneous equation.
The transformation z = xmeay, w = xy'x leads to an (n — 1) th order equation.
© 1995 by CRC Press, Inc.

Supplement 1
Some Elementary Functions
and Their Properties
In Supplement 1, n is a positive integer, unless otherwise specified.
1.1. Trigonometric Functions
1.1.1. Simplest Relations
sin2 ж + сов2ж = 1, tanжcotж = l,
вт(-ж) = -sin ж, сов(-ж) = cos ж,
sin ж cos ж
taпж= , аЛж= — ,
cos ж sin ж
-ж) = - tan ж, cot (-ж) = - cot ж.
1.1.2. Relations Between Trigonometric Functions
of Identical Argument
, f.
= ±v 1 ~~
tan ж
sin ж = ±v 1 ~~ cos x =
л/l + tan2 ж л/l + cot2 ж '
, A r^— , 1 cot ж
cos ж = ± v 1 — sin ж = ± — — = ± -
л/l + tan2 ж \/l + cot2 ж '
sin ж л/1 — cos2 ж 1
taпж = ±— ^r- = ±
- sin2ж С08Ж cota;'
V 1 — sin2 ж cos ж 1
cot Ж = ± ¦ = ±-
sin ж л/1 — cos21 tan ж
1.1.3. Reduction Formulae
sin (ж ± П7г) = (-l)nsiM, СО8(ж±П7г) = ( —1)ПС08Ж,
. / 2n+l \ , ,„ / 2n + l \ , w .
Sinl Ж ± 7Г 1 = ±( — 1) СОвЖ, COsI Ж ± 7Г 1 = =F( — 1) SIM,
tan^ ± птг) = tan ж, cot (ж ± птг) = cot ж,
/2n+l\ /2n + l\
tanI ж ± 7Г 1 = — cotж, cotI ж ± тг 1 = —tanж,
\ Zi J \ Zi J
© 1995 by CRC Press, Inc.

sin f ж ± — J = (sin ж ± cos x), cos (x ± — j = (cos x =F sin x),
/ 7Г \ tan ж ± 1 / 7Г \ cot x =F 1
tan ж ± — = , cot [x ± —- = — .
V 4 У lTtanrr V 4 У licotrr
1.1.4. Addition Formulae
вт(ж ± y) = sin x cos у ± sin у cos x, cos(rr ± y) = cos ж cos у =F sin ж sin y,
tan ж ± tan v . , . 1 =F tan ж tan у
cot(x±y) =
, cot(x±y) =
tan ж tan у tan ж ± tan у
1.1.5. Addition and Subtraction of Trigonometric Functions
, . о ¦ f Х±У\ (ХТУ\
sin ж ± sm у = 2 sin I —-— J cos I —-— J,
/ iy> I 1] \ / X XI \
cos ж + cos у = 2 cos ( —-— j cos ( —-— j,
о • (x + y\ . (x-y\
cos ж — cos у = — 2 sm I —-— 1 sin I —-— 1,
\ Z J \ Z J
a cos ж + b sin ж = гвт(ж + </?)= гсов(ж — ф),
where r = л/a2 + b2, sin ip = a/r, cos ip = b/r, sin ф = b/r, cos ф = a/r,
sin2 ж — sin2 у = cos2 у — cos2 ж = вт(ж + у) sin (ж — у),
sin2 ж — cos2 у = — сов(ж + у) сов(ж — у),
sin (ж ± у) sin (ж ± у)
tan ж ± tan у = v ~~i. ~ _i_—i
, у
cos ж cos у sin ж sin у
1.1.6. Product of Trigonometric Functions
sin ж sin у = y[cos(a; — у) — сов(ж + у)],
cos ж cos у = y[cos(a; — у) + сов(ж + у)],
sin ж cos у = \ [sin (ж — у) + sin (ж + у)].
1.1.7. Powers of Trigonometric Functions
cos2 ж = 4- cos 2ж + 4-, sin2 ж = — 4- <
cos3 ж = -|- cos Зж + -|- cos ж, sin3 ж = — 4^ sin Зж + -|- sin x,
cos4 ж = -|- cos4ж + у cos 2ж + I", sin4 ж = -|- cos 4ж — у cos 2ж + |-,
cos5 ж = -jq cos 5ж + ^- cos Зж + -|- cos x, sin5 ж = -4?- sin 5ж —^- sin Зж + -|- sin x,
© 1995 by CRC Press, Inc.

os2n x = ^=T Ё C"n «*[2(n - k)x] + -^C2n,
k=0
os2n+1 x = i E ^n+i cos[Bn - 2k + l)x],
Z k=o
in2n x = ^rr E(-1)""feC'2n cos[2(n - k)x] + -^C^,
fe=O
giti "*" 7* = \ ( X) "-/'"- sin BiTi 2iK -\~ X)x\
k=0
where C^ = щ^-ку. are binomial coefficients.
1.1.8. Trigonometric Functions of Multiple Arguments
cos 2ж = 2 cos2 ж — 1, sin 2ж = 2 sin ж cos ж,
cos Зж = —3 cos ж + 4 cos3 ж, sin Зж = 3 sin ж — 4 sin3 ж,
сов4ж = 1 — 8cos2 ж + 8cos4 ж, вш4ж = 4cosa;(sma; — 2sin ж),
cos 5ж = 5 cos ж — 20 cos3 ж + 16 cos5 ж, sin 5ж = 5 sin ж — 20 sin3 ж + 16 sin5 ж,
совBпж) = 1 + yj(—l)fe— 4fe sin2fe ж,
cos[2(n + 1)ж] = cos ж< 1
^ 1 ,k [Bn + IJ - 1] [Bn + IJ - 32]... [Bn + IJ - Bfc - IJ] . 2fe
, ^ Г ^, ,fe(n2-l)(n2-22)...(n2-)fc2) k 2k ,
вшBпж) = 2псовж 8тж + 2^(-1)/?- /v —^-^ -4/? sin* L x
L fe=i ^ ^
sin[2(n + 1)ж] = Bn + 1) < sin ж
A fc [Bn + IJ - 1] [Bn + IJ - 32]... [Bn + IJ - Bfc - IJ] 2fc+1
+ ^( lj Bjfc + l)!
2 tan ж 3 tan ж —tan3 ж , 4 tan ж —4 tan3 ж
2 tan ж
tan2ж = - —к—, taпЗж=
1 taпzж
1 — tan ж 1 — 3 tan ж 1 — 6 tan ж + tan ж
1.1.9. Euler and de Moivre Formulae, Relation to Hyperbolic Functions
ez+lx = ez(cosx + isinx), (совж + г sin ж)" = сов(пж) + isin(ra), г2 = —1,
вш(гж) = гвтЬж, сов(гж) = cosh ж, tan(^) = гtanhж, cot(^) = — гсоШж.
© 1995 by CRC Press, Inc.

1.1.10. Differentiation and Integration Formulae
d sin x d cos x . d tan x 1 d cot x
— =cosrr, = -smi, = r—, = =—
dx dx dx cos^ x dx sin x
sin x dx = — cos x + C,
/ cos x dx = sin x + C,
f
/ tanx dx = — In | cosx\ + C,
/ cot x dx = In | sinx\ + C,
sin2™ ж da; = —^С^ж + 2n-i У"^"!)*^^ + С,
fe=i
? ^^ + С,
k=o ZK + L
co^xdx- l Cnx+ l VC* sin[B"~2^] | c
cos xdx-^rC2nx+ 22n_1 2^°2n 2n-2/c '
cos2"+1 ^ ^
tan»»*1 *<fc = (-1)"+1 In |
fe=i
/cot2"
fe=l
fe=l
where C^ = —— ' are binomial coefficients.
1.2. Hyperbolic Functions
1.2.1. Simplest Relations
e^ — e *^
sinh ж = , cosh x =
2 ' 2
tanha;= , cotha; =
sx — e~x '
© 1995 by CRC Press, Inc.

cosh ж — sinh x = 1, tanh ж • coth ж = 1,
sinh(—x) = — sinh ж, cosh(—x) = cosh ж,
sinh ж cosh ж
tanh x = :—, coth x =
cosh x ' sinh x
tanh(—x) = — tanh ж, coth(—x) = —coth ж.
1.1.2. Relations Between Hyperbolic Functions of Identical Argument
• u . / п2 7 j_ tanh ж 1
inn ж = ± v cosh ж — 1 = ± — = ±
sin.
I 2— I о '
V 1 — tanh ж vcoth ж —1
, ЛГТг ГТ , 1 , coth ж
совЬж = vsmh ж + 1 = ±— =¦ = ±— ^,
V 1 — tanh2 ж vcoth2 ж — 1
sinh ж Vcosh2 ж — 1 1
tanh ж = — — = ±-
\/sinh2 ж + 1 cosha; coth ж'
Vsin2 ж +1 , cosh ж 1
coth ж = — = ±-
\/cosh2 ж - 1 tanh ж'
1.2.3. Addition Formulae
sinh (ж ± y) = sinh ж cosh у ± sinh у cosh ж,
совЬ(ж ± у) = cosh ж cosh у ± sinh ж sinh y,
tanl^itanhw
tanh(ж ±y) = , сош(ж ±y) =
v ' 1± tanh ж tanh у y '
1 ± tanh ж tanh у ' coth у ± coth ж
1.2.4. Addition and Subtraction of Hyperbolic Functions
. , , . , _ . /ж±у\ /хту\
sinh ж ± sinh у = 2 sinh I 1 cosh I 1,
\ Z J \ Z J
/ /y I f]\ /IT 11 \
cosh ж + cosh у = 2 cosh ( —-— J cosh ( —-— J,
/ iy> I 1] \ / X U \
cosh ж — cosh у = 2 sinh ( —-— j sinh ( —-— j,
sinh ж — sinh у = cosh ж — cosh у = sinh (ж + у) sinh (ж — у),
sinh ж + cosh у = cosh (ж + у) cosh (ж — у),
втп(ж ± у) втЬ(ж ± у)
tanh ж ± tanh у = — -J—, coth ж ± coth у = ±—г
cosh ж cosh у sinh ж sinh у
1.2.5. Product of Hyperbolic Functions
sinh ж sinh у = -j [cosh (ж + у) — cosh (ж — у)],
cosh ж cosh у = у [cosh (ж + у) + cosh (ж — у)],
sinh ж cosh у = у [sinh (ж + у) + sinh (ж — у)].
© 1995 by CRC Press, Inc.

1.2.6. Powers of Hyperbolic Functions
cosh x = -j cosh 2ж + -j > smn x = \ cosh 2ж — 4/-,
cosh3 ж = -j sinh Зж + -|- sinh ж, sinh3 ж = -j sinh Зж — -|- sinh ж,
cosh4 ж = 4- cosh 4ж + 4- cosh 2ж + ¦§-, sinh4 ж = 4- cosh 4ж — 4- cosh 2ж + ¦§-,
cosh5 ж = -рт- sinh 5ж + -4^- sinh Зж + -|- sinh ж,
sinh ж = -jg- sinh 5ж —4^- sinh Зж + -|- sinh ж,
1 ™-1 1
cosh2™ ж = та-1 ^ C2n cosh[2(n - к)х] + ly^C^,
к=0
„„„V,2n + 1 \ Л /~lk nr\ah\(On 91--I-1W1
СОЫ1 X — —г— 7 ^271-1-1 COollllZ/t — Zit\ -p -*-/Ж|,
fe=0
]_ "~^ ( ]\n
sinh2™ ж = 2n-i 2-/(~^к^2п cosh[2(n — k)x] + 2^—C^,
fe=0
sinh2™ ж = —2— У^(—l)feC2n+i sinh[Bn — 2/c + 1)ж],
fe=o
where C^, = -rr^ гтт are binomial coefficients.
1.2.7. Hyperbolic Functions of Multiple Argument
cosh 2ж = 2 cosh2 ж — 1, sinh 2ж = 2 sinh ж cosh ж,
совЬЗж = — 3 cosh ж + 4 cosh ж, sinh Зж = 3 sinh ж + 4 sinh ж,
cosh4ж = 1 — 8cosh2 ж + 8cosh4ж, sinh4ж = 4coshж(втпж + 2sinh ж),
cosh 5ж = 5 cosh ж — 20 cosh ж + 16 cosh5 ж,
sinh 5ж = 5 sinh ж + 20 sinh ж + 16 sinh5 ж.
fe=0
sinh(ra) = 2_, 2™ С„_^_1(с>
fe=0
where Cf, = ; r— are binomial coefficients, \A] stands for the integer part of num-
m\(k — m)\
ber A.
1.2.8. Relation to Trigonometric Functoins
вшЬ(гж) = г sin ж, совЬ(гж) = cos ж, tanh(^) = г tan ж, coth(гж) = —г cot ж,
where г2 = — 1.
© 1995 by CRC Press, Inc.

1.2.9. Differentiation and Integration Formulae
d sinh ж . d cosh ж . d tanh x Id coth x 1
= cosh ж, = sinh ж,
dx dx ' dx cosh x' dx sinh x
sinh x dx = cosh x + C,
/ cosh x dx = sinh x + C,
f
/ tanh x dx = In | cosh ж | + C,
coth ж da; = In | sinh x\ + C,
f sinh2" ж dx - (~1)W Cn x + l Vr-llfeCofe ашц^" ~ '"У^ + с
/-
2 2 2n-2/c
fe=i
sinh2"+1 xdx = J ^fc+Г
coSh2nxdx- l Cnx+ l °VC* со8Ь[Bп-2/с)Ж]
cosh жйж-^-G2nrr+^rr2^(-2n 2n-2/c '
/ cosh2n+1 xdx = ^ ¦
^ fe=o
(tanhrr)
2"^1
^ 2n-2)fc + l ' '
,2n+i , , , ^ ^апЬжJ"^2 ^
tanh^n+1 ж с^ж = In cosh ж - > h C,
t, 2П-2/С + 2
f ,2n , V^ (cotha;Jn-2fc+1 „
/ coth xdx = x — > — ^—; h C,
fe=i
2t?-I-1 t i i X ^ (COtll Ж) _^
coth ж аж = In | sinh ж| — 2_^ ^5 hpr—^ Ь ^>
fe=i
where Cf, = ——. r--r are binomial coefficients.
m fc!(fc)!
1.3. Inverse Trigonometric Functions
1.3.1. Simplest Relations
Principal values of inverse trigonometric functions are defined by the inequalities
< агсвшж < —, 0 < агссовж < тг, where — 1 < ж < 1,
7Г 7Г
< arctanж < —, 0 < агсаЛж < тг, where — оо < ж < +оо.
© 1995 by CRC Press, Inc.

arcsin(—x) = — arcsin ж, arccos(—x) = тг — arccos ж,
arctan(—x) = — arctanж, arccot(—x) = тг — arccot ж,
sin(arcsina;) = x, cos(arccosa;) = x,
tan(arctana;) = x, cot(arccot ж) = x,
arcsinlsina;) =
x — 2птг
if 2птг — 4-тг < x < 2п7г + 4-тг,
arccoslcosa;) =
2(п + 1)тг if Bn + 1)тг - тгтг < ж < 2(n + 1)тг +
x — 2n?r if 2птг < ж < 2п7г,
-ж + 2(п+1)тг if Bn + 1)тг < ж < 2(n + 1)тг,
arctan(tana;) = x — nw if nw — \ж < x < nw + -j7r,
arccot(cot x) = x — П7Г if nw < x < (n + 1)тг.
1.3.2. Relation Between Inverse Trigonometric Functions
arcsm x + arccos x =
arcsm x = <
2 '
arcsin -\/l ~ x2
— arccos \/l — x2
x
arctan —,
arctan ж + arccot ж = —.
if 0 < ж < 1,
if -1 < x < 0,
if -1 < ж < 1,
arccos ж = <
arccot тг
arcsin л/l — x2
тг — arcsin y/1 — x2
'1-х2
arctan ¦
arccot¦
ж
ж
/1-х
X
if -1 < ж < 0,
if 0 < ж < 1,
if -1 < x < 0,
if 0 < ж < 1,
if -К ж < 1.
arctan ж = <
arccot ж = <
arcsin —1=
Vi
arccos p=
Vi
^
1
arccot —
^ ж
vT
тг — arcsin
1
arctan —
ж
тг + arccot
T
1
T
Д
1
T~
j
V
1
ж
ж2
1?
+ Ж2
Ж2"
1
1 ?Г
if
if
if
if
if
if
if
if
ж —
ж>
ж<
ж >
ж
ж
ж
ж
any,
0,
0,
0.
\ U,
>«,
<о.
© 1995 by CRC Press, Inc.

1.3.3. Addition and Subtraction of Inverse Trigonometric Functions
arcsin a: + arcsin у = arcsin(a:\/l — y2 + yyl — x2 ) if x2 + y2 < 1,
arccosx + arccosу = arccos[xy — \/(l — x2)(l — y2) ] if x + у > О,
arccos x — arccos у = — arccos [xy + \J{1 — x2)(l — y2) ] if x — у > 0,
X I 11
arctana; + arctany = arctan if xy < 1,
1-xy
x — у
arctana; — arctan у = arctan if xy > —1.
l + xy
1.3.4. Differentiation and Integration Formulae
d . 1 d 1
¦ arcsm x = —, , —— arccos x = — -
dx л/1-х2 ' dx л/1-х2
did 1
—— arctana; = тг, —г- arccot a; = — ~
/
arcsin xdx = x arcsin x + yl — x2 + C,
arccos xdx = x arccosx — \J 1 — x2 + C,
I arctan xdx = x arctan x —— ln(a;2 + 1) + C,
arccot xdx = x arccot x -\ ln(a;2 + 1) + C.
1.4. Inverse Hyperbolic Functions
1.4.1. Relation to Logarithmic Functions and Simplest Relations
Arsinh x = ln(a; + \J x2 + 1), Arcosh x = ± ln(a; + \J x2 — 1),
л., 1 , 1 + ж Ail 1 , 1 + ж
Artanna;=— In- , Arcotha;=—-In —,
AY X AX 1
Arsinh(—x) = —Arsinh a;, Arcosh(—x) = Arcosh a:,
Artanh(—x) = — Artanha;, Arcoth(—x) = Arcotha;,
1.4.2. Relations Between Inverse Hyperbolic Functions
X
Arsinh x = Arcosh \Jx2 + 1 = Artanh ¦
yr ¦
Arcosh x = Arsinh yx2 — 1 = Artanh
x
X 11
Artanha; = Arsinh —, = Arcosh — т. = Arcoth —.
л/х2 -1 л/1-Х2 X
© 1995 by CRC Press, Inc.

1.4.3. Addition and Subtraction of Inverse Hyperbolic Functions
Arsinh ж ± Arsinh у = Arsinh (ж л/1 + у2 ± yyl + x2 ),
Arcoshж ± Arcoshу = Arcosh[xy ± \/(x2 — l)(y2 — 1) ],
Arsinh x ± Arcosh у = Arsinh [:
Artanh x ± Artanh у = Artanh ¦
Arsinh x ± Arcosh у = Arsinh [xy ± \/(x2 + l)(y2 — 1) ],
x±y
Artanh x ± Arcoth у = Artanh
l±xy '
xy ±1
1.4.4. Differentiation and Integration Formulae
¦ Arsinh x = —, , —— Arcosh x =
dx y/x2 +1 ' dx Vx2 -1
d 1 d . 4, 1
Artanh x = 7Г, —— Arcoth x = — -
1
dx " ~1-X2' dx —" - 1 _ X2 '
Arsinh xdx = x Arsinh ж — ух2 + 1 + С,
Arcosh xdx = x Arcosh ж =F yx2 — 1 + C,
Artanh ж dx = ж Artanh ж H ln(l — ж ) + С,
/ Arcoth ж dx = x Arcoth ж + — 1п(ж2 - 1) + С.
1.5. Some Conventional Symbols
1.5.1. Factorial
0! = 1! = 1, n! = 1 • 2 • 3... (n - l)n, n = 2, 3, 4, ...,
Bn)!! = 2 • 4 • 6... Bn - 2)Bn) = 2nn\,
ОП + 1 у О ч
Bn+l)!! = l-3-5...Bn-l)Bn + l) = ——Г(п+ — ),
V?r V 2 /
ГB/с)!! ifn = 2fc,
I fOt _i_ 1 "\11 if n — 9k -4- 1
1.5.2. Binomial Coefficients
a T(b + l)T(a-b+l) '
where Г (ж) is the gamma-function.
к_( 1лк(~а)к _ a(a-l)---(a-k + l)
© 1995 by CRC Press, Inc.

* = 1, 2, 3, ...,
, 3,
п = к\(п-к)\'
° = 1, Скп = 0 for /с = -1, -2, -3, ... or jfc > n,
i ]_ 4-1 — ^ .
1 Bп-3)!!
1 / л \nc\—An — \f~m
/2 ~~ \~L) z °2n>
22n+l 22«
^
?r(n-l)/2_
In
1.5.3. Pochhammer Symbol
(o)n = a(a + 1)... (a + n - 1) = Г^ + n) = (-1)"- ГA " a)
Г(а) v ; ГA - a - n) '
Г(а - n) (-1)" . . (n + fc-1)!
(aH = 1, (a)-n = y = -rf—'-—, (n)fe = ±— —-!-
Г (а) A-а)„ (n-1)!
Г fa — ni (—l)n
(a)_n = l J = ) J , where а ф 1, 2, ..., n; /c = 1, 2, 3, ...,
Г(а) A - a)n
n!
(a + n)n = . ,2" , (a + n)fe = ^
© 1995 by CRC Press, Inc.

Supplement 2
Some Special Functions
In Supplement 2, n is a positive integer, unless otherwise specified.
2.1. Gamma-function
The gamma-function T(z) is an analitic function of z everywhere, except for the points
2 = 0, -1,-2,...
For Re z > 0,
/•CO
T(z)= / t^e-Ut.
Jo
For —(n + 1) < Re z < —n, where n is an integer,
The gamma-function possesses the following properties:
T(z + l) = zT(z), Г(п + 1)=п!, ГA)=ГB)
T{z)T{-z) = V
y> y ' zsi
rV^, Г(г)ГAг) =^, rf + ^rf *) =y^r
zsinGrz)' y> y ' sin(Trz) ' V2 J \2 J cos(ttz)
И )
к=о
!)-#*•-«п.
Г(г) l Jn> Г(«л-
© 1995 by CRC Press, Inc.
г

2.2. Bessel functions Ju and Yu
2.2.1. Basic Formulae
The Bessel functions of the first and second kind Jv and Yv (function Yv is also called
the Neumann function) are solutions of the Bessel equation 2.1.2.121 and are defined by
the formulae
v
v{X)~
К—(J
The formulae for Yv(x) is valid for v ф 0, ±1, ±2, ...; (see below for the integral represen-
representation of Yv(x), as well as equation 2.1.2.121 for v = 0, ±1, ±2, ...).
Function Zv(x) = C\Jv{x) + C2Yv[x) is reffered to as the cylindric function.
The Bessel functions possess the following properties:
2vZv{x) =
~i~
- Zv+1{x) = ±—[vZv{x) - Zv±{\,
ax
ax
= x"-nJv-n(x),
7, fril — — r~v7, ,,(r\
n[x-vJv{x)) = {-l)nx-v-nJv+n{x),
J_n{x) = (-l)nJn(x), Y_n(x) = (-l)nYn(x) n = 0, 1, 2, ...
2.2.2. Bessel functions for v = ±n ± \; n = 0, 1, 2, ...
Л/2 (ж) =
sinrr,
( — sin x — cos ж I,
тга; \ ж /
/ 2
J_i/2(a;) = \ cosrr,
' V 7ГЖ
J_3/2(a;) = a/ I cos ж — sin ж I,
' V тгж \ ж /
V (~l)fc(n + 2fc)!
+ cos [x —
-2jfc-l)! BxJk+1 У
^ B/c)!(n-2/c)!Ba;Jfe
(-l)k(n + 2k + l)\
l)!(n-2Jfc-l)!Ba;
2fc+1J'
cosrr,
/ 2
У_1/2(ж) = a/ sinrr,
V 7ГЖ
Y_n_1/2(x) = (-l)nJn+
© 1995 by CRC Press, Inc.

2.2.3. Wronskians an Similar Formulae
Notation: W(f, g) = fg'x - f'xg.
W{JV, J_v) = -— sinGri/), W(JV, Yv) = —,
TTX TTX
Jv{x)J_v+1(x) + J_v(x)Jv_1(x) = ——-—, Jv(x)Yv+1(x) - Jv+1(x)Yv(x) = .
7ГЖ 7ГЖ
2.2.4. Integral Representation
Functions Jv and Yv may be expressed in terms of definite integrals (with x > 0):
/*7T /*OO
7Г Jv(x) = / cos(rr sin0 — vff) d0 — sin txv I exp(—x sinh t — vt) dt,
Jo Jo
/*7T /*OO
7rYv(x) = / sin(a;sin0-^)d0- / (evt + e~vtcosnv)e~xsinhtdt.
Jo Jo
For \u\ < \, x > 0,
г1-^^-^ Z0 sin(rri) dt _ 21-vx~v f°° cos(xt) dt
tt1/2T(\-v) к (t2 - l)^+!/2 ' v[-X'~ n1/2^ - v) к (i2-l
For v = 0, ж > 0,
2 Z0 2 Z0
Jo(x) = — / sin(xcosh t) dt, Yq(x) = / cos(x cosh t) dt.
^ Jo ^ Jo
2.2.5. Integrals with Bessel Functions on Closed Intervals
where Re(A + v) > —1, F(a,b,c;x) is the hypergeometric series (see equation 2.1.2.158),
„av /„л ^ _ со8(гутг)Г(-гу) ^x+v+1 ( A + v + 1
A_,+1 /
1 V
where Re A > | Re v\ — 1.
2.2.6. Asymptotic Expansion, as |ж| —> ос
Л О гМ — 1
'
(_1ГA/> 2то
© 1995 by CRC Press, Inc.

Л О гМ—1
)
ЯМ-1
2J (-1)m{v, 2m + 1)Bх)~2т~1 + i
т=0
where (u,m) =
2.3. Modified Bessel Functions /^ and Ku
2.3.1. Basic Formulae
The modified Bessel functions of the first and second kind Iv and Kv (function Kv is
also called the Basset function) are solutions the modified Bessel equation 2.1.2.122 and are
defined by the formulae
j^)=Efc,rr:fc+ir ад=у
fe=0 *• '
(see equation 2.1.2.122 with v = 0, 1, 2, ... for the representation of Kv{x)).
The modified Bessel functions possess the following properties:
K-V{x) = Kv{x); I.n{x) = (-l)nIn(x), n = 0, 1, 2, ...
Oj.T („\ _ -[Г /«Л _ Г («Л] 9ry^ fri — — гГ/^ -. Гг"! — /^ , -. (
2.3.2. Modified Bessel Functions for v = ±n ± -|-! n = 0, 1, 2, .
/ 2 / 2
Л/2(ж) = A/ sinha;, I_i/2(x) = \ cosh ж,
' V 7ГЖ V 7ГЖ
/3/2(ж) = \ ( sinh x + cosh x ), I_3/o(x) = \ ( — cosh x + sinh x ),
' \ tvx \ x / ' \ tvx \ x /
-1/2И "
kl (» - fc)! Bж)' to kl (» " fc)! Bж
(-l)fc(n + fc)! , 1ЛП _а^л (n + fc)!
+ ( lj e
t0 *! (n - fc)! (to)* + ( lj e t k\ (n -
(П+fc)!
Efc!(n_fc)!B
© 1995 by CRC Press, Inc.

2.3.3. Wronskians and Similar Formulae
Notation: W(f, g) = fg'x - f'xg.
— sinGr^)) W(IV,KV) = --,
7Г' X X
I) lv(x)Kv+1(x) + Iv+1{x)Kv{x) = —.
ЖХ X
2.3.4. Integral Representation
Functions/;, and Kv may be expressed in terms of definite integrals (withrrXD, v>
1) L %
f-OO
Kv(x) = I exp(—a
Jo
2.3.5. Integrals with Modified Bessel Functions on Closed Intervals
xx+v+1
x2\
1; "Г)'
where Re(A + v) > —1, F(a,b,c;x) is the hypergeometric series (see equation 2.1.2.158),
2—
-^ /A + ^+l
l V 2 + '
2 ' 4 Г
where Re A > | Re v\ — 1.
2.3.6. Asymptotic Expansion, as x —> oo
^— + О(ж-М)],
m!
M (л,.2 1\(л,.2 q2
V^
m=l
/2 - 32)... [4^2 - Bm - IJ
m! (8rr)m
2.4. Degenerate Hypergeometric Functions
2.4.1. Definitions
The degenerate hypergeometric functions Ф(а, b;x) and Ф(а, b;x) are solutions of the
degenerate hypergeometric equation 2.1.2.65.
© 1995 by CRC Press, Inc.

If Ь Ф 0, —1, —2, —3, ..., function Ф(а, b; x) is expressed in terms of Kummer's series:
CO / \ fc
\b)k k\
where (а)к = a(a + 1)... (а + к — 1), (a)o = 1. Function Ф(а, 6; ж) is denned as follows:
Table 2.1 represents some special cases where Ф is expressed in terms of simpler func-
functions.
2.4.2. Basic Properties
Rummer's transformation:
Ф(а, b; x) = ехФ{Ь - a, b; -x), Ф(а, b; x) = ^^(l + a - b, 2 - b; x).
Linear relations for function Ф:
(b - а)Ф(а - 1, b; x) + Ba-b + х)Ф(а, b; x) - аФ(а + 1, b; x) = 0,
ЫЬ - 1)Ф(а, b - 1; x) - b(b - 1 + х)Ф(а, b; x) + (b - а)хФ(а, b + 1; x) = 0,
(a - 6 + 1)Ф(а, 6; ж) - аФ(а + 1,6; ж) + F - 1)Ф(а, 6 - 1; х) = 0,
ЬФ(а, 6; х) — ЬФ(а — 1,6; ж) — жФ(а, 6 + 1; ж) = 0,
Ь(а + ж)Ф(а, 6; ж) — (Ь — а)хФ(а, b + 1; ж) — а6Ф(а + 1,6; ж) = 0,
(а - 1 + ж)Ф(а, 6; ж) + (Ь - а)Ф(а - 1,6; ж) - F - 1)Ф(а, 6 - 1; ж) = 0,
Linear relations for function Ф:
Ф(а - 1, b; x)-Ba-b + ж)Ф(а, 6; ж) + а(а - Ь + 1)Ф(а + 1,6; ж) = 0,
(Ь - а - 1)Ф(а, 6 - 1; ж) - (Ь - 1 + ж)Ф(а, Ь; ж) + жФ(а, Ь + 1; ж) = 0,
Ф(а, 6; ж) - аФ(а + 1,6; ж) - Ф(а, 6 - 1; ж) = 0,
F - а)Ф(а, 6; ж) - жФ(а, Ь +1; ж) + Ф(а - 1,6; ж) = 0,
(а + ж)Ф(а, Ь; х) + а(Ь-а- 1)Ф(а + 1,6; ж) - жФ(а, Ь + 1; ж) = 0,
(а - 1 + ж)Ф(а, 6; ж) - Ф(а - 1, Ь; ж) + (а - с + 1)Ф(а, 6 - 1; ж) = 0.
Differentiation formulae:
-—Ф(а, 6; ж) = |-Ф(а + 1,6 + 1; ж), -^-Ф(а, 6; ж) = -аФ(а +1,6+1; ж),
ах о ах
dn (a) dn
-^¦Ф(а, Ь; ж) = ——Ф(а + п,Ь + п; ж), __ф(а) Ь; ж) = (-1)п(а)пФ(а + п, b + n; ж),
Ф(а + п,Ь + п; ж), _
Wronskian:
© 1995 by CRC Press, Inc.

2.4.3. Integral Representation
T(a)T(b-a) Jo
Г(с
1 f°° _ . _л ,_ _л
Ф(а, Ь;Х) = , ч / е xtta (I +1)° ° dt, (if a> 0,X > 0)
Г (a) Jo
where Г (a) is the gamma-function.
2.4.4. Integrals with Degenerate Hypergeometric Functions
f b-1 f 1
J a-1 J 1-a
(CL U' Xi dx — ?7' / ^^^^^^^^^^^^^^^~^^^^^^^^^^^^^^^^ m/ ( CL к 0 к' X i I CJ
n+1 (_л\к+1хп-к+1
(a,b;X)dX = n! 2_. 7 r~7 г ТГ^(а ~ kib — k\X) + С.
2.4.5. Asymptotic Expansion, as |cc| —> 00
Ф(а, b; X) = lH^-ь [g (Са);(,1а)"Ж- + O(,—)], X > 0,
I N (a)n(a-b + l)n
Ф(а,Ь;Х) = —1S2.— (—x) a y^ —(—X) n -\
T(b -a) [^ n!
,т,/ ;, ^ -ol^/ 1Лп(а)п{а-Ь+1)п -ЛГ-1М ^ ^ ,
Ф(а,о;ж) = ж > (—1) ; ж + O(\x ) , — 00 < ж < +oo.
1 ^^^ ni 1
2.5. Legendre Functions
2.5.1. Definitions
The associated Legendre functions Р„(г) and Q^(z) of the first and second kind are
linearly-independent solutions the Legendre equation (see 2.1.2.212):
A - z2)y';z - 2zy'z + \v(y + 1) - M2(l - zY^V = 0,
where parameters v, /i and variable z may be arbitrary real or complex numbers.
With |1 — z\ < 2, the following formulae may be used:
' 2 У
2 '
© 1995 by CRC Press, Inc.
L ^ 2 У
. ГС/Л
В = ег

where F(a,b,c; z) is the hypergeometric series (see equation 2.1.2.158).
With \z\ > 1, the following formulae may be used:
B j I
2 '
The functions
Pv{z)=PS{z), Qv(z) = Ql(z)
are called the Legendre functions and are solutions of the Legendre equation 2.1.2.148.
The modified associated Legendre functions on the cut z = x, —1 < x < 1, are defined
by the formulae
гО) + e-H
a; + гО) + е^<Х(ж - гО)].
2.5.2. Trigonometric Expansions
With — 1 < x < 1, the modified associated Legendre functions can be expressed in terms
of trigonometric series:
^ /c!(ry+f)fe
where 0 < в < тт.
2.5.3. Some Relations
n = 0, 1, 2,
Р-Л*)!-
q(z) e \p(z)
Qv {Z) ~ 2 sin(M7r) [Л ' ГA + v -
For -1 < ж < 1,
Рм () 2гУ+1
—PM (x)
vx
© 1995 by CRC Press, Inc.

For 0 < ж < 1,
P?{-x) = P?(x) cos[7r(rv + /л)} - 2n-1QZ{x) sin[7r(ry + /л)},
Wronskians:
W(PV, Qv) =
where к = 22м-
1 — ж2 ' 1 — ж2 '
2 / V 2 /
(f)
For n = 0, 1, 2, ...,
The Legendre polynomials Рп(ж) and the Legendre function Qn(a;) are given by the
formulae
1 1-1- n 1
Qn(a;) = -Pn(x)ln-^ - ^ — Рт_1(Ж)Рп_т(Ж).
m=l
Functions Pn = Pn{x) can be conveniently calculated by the recurrence relations
P0 = l, P1=x, P2 = ±-Cx2-l), ..., Pn+1 = 2H±±xPn--!l-P
2 n+1 n+1
Three leading functions Qn = Qn{x) are
1 1 + ж ж 1 + ж Зж2 - 1 , 1 + ж 3
2.5.4. Integral Representation
For n = 0, 1, 2, ...
C(Z)= Тг^+У ?(z + cost^^lYcoS(nt)dt, Rez>0,
Note that in the latter formulae z ф x, — 1 < ж < 1.
2.6. The Weierstrass function p
2.6.1. Definitions
The Weierstrass function p = р(г,д2,9з) is defined implicitly by the elliptic integral
dt
J оо
2 =
/оо \/4i3 — 92t — :
and satisfies the first order differential equation
{p'zf = 4P3 -92P-93-
© 1995 by CRC Press, Inc.

2.6.2. Some Properties
Below p(z) stands for р(г,д2,дз)-
Z2) = -p(Zl) - p(Z2
In the vicinity of the point z = 0, the Weierstrass function can be expanded into the
series
© 1995 by CRC Press, Inc.

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