OEDINAEY
DIFFERENTIAL EQUATIONS
AN ELEMENTARY TEXT-BOOK
WITH AN INTRODUCTION TO
LIE'S THEOBY OF THE GBOTJP OF ONE PABAMETEB
JAMES MORRIS PAGE
MACMILLAK AND CO., Limited
NEW YORK : THE MACMILLAN COMPANY
1897
All rigUi reserved

PREFACE.
This elementary text-book on Ordinary Differential
Equations, is an attempt to present as much of the
subject as is necessary for the beginner in Differential
Equations, or, perhaps, for the student of Technology
who will not make a specialty of pure Mathematics.
On account of the elementary character of the book,
only the simpler portions of the subject have been
touched upon at all; and much care has been taken
to make all the developments as clear as possible—
every important step being illustrated by easy examples.
In one material respect, this book differs from the
older text-books upon the subject in the English
language: namely, in the methods employed. EveT
since the discovery of the Infinitesimal Calculus, the
integration of differential equations has been one of the
weightiest problems that have attracted the attention of
mathematicians. It is not possible to develop a method
of integration for all differential equations; but it was
found possible to give theories of integration for certain
classes of these equations; for instance, for the homo-
homogeneous or for the linear, differential equation of the
first order. Also, important theories for the linear differ-
differential equations of the second or higher orders, have

vi ORDINARY DIFFERENTIAL EQUATIONS.
been developed. But all these special theories of in-
integration were regarded by the older mathematicians as
different theories based upon separate mathematical
methods.
Since the year 1870, Lie has shown that it is possible
to subordinate all of these older theories of integration
to a general method: that is, he showed that tbe older
methods were applicable only to such differential equations
as admit of known infinitesimal transformations. In this
way it became possible to derive all of the older theories
from a common source: and at the same time, to develop
a wider point of view for the general theory of diffeiy
ential equations.
Only a very small part of Lie's extensive and im-
important developments upon these subjects could, however,
be presented in a text-book intended for beginners. The
memoirs published by Lie on differential equations are
to be found in the " Verhandlungen der Gesellschaft
der Wissenschaften zu Christiania," 1870-74; in the
Mathematische Annalen, "Vol. II., 24 and 25; and in
his Vorlesungen ilber Differentmlgleichungen mit Be-
lannten Injlnitesimalen Transformationen, edited by Dr.
G. Scheffers, Teubner, 1891. Besides these sources of
information, the writer had the advantage of hearing,
in 1886-87, at the same time with Dr. Scheffers, Prof.
Lie's first lectures upon these subjects at the University
of Leipzig.
All the methods, depending upon the theory of trans-
transformation groups, employed in Chapters III.-V., and
IX.-XII. of this book, are due escclicsively to Prof. Lie.
Lie has also developed elegant theories of integration
for Clairaut's and Eiccati's equations, as well as for the

PREFACE. vii
general linear equation with constant coefficients; but,
as an exposition of these theories requires a more ex-
extensive preparation than it was considered advisable to
give in a purely elementary text-book, the author deter-
miued to follow, in the treatment of the above-mentioned
equations, the older methods—hoping to present Lie's
methods for these equations, as well as some of his
more far-reaching theories, in a second volume.
In the preparation of this book the author has made
free use of the examples in the current English text-
textbooks : and he is under special obligations to the works
of Boole, Forsyth, Johnson, and Osborne. The treatment
of Riecati's equation, Chapter VII., is substantially that
given by Boole.
The arrangement of the matter will be found suffic-
sufficiently indicated by the table of contents; and an index
is given at the end of the book.
The articles in the text printed in small type may
be omitted by the reader who is going over the subject
for the first time.
JAMES MOERIS PAGE.
Johns Hopkins University,
Baltimore, U.S.A.,
July, 1896.

CONTENTS.
CHAPTER I.
GENESIS OF THE ORDINARY DIFFERENTIAL EQUATION
IN TWO VARIABLES.
§ 1. Derivation of the Differential Equation from its Complete
Primitive. Order and Degree of a Differential Equation, 1
DeBnition of General Integral, 3
Particular Integrals, 4
§ 2. Geometrical Interpretation of the Ordinary Differential
Equation in Two Variables, 6
Examples to Chapter I., 9
CHAPTER II.
THE SIMULTANEOUS SYSTEM, AND THE EQUIVALENT
LINEAR PARTIAL DIFFERENTIAL EQUATION.
§ 1. The Genesis of the Simultaneous System, 10
§ 2. Definition of a Linear Partial Differential Equation, - ¦ 13
The Linear Partial Differential Equation and the Simul-
Simultaneous System represent fundamentally the same
Problem, 14
Geometrical Interpretation of the Simultaneous System in
Three Variables, 17

x ORDINARY DIFFERENTIAL EQUATIONS.
§ 3. Integration of Ordinary Differential Equations in Two
Variables in which the Variables are Separable by
Inspection, 19
Integration of a Special Form of Simultaneous System in
Three Variables, 21
Examples to Chapter II., 24
CHAPTER III.
THE FUNDAMENTAL THEOREMS OF LIE'S THEORY OF THE
GROUP OF ONE PARAMETER.
§ 1. Finite and Infiniteaimal Transformations in the Plane, The
Group of one Parameter, 25
Definition of a Transformation, 26
DeBnition of a Finite Continuous Group, 27
Derivation of the InBnitesimal Transformation, 29
Kinematic Illustration of a G1 in the Plane, .... 32
The Increment 5/of a Function/(a^, yj under an Infinitesi-
Infinitesimal Transformation, 36
The Symbol of an Infinitesimal Transformation, 37
The Form of the Symbol when New Variables are Introduced, 38
The Development
Лъ> У,) =Ж У)+ U/. t+ U{Uf)^+. . ,
and the Equations to the Finite Transformations of a (?lt 40
§ -2. Invariance of Functions, Curves, and Equations, 42
Condition that the Function ii{x, y) shall be Invariant under
the G1 U/, 42
The Path-Curves of a G± in the Plane, 44
Condition that a Family of Curves shall be Invariant under
a Gx in the Plane, 47
Condition that the Equation Й-0 shall be Invariant under
the Gj V/, 50
Method for Finding all Equations which are Invariant under
a given (?i in n Variables, 51

CONTENTS. xi
§ 3. The Lineal Element. The Extended Group of One Para-
Parameter, 54
The InBnitesimal Transformation of the Extended Gv ¦ - 59
Examples to Chapter III., 59
CHAPTER IV.
CONNECTION BETWEEN EULER'S INTEGRATING FACTOR
AND LIE'S INFINITESIMAL TRANSFORMATION.
§ 1. Exact Differential Equations of the First Order in Two
Variables, 62
Condition that a Differential Equation of the First Order
shall be Exact, 63
Definition of Euler's Integrating Factor, .... 66
§ 2. Invariant Differential Equation of the First Order may be
Integrated by a Quadrature, 67
Definition of an Invariant Differential Equation, 67
To 6nd all Differential Equations which are Invariant under
a given G^, 68
The Integral Curves of an Invariant Differential Equation
of the First Order constitute an Invariant Curve-Family, 73
Proof of the Theorem that every Differential Equation of
the First Order which ia Invariant under a known (?x
may be Integrated by a Quadrature, 73
Definition of a trivial Glt 78
§ 3. Classes of Invariant Differential Equations of the First Order
in Two Variables.
The Equations Invariant under Uf=!*L, • - - - 79
The Equations Invariant under Uf=x~-, 80
All homogeneous Differential Equations of the First Order
are Invariant under О/=хЖ + у^-, 81
да оу

i ORDINARY DIFFERENTIAL EQUATIONS.
The Equation
(ax + by+ c)dx - (a'x+ b'y+ c')dy = 0
may usually be reduced to ft homogeneoua form, ¦ - 83
The Equations Invariant under Uf= ~У%. + Х%.> ' ' 8i
The Equations
fi(xy)xdy -Mxy)ydx ~ 0
unde
The Linear Equation
e Invariant under Uf-x^-уЩ,
[tb(x}dx 7=if
is Invariant under Uf=e> .Ш,- - . . 88
The Equation
y'-<p(x)y-f(x)yn = 0
may be reduced to a Linear Form, 90
The Classes of Invariant Differential Equations of the First
Order may be Multiplied IndeBnitely, .... 91
To Find the (?x of which a given Differential Equation of
the First Order admits, 92
The Development of the General Integral of a Differential
Equation of the First Order in a Seriea, 93
Table of Classes of Invariant Differential Equations of the
First Order, 96
Examples to Chapter IV., 97
CHAPTER V.
GEOMETRICAL APPLICATIONS OF THE INTEGRATING
FACTOR. ORTHOGONAL TRAJECTORIES AND ISO-
ISOTHERMAL SYSTEMS.
Geometrical Meaning of the Integrating Factor, - 100
Application to Parallel Curves, 101
Orthogonal Trajectories, 103
Isothermal Systems, 104
Examples to Chapter V., Ю7

CONTENTS.
CHAPTER VI.
DIFFERENTIAL EQUATIONS OF THE FIRST ORDER, ВОТ
NOT OF THE FIRST DEGREE. SINGULAR SOLU-
SOLUTIONS.
PAOE
§ 1. Differential Equations of a Degree Higher than the First, 109
Decomposable Equations, 110
Equations which may be solved with respect to y, - - III
Equations which may be solved with respect to x, • ¦ 112
§ 2. Method for Finding the Singular Solution of an Invariant
Differential Equation of the First Order, - - - 113
Examples to Chapter VI., 118
CHAPTER VII.
BICCATI'S EQUATION, AND CLAIRAUT'S EQUATION.
§1. Riccati's Equation, 119
The Equation
is Integrable when n = 2a, 120
This Equation is also Integrable when ~- is a Positive
Integer, 123
The Forms of the General Integral, 125
Application to the Equation
^ + bv? = czm, 125
§ 2. Clairaut's Equation, 128
The Equation
у ~x\f/{y') + tfriy1), - • - - 129
Examples to Chapter VII., - 131

ORDINARY DIFFERENTIAL EQUATIONS.
CHAPTER VIII.
total Differential equations of the first order
and degree in three variables which are
derivable from a single primitive.
PAGE
§ 1. The Genesis of the Total Differential Equation in Three
Variables, 132
The Condition that this Equation shall be Derivable from a
Single Primitive, 133
§ 2. Integration of the Total Differential Equation in Three
Variables, 134
Geometrical Method for Reducing the Integration of this
Equation to the Integration of One Ordinary Differential
Equation of the First Order in Two Variables, - - 137
Examples to Chapter VIII., 139
CHAPTEK IX.
ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND
ORDER IN TWO VARIABLES.
§ 1. Exact Differential Equations of the Second OrJer, - - 140
Definition of Fimt Integrals, 141
Condition that a Differential Equation of the Second Order
shall be Exact; Method of Integration, 143
§ 2. Invariant Differential Equations of the Second Order, - - 144
Method for Finding all Differential Equations of the Second
Order which are Invariant under a given (?„ - - - 145
Method of Integrating Invariant Differential Equations of
the Second Order, 146

CONTENTS.
xv
PAGE
§ 3. Classes of Invariant Differential Equations of the Second
Order, 150
The Equations Invariant under Uf= Щ-. ¦ ¦ - 150
The Equations Invariant under Uf=W, - 152
oy
The Equations Invariant under Uf=x*~, ¦ ¦ 154
The Abridged Linear Equation is Invariant under U/= yW, 155
The General Linear Equation is Invariant under
и/^ф(х).'^> --. 156
The Equations Invariant under Uf= xl?+y$L>" ¦ - 157
To find the Ог of which a given Differential Equation of the
Second Order admits, 159
Table of Classes of Invariant Differential Equations of the
Second Order, 162
Examples to Chapter IX., 162
CHAPTER X.
ORDINARY DIFFERENTIAL EQUATION OF THE mlb ORDER
IN TWO VARIABLES.
§ I. The General Invariant Differential Equation of the m'L
Order, 164
§ 2. Classes of Invariant Differential Equations of the m'h Order, 169
Equations Invariant under Uf=^t 169
Equations Invariant under Uf= $-, 170
The Abridged Linear Equation of the m Order is Invariant
under the G-iUf= уЖ. 170

xvi ORDINARY DIFFERENTIAL EQUATIONS.
The General Linear Equation of the mlh Order is Invariant
under the Gj Uf= 0(*)^> *7I
The Equations of the Forma
Examples to Chapter X., 174
CHAPTER XI.
THE GENERAL LINEAR DIFFERENTIAL EQUATION IN
TWO VARIABLES.
§ 1. The Abridged Linear Equation of the ma Order with
Constant Coefficienta, 175
Case of Equal Roots, 177
Case of Imaginary Roots, 178
§ 2. The Linear Equation of the mth Order with Constant
Coefficienta and the Second Member a Function of x, - 179
First Method for Finding the Particular Integral-Function, 180
Method of Variation of Parameters for Finding the Par-
Particular Integral-Function, 183
§ 3. The General Linear Equation of the m'h Order in which the
Coefficients are Functions of x, 185
Reduction of the Order of the Equation when a Particular
Integral-Funetion -of the Correaponding Abridged Equa-
Equation is known, - - 185
The General Linear Equation of the Second Order,
may be completely Integrated by two Quadratures, when
a Particular Integral-Function of the Corresponding
Abridged Equation,
y'' + X^y' + Х^^О,
is known,

CONTENTS. xvii
Thus, the Linear Equation of the Second Order with
Constant Coefficients may alwaye be Integrated by Two
Quadratures, 188
Examples to Chapter XI., 189
CHAPTER XII.
METHODS FOR THE INTEGRATION OF THE SIMULTANEOUS
SYSTEM.
§ 1. Special Methods for Certain Forms of Simultaneous Syetems, 190
A Method Peculiarly Applicable to a System of Linear
Equations with Constant Coefficients, - 192
The Ordinary Differential Equation of the ma Order in
Two Variables is Equivalent to a Simultaneous System
of m Differential Equations of the First Order in (m+ 1)
Variables, 194
Simultaneous Systems of Equations of an Order Higher than
the First, 196
§ 2. Theory of Integration of a Simultaneous System in Three
Variables which is Invariant under a known Gv - - 196
What it Means for a Simultaneous System to be Invariant, 197
The Meaning of the Expression U(Af) -A(Uf)t - ¦ -198
Condition that the Linear Partial Equation
shall be Invariant under the Ou
is (P, А) = Цх, у, z)Af,
Two Linear Pwtial Equations
Л,/=0, A,f=0
which gatiefy a Condition of the Form
may always be put into such Forms that they satisfy
the Condition,
(AltAJ = Q, 201

xviii ORDINARY DIFFERENTIAL EQUATIONS.
To Find the Common Solution which Af= 0,' Uf= 0 Possess
when (U, A) = 0,
This Solution is the General Integral of a Total Equation in
Three Variables,
When this Solution is known, the Invariant Linear Partial
Equation may be completely Integrated by a Quadrature,
Resume of the Results of the Section,
§ 3. Application of the Method of § 2 to the Ordinary Differential
Equation of the Second Order in Two Variables, which
is Invariant under a known Gu
Examples to Chapter XII.,

CHAPTER I.
THE GENESIS OF THE ORDINARY DIFFERENTIAL
EQUATION IN TWO VARIABLES. GEOMETRICAL
INTERPRETATION.
1. In the first section of this Chapter, we .shall
explain what is meant by an ordinary differential
equation in two variables, and show how to derive a
differential equation from its complete primitive.
In the second section, we shall show how ordinary
differential equations in two variables may be interpreted
geometrically.
SECTION I.
Complete Primitive. Order and Degree of an Ordinary
Differential Equation.
2. An equation of the form
<o(a,3/) = 0 A)
is ordinarily used to express in algebraic language the
fact that one of the two variables x and у is a function of
the other. If this equation also contains an arbitrary
constant c, its presence is indicated by writing the equa-
equation in the form
w(x,y,c) = 0 (Г)

2 ORDINARY DIFFERENTIAL EQUATIONS.
By differentiating A'), we obtain
d,c^dx + ^dy=Q; B)
and the constant с 'may have been removed by the pro-
process of differentiation. If, however, B) still contains c,
it may be eliminated by means of (V); so that we find,
either immediately after the differentiation, or after the
elimination, an equation involving x, y, and -Л, of the
general form x
If we make use, as we shall often do, of the customary
abbreviations,
y ~dx' " ~ dx2' '"' " ~dxn'
the last equation may be written
F(x,y,y') = O; C)
and C) is called an ordinary differential equation of
the first order in two variables.
3. If the equation A) contains two independent arbi-
arbitrary constants, so that it may be written in the form
w(x, y, c, d) = Q, A")
(c, d, consts.);
two successive differentiations of A") will give an
equation containing y", from which, by means of A")
and the equation obtained from A") by a first differenti-
differentiation, both arbitrary constants, c, d, if they are still
present, may be eliminated. We obtain thus an equation
of the general form
which is called an ordinary differential equation of the
second order in two variables.

COMPLETE PRIMITIVE. 3
4. The equations A') and A") from which the differ-
differential equations C) and D) are obtained, are called the
complete primitives of C) and D), respectively. It is
clear that if A) contained three independent arbitrary
constants it would give rise to a differential equation
of the third order; and, in general, we see that the
order of a differential equation, which is defined as
that of the highest derivative in the equation, is
the same as the number of independent arbitrary
constants in the complete primitive. Thus, if the com-
complete primitive contains n independent arbitrary con-
constants, it will give rise to a differential equation of the
nth order.
The degree of a differential equation is the same as
the degree of the derivative of the highest order in
the equation, after the equation has been put into a
rational form, and cleared of fractions. Thus the
equation
is of the second order, and of the second degree.
From what has been said, it is seen that to find the
differential equation of the ntb order corresponding to
a primitive containing n arbitrary constants, it is
necessary to differentiate the primitive n times succes-
successively, and eliminate, between the тг + l equations thus
obtained, the n arbitrary constants.
The resulting equation will be the required differential
equation of the ntb order.
5. The inverse process—usually involving one or more
integrations—of finding from a differential equation its
complete primitive, is called solving, or integrating, the
differential equation, and the arbitrary constants, which
were formerly made to vanish by differentiation and
elimination, now reappear as constants of integration.
When the equation thus obtained contains exactly n
independent arbitrary constants, it is called the general
integral, or the complete primitive, of the differential

4 ORDINARY DIFFERENTIAL EQUATIONS.
equation of the nib order. Thus, if
F(x,y,y', ...,3/<»)) = 0 E)
be a differential equation of the ntb order, its general
integral will be an equation of the form
w(x, y,cx, ...,cn) = Q, F)
where the cv ..., cn are independent arbitrary constants.
It may be noted that F) is usually referred to as the
general integral of E), when F) is considered as having
been derived from E); if, however, E) is considered
as having been derived from F), F) is referred to as
the complete primitive of E).
It is evident from the method of deriving from a
complete primitive its corresponding differential equation
that the general integral cannot contain more than n
independent arbitrary constants; for the general integral
would then, being treated as a complete primitive, give
rise to a differential equation of an order higher than
the «*4
6. If a special numerical value is assigned to each
of the arbitrary constants, respectively, of a known
general integral of a given differential equation, the
resulting equation is called a particular integral of
the given differential equation. Thus the particular in-
integral is free from all arbitrary constants of integration.
For example, if the general integral has the form
y — mx — 71 = 0,
then the equations
will be particular integrals of the given differential
equation.
7. We shall now apply to two simple examples the
method of finding the differential equation corresponding
to a given complete primitive.

COMPLETE PRIMITIVE. 5
Example 1. It is required to find the differential equation of
the first order corresponding to the complete primitive
y-cx=O, G)
where с is an arbitrary constant
By differentiation, we obtain,
dy-cdx=%
(8)
v
dx
Hence, from the first equation,
d3-JL
dx x
This is the differential equation required. If we consider (8) as
given, and G) as having been derived from it—by methods to be
explained later—G) is called the general integral of (8). By
assigning to с in G) different numerical values, different particular
integrals are obtained.
Example 2. It is required to find the differential equation of
the second order corresponding to the complete primitive,
x2+2ax+y2 + 2by=0. (a, b, consts.)
By two successive differentiations, we obtain the equations
If a and 6 are eliminated from these three equations, we find,
as the differential equation required,
8. It has been shown that to pass from a complete
primitive to the corresponding differential equation
involves merely the processes of differentiation and
elimination; but since the steps of an elimination
cannot be retraced, it is a matter of much greater
difficulty—if possible at all—to pass from the differen-
differential equation to the corresponding complete primitive,
or general integral. It will be our object to show how,
in a number of the simplest and most important cases,
we may, from a given differential equation, deduce ite
general integral.

ORDINARY DIFFERENTIAL EQUATIONS.
SECTION II.
Geometrical Interpretation of Ordinary Differential
Equations in Two Variables.
9. If the ordinary differential equation of the first
order in x and y,
?(x,y,y') = 0 A)
be written in the solved form,
„Zfej B)
where X and Y are supposed to be one-valued functions,
it is clear that to any pair of values ascribed to x and
y, a fixed value of ?/ will correspond.
If we consider x and у to be the rectangular
coordinates of a point in the plane, y' will represent
the numerical value of the tangent of the angle made
with the ж-axis by the straight line connecting the
point (x, y) with the origin of coordinates. Now
suppose the point (x, y) to move a short distance in
the direction given by y'; in the new position of the
point, if will generally have a new value. Suppose
the point to move a short distance in the direction
now given by y'; in this third position of (cct y) there
will be in general a third value ascribed to y'\ the
point (x, y) can now be supposed to move a short
distance in this last direction—and so on. By this
means a figure will be traced of which the limit will
be a curve of some kind, when the distances through
which the point (x, y) is moved are indefinitely
diminished. At every point on this curve the equation
y=j <2>
is satisfied; that is, if
Ф,у)=о C)
be the equation to this curve, the equation <o = 0 must

GEOMETRICAL INTERPRETATION 7
be a particular integral of equation B), or of the equi-
equivalent equation A).
The curve traced by a point moving under the above
restrictions is therefore called an integral curve of the
ordinary differential equation A). If we start from
any point not on the curve C), it is evident that by
proceeding as before we get a new integral curve. We
might, for instance, take as successive starting points
the points on the ж-axis—provided that the ж-axis does
not happen to be itself an integral curve—and it is
evident that, in all, oo1 different integral curves would be
obtained, one passing through every point of ordinary
position in the plane. These curves must be represented
by an equation of the general form,
co(x,y,c) = 0, D)
where с is an arbitrary constant, or parameter, which
assumes different numerical values according as D) is
made to represent the different individual curves of the
whole system of integral curves belonging to equation
A). In other words, D) is the general integral of A).
Example. The differential equation of the first order
•-*¦
..E)
О JC
represents a system of oo1 straight lines through the origin. For
^ is the numerical value of the tangent of the angle between the
дг-axis and the line joining the point (x, y) with the origin ; and

8 ORDINARY DIFFERENTIAL EQUATIONS.
as У ffives the direction in which the point (x, y) is to be moved,
equation E) asserts that the point (x, у) always moves on the
straight line connecting that point with the origin. Since there-
therefore each point of the plane moves on one line of a system of
straight lines through the origin, equation E) represents the family
of oo1 straight lines
с being the arbitrary parameter. Thus F) is the general integral
of E) : and the particular integrals are obtained by assigning to о
different numerical values.
10. Since the complete primitive, or the general
integral, of a differential equation of the second order
must contain two independent arbitrary constants, or
parameters, it is clear that this general integral, or, as
we may say, the differential equation of the second order
itself, represents geometrically a doubly infinite system
of curves in the plane.
Similarly, a differential equation of the third order
represents a triply infinite system of curves, etc.
Example. The ordinary differential equation of the second order
У'=0 G)
asserts that the curvature of the path along which the point (x>y)
is to be moved is everywhere zero. Hence the point (x, y) must
always describe a straight line, that is, the doubly infinite system
of curves which satisfy the above differential equation must be the
да а straight lines of the plane
y-mx-n = Q. (8)
It may at once be verified that (8) is the general integral of G).
EXAMPLES.
Form the differential equations of which the following are the
complete primitives, a, b, с being arbitrary constants.
A) y-<ar.
B) y=

EXAMPLES.
D) /-2<a:-c2 = 0.
E) y = «-*»"+taii-
F) y = (cx + \ogx+V
G) !/=о1-+с-Л
(8) ег» + 2ете»+са=0.
(9) 3.=<rf+te.
A0) у=с<хш(жс+Ъ).
A3) y
A4) 2/=(a+fe+yje4-c.
A5) Form the differential equation of the oo1 circles having their
radii equal to r :
A6) Form the differential equation of all circles having their radii
equal to r.
A7) Find the differential equation of the family of straight lines
which touch the circle
and show that the circle itself also satisfies the differential
equation. The equation to the tangents is
where the constants a and Ь must satisfy the condition
as+6!=l.
A8) Find the differential equation of all the conic sections whose
axes coincide with the coordinate axes :
A9) Find the differential equation of all logarithmic spirals around
the origin :

CHAPTEK II.
SIMULTANEOUS SYSTEMS OF ORDINARY DIFFER-
DIFFERENTIAL EQUATIONS, AND THE EQUIVALENT
LINEAR PARTIAL DIFFERENTIAL EQUATIONS.
11. We shall reserve for a later chapter the con-
consideration of the genesis of an ordinary differential
equation in three or more variables, when that equation
is obtained from a single primitive by methods similar
to those of Chapter I. It will be necessary, however, to
give in Sees. I. and II. of this chapter a few propositions
relating to simultaneous systems of ordinary differential
equations, and the equivalent linear partial differential
equations, in order to develop in the next chapter as
much of the Theory of Transformation Groups as we
shall need.
The third section of this chapter is intended as a
supplement to this chapter and to the preceding one.
We there indicate, for convenience of reference in
Chapter III., the method of integrating the simplest
form of an ordinary differential equation in two variables,
a problem which really belongs to the Integral Calculus;
and we also make a remark upon the integration of
the simplest form of a simultaneous system in three
variables.
A theory of integration for the general simultaneous
system will not be given until Chapter XII.

SIMULTANEOUS SYSTEMS. 11
SECTION I.
The Simultaneous System of Ordinary Differential
Equation*.
12. Suppose two equations of the form
U(x,y,z) = a, V(x,y,z) = h A)
are given, where U and V are independent functions of
x, y, z, and a and b are arbitrary constants. By differ-
differentiating A) we find
bU, 30, Ъ11, „1
—dx+-^dy + -5- dz=0,
дх Ъу 1 Эг I
as resulting equations.
But from the equations B) we find that relations of
the form
<fe dydz
dy dz dz dy dz дх Ъх dz dx dy dy dx
must hold: and, if we denote the denominators of these
ratios, which are known functions of x, y, z, by X(x, y, z),
Y(x, y, z) and Z(x, y, z), respectively, the equations C)
may be written
X Y Z w
Thus the system of equations A), treated as simultaneous
complete primitives, gives rise to the so-called simul-
simultaneous system of ordinary differential equations of the
first order, D).
13. This result in three variables is entirely analogous
to that of Art. 2 in two variables. The differential

12 ORDINARY DIFFERENTIAL EQUATIONS.
equation derived in that article from #one primitive of
the form U(x, y) = a may, of course, be written in a
form symmetrical with D),
dx _ dy
14. It is obvious that the results of Art. 12 may be
extended to n variables.
If
U1(xvx2>...,xn) = av U2(xv ...,xn) = a2t ...,
Un-i(xi, ...,xn) = xn.i E)
be a system of n — 1 equations in the n variables
oh, •••,&n, the Z7"i, ..., JIn_\ being independent functions
of those variables, and the a\, ..., an_i being arbitrary
constants, the system of equations E), being treated as
simultaneous complete primitives, will evidently give
rise to a so-called simultaneous system of ordinary
differential equations of the first order, which may be
written in the form
dxl_dx2_ _dxn . .
xx-x%---xn w
Here the Xlt..., Xn are known functions of x^, ..., av
In the next section we shall see how the simul-
simultaneous system in three variables may be interpreted
geometrically.
SECTION II.
Simultaneous Systems and the Equivalent Linear
Partial Differential Equations.
15. Equations are of frequent occurrence by means
of which a relation between the several partial deriva-
derivatives of a function of two or more variables is expressed.
If / be any function of x, y, z, the general form of such

SIMULTANEOUS SYSTEMS. 13
an equation, involving only partial derivatives of / of
the first order, and the variables x, y, z, will be
\ ' V' "дх' ду' bzJ '
and if / be known, the values of its partial derivatives
substituted in this equation must satisfy the equation
identically.
An equation which expresses a relation between the
partial derivatives of a function of two or more inde-
independent variables—and which may also contain the
independent variables themselves explicitly—is called a
partial differential equation; and the function /, whose
partial derivatives satisfy the equation identically, is
called the solution of the equation.
The order and degree of a partial differential equation
are determined just as are the order and degree of an
ordinary differential equation. A partial differential
equation of the first order and degree is said to be linear
of the first order; the term linear having reference only
to the manner in which the partial derivatives of the
solution / enter the equation.
Thus the general form of a linear partial differential
equation of the first order in n variables is
where the Xlt ..., Xn are certain known functions of the
independent variables x^, ..., xn.
We shall hereafter limit ourselves to the consideration
of such partial differential equations as are linear and of
the first order; since this class of equations is, as we
shall see, intimately connected with ordinary differential
equations.
16. The ordinary differential equation of the first order
in two variables may be written in the solved form,
dx _ dy m
T(^y)-Y{x,yy KX)

14 ORDINARY DIFFERENTIAL EQUATIONS.
and an intimate relationship may be shown to exist
between A) and the linear partial differential equation
in two variables,
X(x, y)-^-+ Y(xt y)~^- = 0 B)
For, if co(x,y) = const, be the integral of A), we find by
differentiation,
Now eliminating -Ж between C) and A), we find as
a necessary consequence of these equations the identity,
дх Ъу
That is to say, if the equation w(x, y) = c is an integral
of the ordinary differential equation A), w is also a
solution of the linear partial differential equation B).
Conversely", it may be readily seen that if the function
ю is a solution of the linear partial equation B), w = c
will also be an integral of A). Thus the equations A)
and B) represent fundamentally the same problem, since
to find an integral of A) is the same as to find a solution
of B), and vice versa.
17. If the general integral of a given differential
equation of the first order A) has been put into the form
?l(x, y) = c, (c = const.)
we shall call the function Q(xr y) the integral-function
of the given differential equation.
It is a proposition of the Theory of Functions, which
we shall here assume without proof, that an integral-
function of a differential equation of the first order
always exists, and that all integral-functions of a given
differential equation of the first order must be functions
of any one of the integral-functions; that is, that no
differential equation of the first order, A), can have two

SIMULTANEOUS SYSTEMS. 15
independent integral-functions. Thus if U and V be
two integral-functions of a given differential equation
of the first order, we must be able to express the one
as a function of the other, say
From this it follows that if we know any integral of
a differential equation of the first order, containing an
arbitrary constant, we may regard all possible integrals
of that equation as known.*
Also, since A) always has an integral-function, though
it cannot have two independent integral-functions, the
linear partial differential equation of the first order B)
must always have one solution, although it cannot have
two independent solutions. The whole number of solu-
solutions of B), or of integral-functions of A), is evidently
unlimited; for if w be a solution of B), it is easy to
see that any function of w, as Ф(а>)> is also a solution
of B).
For, substituting Ф(а>) in place of / in B), we find
for that equation
<%Ф{уди> vdw\
-7—1 Л ——H ¦* ^r~ I — " »
dw\ dx dy/
but as the expression in parenthesis is zero on account of
w being a solution of B), the left-hand member of the
last equation is zero, that is, Ф(ьи) is also a solution
of B).
Since every solution of B) is an integral function of
A), it also follows from this that the most general
integral of the ordinary differential equation A) has
the form
Ф((о) = const.,
where w is any integral-function of A).
* The fact that an ordinary differential equation always has a general
integral is illustrated by the types of integrable equations, Chapter IV.,
as well as by the development, Art. 72, of the general integral in a
series.

16 ORDINARY DIFFERENTIAL EQUATIONS.
18. The linear partial differential equation in three
variables has the form
X(x, y, z)&+ Y(x, у, *)Ц+2(х, у, г)^ = 0 D)
and it is easy to see that the same relation exists
between D) and a system of equations of the form
dx dy da
that was seen to exist between A) and B).
It is shown in the Theory of Functions that there
are always two, and only two independent functions of
the form U(x, y, z), V(x, у, г), which, when written
equal to two arbitrary constants a, b, respectively,
U(x,y,z) = a, V(x,y,z) = h F)
will give, when these equations are differentiated as in
Art. 12, values for the ratios dx, dy, dz, which satisfy
the simultaneous system E). When the equations F)
are derived from E)—by methods to be explained later—
they are called the integrals of E).
By differentiation, we find from F)
ЭР, ЪТТ ЭР,
ЭК, , ЭК, , ЭК, „
and these equations, by means of E) may be written,
But the last two equations show that the functions U

SIMULTANEOUS SYSTEMS 17
and V, which in accordance with Art. 17 we shall call
the integral-functions of E), must be solutions of the
linear partial differential equation D). It is thus obvious
that any integral-function of E) must be a solution of
D), and vice versa. Hence we see that D) cannot
have more than two independent solutions; that is, that
every solution of D) must be capable of being expressed
as a function of any two independent solutions of D).
The whole number of solutions is, however, unlimited;
for if U and V are solutions, it is easily seen that any
function of II and V, as Ф( U, V) is also a solution of
D). For, substituting Ф for /in D) there results
but since tTand Fare solutions of D), the expressions in
the parentheses are zero; ;that is, the last equation is
identically satisfied, or Ф( U, V) is a solution.
Since the solutions of D) are also integral-functions of
E), the most general integral of E) has the form
&(U, V) = const.,
where U and V are any two independent integral-
functions.
19. The equations of the preceding article,
U{x,y,z) = a, V(x, y, z) = b F)
represent two families of со1 surfaces in space; these are
the so-called integral surfaces of the simultaneous system
E). Also the system of equations F) may obviously be
said to represent a family of со2 curves in space—the
curves of intersection of the two families of surfaces—
each particular curve being obtained by assigning a pair
of special numerical values to the arbitrary constants a
and b. One of these curves evidently passes through
every general point in space; and at every general point
P, on one of these curves, the equations E) must be
satisfied. That is to say, the tangent at the point P

18 ORDINARY DIFFERENTIAL EQUATIONS.
to the curve passing through that point must have a
direction of which the direction cosines are proportional
to X, Y, Z respectively.
These со2 curves, at every point of which the equations
E) are satisfied, are sometimes designated as the char-
characteristics of the linear partial differential equation D),
which is equivalent to the simultaneous system E).
Example. As a simple example, we may suppose the equations
6) to have the forms
x+y+z=a, x4f+s2=b% F')
(a, b2 consts.)
the first equation representing a system of parallel planes, the
second a system of concentric spheres around the origin. Thus the
simultaneous equations (&) represent the oo2 circles cut from the
oo1 concentric spheres by the да1 parallel planes.
By the method of Art. 12 we find the simultaneous system to
which F') give rise by differentiation in the form,
xdx+ydy + zdz=O;
. dx dy dz
whence = —aL= .
z—y x~z y — x
This is of course equivalent to the linear partial differential equation
and it may be readily verified that the most general solution of
this partial differeutial equation has the form
Ф(х+у+г, а?+у2+гг).
20. In a manner entirely analogous to that of Art. 18,
it may be seen that the linear partial differential equation
of the first order in n variables,
*&*%+¦¦¦+*•?-* m
where the Xv ..., Xn are certain functions of xv ..., scn,

SIMULTANEOUS SYSTEMS. 19
represents the same problem as does the simultaneous
system of ordinary differential equations
dx1 dx2 dxn ,ЙЧ
г==()
Also it follows from considerations similar to those oi
Arts. 17 and 18, that G) cannot have more than n — ]
independent solutions. If these are of the form
the U'b being put equal to arbitrary constants,
will give the integrals of (8). Moreover the most
general solution of G), or the most general integral
function of (8), has the form
SECTION III.
Integration of Ordinary Differential Equations in Two
Variables, in whim the Variables can be separated
by Inspection; and of a Special Form of a Simul-
Simultaneous System in Three Variables.
21. Although we are not yet ready to present any
general theory of integration of ordinary differential
equations, it will be necessary for us to call attention
here to the fact that when the variables can be separated
by inspection in an ordinary differential equation of the
first order in two variables, so that the equation may
be written
dx _ dy
X(S)-Y(y) W
its complete integration, which is virtually a problem
of the Integral Calculus, may be immediately accom-

20 ORDINARY DIFFERENTIAL EQUATIONS.
plished. The general integral will have the form
and B) is considered the general integral of A), whether
the functions in equation B) can be expressed in a
form free from the sign of integration or not.
Of course the differential equations in which the
variables may be separated by inspection constitute only
a very small class of all ordinary differential equations
of the first order in two variables; but we shall see that
the integration of these, the simplest possible differential
equations of the first order, will, in a future chapter,
furnish us with the means of integrating whole classes
of very complicated equations.
Example 1. The ordinary equation in two variables
{\ + x)ydx+{\ -y)xdy=0
may be written dx-\ *ч$у=о.
The general integral will therefore have the form
which is seen to be log(#^)+.r—y=const.
The given ordinary differential equation is, moreover, equivalent
to the linear partial differential equation
and it may at once be verified that if
or any function of this function, be put in place of / in the
liuear partial differential equation, that equation will be satisfied
identically.
Example 2. Given the equation

SIMULTANEOUS SYSTEMS. 21
Here the variables are already separate, and the general integral is,
em^ + sin"^ = a, (a = const.)
But a function of an arbitrary constant is itself an arbitrary con-
constant : hence, taking the sine of both members of the last equation,
and replacing sin a by c, we see that the general integral may be
written __
Wl -у2+учЛ -а? = с. (e=const.)
It may be readily verified that any function of the integral function
х*1\-у2+уЛ 1 - зР-
is a solution of the linear partial differential equation
22. Similarly, if a given simultaneous system in three
variables has the very special form
dx _ dy _ dz
its integrals may also at once be written in the forms
dx
f dy f dz
Example. The simultaneous system
dx _ dy _ dz
!c~ y~T
evidently has for its integrals
logx — logy = const., logy — log s = const. ;
or, as they may be written,
-=af ¦- = &. (a, b const)
It may at once be verified that any function Ф ( -, - ) is a solution
of the linear partial differential equation, equivalent to the above
simultaneous system,
3/ ы df

22 ORDINARY DIFFERENTIAL EQUATIONS.
23. It may, finally, be noticed that if the given
simultaneous system has the particular form
dx _ dy da ,
X(x, y)~ Y(x, y)~Z(x, y, z)'
and if the integral of the ordinary differential equation
in two variables,
dx dy
X@-y~)-Yfalj)'
has been found, either by separating the variables, or by
methods to be explained later, in the form
U(x,y) = c, (c = const.)
then the last equation may be used to eliminate either
of the variables x or y, as may be desired for the
purpose of integration, from X, Y, or Z. If, for instance,
we find from the last equation
у = ф(с, x),
the second integral of the given simultaneous system
may be found by integrating an ordinary differential
equation in two variables of the form
dx _ da
X(x, ф)~Z(x, ф, z)'
where, of course, the value of ф in terms of x and с
has been substituted in place of y.
If the integral of this equation has been found in
the form
W(x, z,c) = b, (b = const.)
we now substitute for с its value U(x, y), finding the
second integral required in the form
V(x,y,z) = b.
The reader will bear in mind that the above is only a
very special form of simultaneous system in three

SIMULTANEOUS SYSTEMS. 23
variables. A general theory of integration of such
differential equations will be given later; but it is
convenient to notice these simplest forms now, in order
to make use of them in the next chapter.
Example. Given the simultaneous system
dx dy dz
яг xy z*
An integral of -5 = -^
яг xy
is found to be ~—c. (c=const.)
Hence, in the equation =~f
we may put for x} -. Thus we find
dz
of which the integral is —- = b. . (b = conat.)
Now put for с its value, ^, and we find as the second integral
• i 11,
required =6,
Of course this result might have been obtained directly from
dx_dz
without any intermediate steps.
It may readily be verified that any function of the form
is a solution of the linear partial differential equation
which is equivalent to the given simultaneous system.

24 ORDINARY DIFFERENTIAL EQUATIONS.
EXAMPLES.
Integrate the following ordinary differential equations of the first
order in which the variables may be separated by inspection, giving
in each case the equivalent linear partial differential equation in
two variables, and verifying that the integral-function of the
ordinary equation is a solution of the linear partial equation :
0) %=Щ**-
1 ' 1+y l+x
E) smxcoaydx=coaxainydy.
(8)(
CO'
Give the linear partial differential equations equivalent to the
following simultaneous systems ; integrate the simultaneous
systems, and show that any function of the integral functions of
each simultaneous system is a solution of the corresponding linear
partial equation.
dx _dy _ dz , . dx ^dg_ds
' ' x ~~ у ~ ~z ' yz хг осу
... dx dy dz . dx_dy_ dz
A0) **=«&*.
4 ' -y X 1+22
dx
In A2) the symbol — is used merely to show that the coefficient
of J- in the linear partial differential equation equivalent to A2) is
zero. That partial differential equation is
3
and since J- does not occur at all, it is clear that x ia a solution of
the equation: that is, #=const. is one integral of A2).

CHAPTER III.
THE FUNDAMENTAL THEOREMS OF LIE'S THEORY
OF THE GROUP OF ONE PARAMETER.
24. We propose to develop in the present chapter
such of the propositions which Lie has established with
reference to the transformation group of one parameter,
as we shall need subsequently in the integration of
ordinary differential equations. The theory of the
group of one parameter in two variables is minutely
explained in order to enable the reader to make use of
the group as an instrument for investigation.
For the sake of greater clearness, we shall generally
limit ourselves to two variables in establishing the
necessary fundamental propositions; but the method of
extending the results to n variables, in such cases as it
is desirable, will be sufficiently indicated.
SECTION L
Finite and Infinitesimal Transformations in the Plane.
The Group of one Parameter.
25. By a transformation of the points of the plane, we
understand an operation by means of which every point
of the plane is conveyed to the position of some point of
the same plane.

26 ORDINARY DIFFERENTIAL EQUATIONS.
The general form of a transformation of the points of
the plane is given by the system of equations
Я1 = ф(я, у), Ух = ^{х, у), A)
where ф and \fr are independent functions of x and y.
We suppose here that the coordinate axes remain un-
unchanged ; but every point of general position (x, у) is
conveyed to a new position of which the coordinates are
ф(х, у) and yjs(x, y). If, now, A) be solved in the form
x = $(xvVl), y = ?(xvyi) B)
a transformation is obtained which will evidently carry
the point ф(х, у), yjr(x, у) back to its original position
(x, y). The transformation B) is thus said to be inverse
to the transformation A).
The successive performance of the transformations B)
and A) will evidently give a transformation of the form
хг = х,уг = у;
and the last is called the identical transformation. This
transformation, therefore, really leaves the position of
the point {x, y) unchanged.
26. Suppose that a family of oo1 transformations is
given by the equations
х1 = ф(х,у,а), уг = у1г(х, у, а), C)
where a is a parameter which can assume oo1 continuous
values. In general, then, it will not be the case that
the performance of any two transformations of the
family C) successively upon the points of the plane will
be equivalent to the performance of a third transforma-
transformation of the family C) upon those points. For instance,
the equations
x^a-x, ^ = 1/
represent a family of transformations which do not
possess the above peculiarity. For if
xz = a1-xv У2 = уг
be a second transformation of the family, we find, when

THE GROUP OF ONE PARAMETER. 27
the two transformations are successively performed
upon the point (x, y), that this point assumes a position
given by
aE = o1-a+aj, yz = y.
But the transformation given by the last equations does-
not belong to the original family, of the general form,,
xx — const.—x, yx — y.
If, now, Xl = <f>(x, у, а), ух = у1г{х, у, a)
be any given transformation of the family C), and if
be a second transformation of that family, then the
transformation which, results from performing these two-
successively evidently has the form
> У> a)> VK^, У> »), «Л,
х, У, a), yfr(x, y, a), c^}.
If it happens that the right-hand members of th.es&
equations have the general forms
ф{ф(х, у, a), yf,(x, y, a), aj = ф{х, у, Х(а, а^},
yjs{<f>(x, y, a), yj,(x, y, a), aj = yjr{x, y, X(a, c^)},
where X is a constant, or parameter, depending upon a
and ax alone, then the family of oo1 transformations C)-
are said to form a finite contvnuous group.
Expressed in words, this condition evidently is that
tfte result of performing successively any two trans-
transformations of the family C) upon the points of the
plane must be equivalent to the result of performing a
third transformation of that family upon those points.
We shall add to the above, in the groups which we shall consider^
the further condition that to every transformation with the para-
parameter a, of the family C), we must be able to assign a certain
transformation of the family with the parameter a, such that the
latter transformation shall be the inverse of the former, a being
a function of a alone. From this further condition follows imme-
immediately that the family C) must contain the identical transformation.

28 ORDINARY DIFFERENTIAL EQUATIONS.
The above can hardly be considered aa a new condition, however,
for it will be seen to be satisfied eo ipso in the groups of which
we shall make use; and, in fact, this condition is always satisfied
eo ipso in the groups which are of importance in practical investi-
investigations.
Since the family of transformations C) contains one
arbitrary parameter, we call it, under the above con-
conditions, a group of one parameter; or, symbolically,
It is clear that the above definition of a (?j is in-
independent of the number of variables; that is, if in n
variables we have oo1 continuous operations given which
satisfy the above conditions, these oo1 operations are
said to form a group of one parameter.
Example 1. As an example of a group of one parameter,
or a <?!, in two variables, we may take the family of cc1 translations
along the #-axis given by the equations
x1=x+a, уг=у. D)
{a = parameter.)
A second transformation of this family is given by equations of the
form
x% = x1 + a1, y2=yi.
By eliminating arj, y\ from these equations, we find
#2=я+(а+аД y2=y;
and these last equations evidently represent a transformation of
the original family D), namely, the transformation for which the
parameter has the value (a + a^). Hence the family of translations
D) form a Gx : since we see that the effect of performing two of
the transformations successively is the same as that of performing
a certain third transformation of the family. Also to the trans-
transformation with the parameter a we may always assign one with
the parameter - a, such that the latter is the inverse of the former :
hence the <?i contains the identical transformation, which is obviously
obtained in this case by assigning to the parameter the value a — a,
or zero.
Example 2. As a second example of a (?j we may take the family
of cc1 rotations of the points of the plane around a fixed point,
which we shall assume as the origin of coordinates. The well-known

THE GROUP OF ONE PARAMETER.
29
equations to these rotations, as given in Analytical Geometry, are
x1=x cos a — у sin a, E)
where a is the angle through which the radii vectors of the points
of the plane are turned.
A second transformation of the family E) is given by the
equations
x2 =#iCO8 o^ — ^sin alr
y2=xx sin Oj+^j cos Oj.
By eliminating хг and yv we find
#2 = {% cos a — у Bin a) cos ax — (x sin a+у cos a) sin Oj
= :r(cos a сое ax — sin a sin Oj)—y (sin a cos ax + cos a sin Oj) ;
Similarly, ya = ^sin(a + ai)+ycos^a + a1).
But the last two equations evidently represent a transformation
of the family E), namely the transformation for which the para-
parameter of the family has the value a + a!. Also the inverse of the
transformation with the parameter a is evidently the transformation
with the parameter -a. We further find the identical trans-
transformation in the family by assigning to the parameter the value
a —a, or zero. Hence, according to the definition, the family of cc1
transformations E) form a group of one parameter, or a Gv
27. Suppose that a Gx is given by the equations
Х1 = ф(х,у,а)> y1 = yjr(x,y>a); F)
we shall show that under the conditions of Art. 26, the
Gx always contains one infinitesimal transformation.

30
ORDINARY DIFFERENTIAL EQUATIONS.
For, let a0 be the value of the parameter for which F)
gives the identical transformation, so that
х = ф(х,у,а0), y = x/,(x, y, a0);
If, now, we assfgn to the parameter a a value which
¦differs from a0 only by an infinitesimal quantity, say
ao + §a, the corresponding transformation
х^ = ф(х, У, aa+Sa), y^xjsix, y, ao+Sa) F')
will differ only infinitesimally from the identical trans-
transformation ; that is, F') will be an infinitesimal transfor-
transformation.
By Taylor's theorem,
1.2
or, from the above value of the identical transformation,
1Г2"-'
Thus we see that ж?, уг really differ from x and у by
infinitesimal quantities.
If the coefficients of all powers of Sa up to the rth
vanish for all values of x and у in the last equations,
we introduce St=Sar as a new infinitesimal quantity, and
so obtain the equations of the infinitesimal transforma-
transformation in the general form
X1 = x+g(x,y)$t+..., yx = y
Here ? and r\ also contain aQ; but since aQ is a mere
number, it is not necessary to write it explicitly in g
and t].

THE GROUP OF ONE PARAMETER. 31
It is true that by this method for finding the infin-
infinitesimal transformation of a given Qx F), it is impossible
to say whether the succeeding terms of the last equations
involve integral or fractional powers of St; this difficulty
is however avoided by a second method given below.
28. Let a fixed value e be assigned to the parameter a in the Qv
3>! = <i>{x,y,a), У! = ^(*,у,а), F)
and suppose that the corresponding transformation, which we shall
designate as the transformation (e), carries the point of general
position P to the new position Px. Then, by hypothesis, the trans-
transformation in the <?! F) which is inverse to (e) will carry the point
P1 back to the position P. Now if the parameter of the last trans-
transformation be designated by ё, it is clear that a transformation with
the parameter ё + Se, where 8e is an infinitesimal quantity, will
carry the point P1 not exactly back to P, but to a position P which
is at an infinitesimal distance from P. If the transformations (e)
and (e + 8e)be performed successively, the result must be equivalent
to the performance of a third transformation of the family F);
one that will take the point P directly to the position P. But
since the distance PP is infinitesimal, the transformation which
carries the point P directly to the position P is called an
infinitesimal transformation.
The above geometrical considerations may be carried out analyti-
analytically. The first transformation is represented by
Х1 = ф(х, у, e), y^^x, y, e) ;
and the second by
where we suppose {x, у), (хъ yt), and {otft y') to be the coordinate
of the three points P, P1( and F respectively. The transformatio
which carries P directly to P' is found by eliminating хъ ух fro
the above equations : we find
Developing in powers of <5ey we have

32 ORDINARY DIFFERENTIAL EQUATIONS.
But since the transformations (e) and (?) are inverse, we have the
identities,
y, e\ 5},
and the last two equations become
nd it is evident that these equations represent an injmiterimal
transformation.
It is easy to see that the coefficients of 8e above do not vanish
identically; for they may be written
Эф(*ь У!» «) Э^иУц •)
respectively : and if these expressions were identically zero, the
equations F) would necessarily be free of any parameter, which is
contrary to hypothesis.
Since ё depends upon e alone, the equations to the infinitesimal
transformation may evidently be written
and it is clear that every Gft in the plane contains at least one
infinitesimal transformation.
29. If t be a parameter, it follows from the last two
articles that the general form of an infinitesimal trans-
transformation in two variables will be
..J v
a, y)St+
We shall, as usual, neglect higher powers than the
first of the infinitesimal quantity Й; and hence the
increments which x and у receive by means of the above
infinitesimal transformation have the forms
Sx = ?{x,y)St, iy = 4(x,y)it (8)

THE GROUP OF ONE PARAMETER. 33
It is clear that this transformation assigns to every
point (x, у) of general position, a direction through which
it is to be moved, given by
and also a distance through which it is to be moved,
given by
JSx' + Sy2 = VfT?. St.
As far as determining a direction through which, a
point of general position is to be moved 19 concerned, the
infinitesimal transformation offers an analogy to the
ordinary differential equation of the first order in two
variables (Chap. I, Sec. II).
We can get a clear and fruitful idea of an infinitesimal
transformation, if we suppose that we put all the points
of the plane into motion simultaneously, by performing
upon them the infinitesimal transformation (8) an in-
infinite number of times. In this manner a point (ж, у)
will assume a simply infinite number of continuous
positions, which form a curve. The whole change of
position of the points of the plane, since it is repeated
from moment to moment, may be called a permanent
motion, and may be compared to the flow of the
molecules of a compressible fluid.
If ? represents the time, and we measure it from a
fixed point, say t = 0 it is clear that] the point of general

34 ORDINARY DIFFERENTIAL EQUATIONS.
position (x, y) will, after the time t, arrive at a new
position (хг, уг), where the coordinates xX) yv are functions
of ж, y, and t. If ? increases by dt*, x1 and y1 will, by
(8), receive the increments
so that x1 and y1 may be found as functions of t by
integrating the simultaneous system
The first of these equations has, as we know, an
integral of the form
U{xb уг) = const.,
and by Art. 23, the second equation has for general
integral,
V{xli yx) — t = const.
Since at the time t = 0 the point (xv уг) must be at the
fixed position (ж, y), we must choose the arbitrary
constants in the last equations in the forms
U(x, y), V(x,y);
so that xlt уг are given as functions of t, x, and y, by
the equations
U(xv y,)= U(x, y), \
v '
These equations obviously represent a Glt with the
parameter (; and that such must be the case was clear,
a priori, from the kinematic illustration. For, if in the
time ( the permanent motion carries the point (ж, у} to
the position (xv уг)—and in the time tx carries the point
(xlt 2/j) to the position (ж2, у^)—it is evident that in the
time t+tt the point (ж, у) will be carried to the position
* We may clearly use either of the symbols S or d, to indicate an
infinitesimal increment. Here we make use of d in order that the
simultaneous system may appear in the usual form.

THE GROUP OF ONE PARAMETER 35
(ж2, y2); that is to say, the successive performance of any
two transformations of the family (9), with the values
t and tx of the parameter, is equivalent to the perfor-
performance of a single transformation of the family, with
the value (?+?j) of the parameter. That is, the trans-
transformations (9) form a Gv
Thus we see that every infinitesimal transformation
(8) in the plane belongs to a G1 of finite transformations
(9); and the Gx (9) may be said to be generated by the
infinitesimal transformation (8), since (9) may be con-
considered as equivalent to the repetition of (8) for an
infinite number of times.
It may be shown by means of the Theory of Functions,
that every group of one parameter in the plane contains
one and only one, infinitesimal transformation. Thus
we may consider the infinitesimal transformation as the
representative of the Gv
30. It is clear that a practical method for obtaining
the infinitesimal transformation of a given Qlt is to
assign to the parameter, in the equations to the finite
transformations of the Glt a value differing only by an
infinitesimal quantity from that value which gives the
identical transformation.
Example 1. The equations to the Gx of rotations,
.% = X cos a — у sin a,
Ух = 05 sin o. + y cos a,
will obviously represent an identical transformation when the
parameter a has the value zero. Thus, if we give to a a value Щ,
where 8t is an infinitesimal quantity, we should obtain the infini-
infinitesimal transformation of the above Ov We find
хх = зв cos bt — y sin 8t,
y1=xsin 8t+y cos 8t;
or, by a well-known trigonometrical reduction,
The last equations, therefore, represent an injmitesimal rotation.

36 ORDINARY DIFFERENTIAL EQUATIONS.
Example 2. Suppose the x1 transformations
Xl=xt, yx=yt
to be given; it is easy to verify that they form a group of one
parameter. Also the identical transformation is obviously given
when the parameter t has the value 1. Hence, to find the infini-
infinitesimal transformation, we assign to the parameter the value 1 + St;
and the infinitesimal transformation of the above Gx is seen to be
31. Since transformations of the nature of those which
we have been discussing are certain operations, upon the
points of the plane, which are independent of the coor-
coordinate axes, it is evident that if equations, representing
a Glt of the form
х1 = ф(х, у, t), Уг = ^(х, у, t), A0)
are given, these equations must still represent a Qx when
new variables are introduced.
32. We shall now derive a very useful symbol to
represent an infinitesimal transformation in the plane.
If х^ф{х,у,1), ^ = it(xtytt) A0)
be the finite equations to a Glt we can evidently consider
fti f th f f( ) fti f
q lt y
any function of the form f(xv уг) as a function of x, y,
and t; and for that value of t which gives the idtil
transformation, say for t = 0, we must have xx = x
d h f( ^f{ ) Si/(
, y , x , yx yt
and hence f(xlt y^)—f{x, y). Since/(ж,, у2) varies when
t varies, we are led to inquire as to what increment, 8f,
the function f(xv yt) receives, when x1 and y^ receive
their respective increments,
We find

THE GROUP OF ONE PARAMETER. 37
and this is the increment of the function f(xlt y^) under
the infinitesimal transformation of the Qv
If *=0, since then xt = x and у^ = у, we must have
From the form of this increment, it is seen at once that
if we know what increment a function f(x, y) receives by
means of the infinitesimal transformation of a Qlt we
may consider the infinitesimal transformation of the G1
to be known, since the increments of x and у under the
infinitesimal transformation are precisely the coefficients
of -^ and -^ in the above expression for 8f. Hence it
Ъх Ъу r J
is quite natural to introduce the symbol
to represent the infinitesimal transformation
хг = х + ?(х, y)St, yx = y + v(x,y)St.
Thus, when we speak of the infinitesimal transformation
a/ a/
"Ъх Ъу
we mean the infinitesimal rotation
x1 = x-ySt, y1=y + xSt.
We shall usually represent the above symbol for an
infinitesimal transformation still more briefly by the
symbol Uf; so that, of course,
Since the infinitesimal transformation of a given G^
may be considered to represent the Glt Art. 29, we shall
often speak simply of the (?„ Uf.
33. It is easy for the reader to convince himself that

38 ORDINA R Y DIFFERENTIA L EQUA TIONS.
an infinitesimal transformation in the variables x, y, z
—that is, a transformation by means of which these
variables receive the increments
may be represented by the symbol
and the remarks of Arts. 29-32, mutatis mutandis, may
at once be extended to this transformation in three
variables. Thus the above transformation assigns to
each point of general position (ж, у, z) a distance,
through which it is to be moved; and a direction, given
by
&в:ЙЗ/:& = ?:,:?
The symbol of an infinitesimal transformation in n
variables is obviously
and the remarks of Arts. 29-32 as to the geometrical
meaning of a transformation, etc., may also be extended
to the transformation in n variables.
34. The symbol U(x) means, put x in place of fin the
identity
and similarly for U(y). It is seen immediately that
so that U(f) = U(x). |ч

THE OROUP OF ONE PARAMETER. 39
35. If new variables x', у are introduced into the
symbol Uf, since
Эж Ъх' Ъх by' Ъх'
Ъу Ъх" ду^гу' Ъу'
Uf becomes
where ? and rj are expressed in terms of x' and y'. But
the last expression may evidently be written
The method of extending this result to n variables is
obvious.
36. We shall now see how convenient the new symbol
for an infinitesimal transformation is.
If the function f(xlt i/j) be developed by Maclaurin's
formula, we find, writing^ iorffa, y^),
Now
(ft Ъх1 dt 'гЪу1 dt '
and writing, as we may, the symbol d for the symbol 5
to express an infinitesimal increment of хг and yv we
have, Art. 29,
Hence

40 ORDINARY DIFFERENTIAL EQUATIONS.
But the last expression is exactly the symbol Uf written
in the variables xv уг; let us call this U-J. Thus
dt ~ U^'
and this identity means that any function fx of x1 and
gives, when totally diflerentiated with respect to t, U
But f/j/is itself such a function; hence
and the law of formation of the coefficients in the
expansion A1) is now obvious.
If we put t = 0 in the coefficients of A1), then хг and
are changed into x and у; also U-if becomes Uf;
s »?¦ a\ т TT" у тт" /*\ j rxn p i J-t»
i becomes U(Uf), etc. Thus we arrive at
important expansion
jo A2)
This holds, of course, when /г has the particular values
xv and 3/j. Thus
xl = x+\u(x)+~V{U{x))+.
and these are evidently the finite equations of the ffj of
which
is the infinitesimal transformation. The equations A3)
are of course only another form of the finite equations

THE GROUP OF ONE PARAMETER. 41
found, Art. 29, by integrating a simultaneous system.
The reader may readily see that the results of this
Article may at once be extended, mutatis mutandis,
to n variables.
Example 1. Suppose the infinitesimal transformation
is given ; and we wish to find the finite equations to the G^.
Here it is seen at once, Art. 34,
V(x) = -y, J%) = *,
ЩЩ*)) =-*, U(U<g)) =-jr,
щщщх))) = у, щщиш »-«.
щщщщх))))= .v, щищиш)* у-
Thus, by A3),
Х-Т 'у- '' Г+ ''" У4- -?- Г
*=*{[- ПГз+-) +3/0" Г2+ГОГ4 --)•
By well-known developments of the Differential Calculus, the
last equations may be written
#i = #cos t— у sin t, ух = хъхп t+y cos t.
Hence the Gx is the G1 of rotations, mentioned Art. 26.
Example 2. Given
to find the finite equations of the G\.
Here, proceeding as above, we find the expansions

42 ORDINARY DIFFERENTIAL EQUATIONS.
Instead of ё we may choose a as the parameter of the Gu and
we find as the finite equations,
In a number of the most important G^'s it will be found that all
the terms in the series A3), after the second, are zero.
SECTION II.
Invariance of Functions, Curves, and Equations.
37. Suppose, now, that we demand that a given
function of x and y, of the form ?l(x, y), shall be
invariant when we perform upon it the transformations
of a given Qv That is, if'the infinitesimal transforma-
transformation of the given G1 be
and the equations to the finite transformations be
х1 = ф(х,у,г), yt = yj,(x,y,t), A)
we demand that when, by means of A), Q is expressed
as a function of xv ylf ?2 must be the same function of
xlt y1 that it was of x, y. Thus we must have, for all
values of t,
Щх1,у1) = Щх,у),
by means of A).
But, from A2) in Sec. I., the last equation may be
written
and we see that a necessary and sufficient condition that
?2(ж, у) shall be invariant under the (?j A) is that
B)
If this condition be fulfilled, Q is called an invariant of
the <?! A).

INVARIAMCE. 43
The condition B) may be written out in full
.га эп .
and this shows that Q, is a solution of the linear partial
differential equation in two variables
or an integral-function of the equivalent ordinary differ-
differential equation
dd
Hence we see that, by Art. 17, a Gx in two variables
always has one invariant; and every invariant can be
expressed aa a function of any one invariant.
Example. The function
is an invariant of the O^ of rotations ;
xx—a; cos t —y ain (,
yi=x sin t +y cos (.
For, from the last equations,
J2(#, ^)sj;4^=(^i coa t+y-y sin tf + {yx cos t-^sin t)%
^^(cos3 i+sin2 t)+y12(sm2t + co3i t)
= x^+yii = U{xvy^.
Hence Q. has tbe same form in the variables xlt уъ for all values of
t, that it haa in the variables x, у ; i.e., J2 is an invariant of the Ov
The infinitesimal transformation of thia O1 is
and we may at once verify the fact that ?f(J2)=O ; for

44 ORDINARY DIFFERENTIAL EQUATIONS.
We see that the verification of the fact that J2 is an invariant is
much simpler when accomplished Ъу means of the infinitesimal
transformation of the Gu than when accomplished by means of the
finite transformations.
38. Every point of general position in the plane
describes, Art. 29, a continuous curve when the infini-
infinitesimal transformation of a given 0г is performed upon
it an infinite number of times. We shall call this curve
the patk-curve of the point under the transformations of
the Q1; and it is obvious that each Q^ may be said to
have oo1 path-curves, one through each point of general
position in the plane.
The direction through which a point (x, y) is moved
by a given Gv of which the infinitesimal transformation
is
is given, Art. 29, by
fy^njz, У)
Now if Q(x, y) be an invariant of the Qv we saw that
Q must satisfy the linear partial differential equation
But this partial differential equation is equivalent to the
ordinary differential equation
№>, y)dy-i(x, y)dx = O.
That is, Q must be an integral function of the last
equation; and the integral curves
0(x, 2/) = const,
have in each point the tangential direction
dy^i(x, У)
dx ((x,y)

INVARIANCE.
45
Hence an invariant, il(x, y), of a (?x in the plane, being
written equal to an arbitrary constant, will represent
that family of go1 curves in the plane which we call
the path-curves of the Gr
39. It should be noticed that any point, or points, in
the plane for which
i(x, У) = ч(х, y) = 0,
are absolutely invariant under the infinitesimal trans-
transformation of the given Glt
since in these points x and у do not receive any in-
increments at all.
Example 1. The infinitesimal transformation of the G1 of
rotations is
Uf= -у?+Ж
Thus, aa we know, the invariant must be a solution of the linear
partial differential equation

46
ORDINARY DIFFERENTIAL EQUATIONS.
That is, 12 is the integral-function of
dx _dy
or of xdx+ydy=O.
The integral-function of this ordinary differential equation may
obviously be assumed to be
Hence the path-curves of the Оъ that Is, the curves which the
points of the plane describe when they are subjected to the trans-
transformations of the G\ of rotations around the origin, are the circles
12 = x2 +y2= const.
This was, of course, geometrically evident a priori. The origin is
obviously an absolutely invariant point.
Example 2. Suppose the infinitesimal transformation
to be giv
The invariant is found as the solution of
#~—+w^—=0,
dx s Ъу ^
or as the integral-function of
xdy — ydx=0.
This integral-function is obviously 12 = -. Hence the path-curvet

INVARIANCE. 47
of the Qx, of which x2~?-+xyJ- is the infinitesimal transformation,
are the straight lines through the origin
У
~ = const.
X
The absolutely invariant points are given by
that is, x=0. Thus the y-axis is an invariant straight line, which
consists of absolutely invariant points.
40. A family of oo1 curves in the plane, considered as a
whole, may be invariant under the transformations of a
given (?! in two ways; each curve of the family may be
separately invariant, when, of course, the family is, as a
whole, also invariant; or the curves of the family may,
by means of the transformations of the Glf be inter-
interchanged among each other, leaving the curve-family as a
whole, however, still invariant.
We have seen that the path-curves of a given Gx are a
family of oo1 curves which is invariant in the first way,
that is, each member of the family is separately invariant.
Usually, however, when a family of oo1 curves in the
plane is invariant under the transformations of a given
Gv the individual members of the family are not in-
invariants, but are merely interchanged by means of the
transformations of the Gv
Let
Q(x, y) = const.
be any family of curves in the plane, which, as a family,
are invariant under a Gx whose finite transformations
are given by the equations
whilst the infinitesimal transformation of the Gx is
Since the curve-family is to be invariant, the equation

48 ORDINARY DIFFERENTIAL EQUATIONS.
to the curves must, in the variables xlt yv have a
functional form either identical with, or equivalent to,
that in x and у; that is, the equation to the invariant
family may be written, in the new variables, in the form
Q(a\, у1) = const.
Now we know that Ф(?2(ж, 2/)) = const. represents the
same family of curves that Q(xt y) = const, does; hence we
may write, as the condition that the family ?l{x, y) = const,
shall be invariant,
If the left-hand member of this equation be developed
by means of A2) in Sec. I., we find that a necessary and
sufficient condition that the curve-family ?1 (x, y) = const,
shall be invariant, is that
When a relation of this form holds, we sometimes say
that the family of curves admits of the transformations
of the Gv For the particular case that F(Q(x,y)) = 0,
the above condition gives, as it should, the family of
invariant path-curves.
Example 1. We saw that the concentric circles, Art. 39,
X2+yi = 7& (r = COnst.)
are the path-curves of the Gt of rotations ; and hence, of course,
they form a family of curves which are invariant under that OY
in such manner that each curve is separately invariant. But the
family of oo1 circles is also invariant under the Gv
with the infinitesimal transformation
For, from the above equations,

INVARIANCE. 49-
and substituting these values in the equation to the circles, we
find
or хг2 +yi = const. ;
which is an equation of the same functional form in xJt yx that the
original equation was in x, y.
Thus, by means of the finite transformations of the Gv we see that
the curve-family as a whole is invariant, while the individual
members are obviously not invariant. We may at once verify the
same thing by means of the infinitesimal transformation Uf. For
here
In this case, therefore,
or the curve-family is invariant.
Example 2. The family of straight lines
- = const.
x
admit of the G, of rotations around the origin. This may be
readily verified by means of the finite equations of the rotations.
But the infinitesimal transformation m
Hence the condition that
F(!2)sT(fi)
holds in this case.
41. The results of Arts. 37-40 may be readily extended,
mutatis mutandis, to three or more variables.

50 ORDINARY DIFFERENTIAL EQUATIONS.
Thus, if
be the infinitesimal transformation of a Gx in three
variables, the points, or curves, for which
are absolutely invariant under the Gv
Also, the necessary and sufficient condition that a
family of oo1 surfaces, ?l(x, у, г) = const., shall be invariant
under the Ог is that
())
42. In a manner entirely analogous to that of Art. 37
it is seen that the necessary and sufficient condition that
an equation of the form
Q(x,y) = 0
shall be invariant under a given Glt Iff, is that the
expression U(Q) shall be zero, either identically or by
means of ?1 = 0. This condition may at once be extended
to n variables.
Example 1. The equation Й = ж2+у2-1=о is invariant under
the <JV
For here
Hence the condition for an invariant equation is satisfied.
Example 2. The equation
is invariant under
For here
Hence the condition is satisfied.
_.m эй ъа

INVARIANCE. 51
43. We shall now find all equations of the general
form Q = 0, which are invariant under, or " admit of,"
a given Gv Uf\ and as this result is very important for
future use in more than three variables, we shall develop
it at once in n variables.
If the given Gx, in the n variables xlf ..., xn, have the
form
it might be possible that the ?lt..., ?n, are such function
that they all become zero by means of an equation which
is invariant under the Gv If we represent the equation
Ъ
it is true that in this case О = 0 is an invariant equation :
but the system of values of the variables which satisfy
0 = 0 is not transformed at all.
For instance, in two variables, the equation
is evidently invariant under the Gv
inasmuch as the infinitesimal transformation of the Gt
vanishes entirely when x2 + y2 — 1 is zero; and the
(?! does not transform at all the system of values of
x and y, which satisfy the equation
We shall, in future, exclude from consideration an
invariant equation which makes all the ?v ..., ?n iden-
identically zero.
Thus we may assume that one at least of the ?, ..., ?n
in Iff does not become zero by means of the equation
?2 = 0. Let us assume that ?n is not zero; then, by

52 ORDINARY DIFFERENTIAL EQUATIONS.
Art. 42, it is clear that if Q = 0 is invariant under the
infinitesimal transformation Uf, it will also be invariant
under the transformation
For, if J7(Q) is zero, either identically, or by means of
Q = 0, it is clear that 7@), which is U(Q) divided by ?„,
will also be zero, either identically or by means of Q = 0.
Now the linear partial differential equation of the first
order in n variables,
17=0,
has (и — 1) independent solutions which are functions of
xv ..., %n, and which we shall designate as
»i, Уг, •.¦,»«-!•
But if we consider xn in connection with these (n — 1)
independent functions, it is clear that the n functions
»i. Уг, ¦¦¦,Уп-ь х«
must also be independent. Otherwise we might express
xn as a function of y1 Уп-л, say in the form
But, Art. 20, the last equation means that xn must be
a solution of the linear partial equation Yf= 0 ; which is
manifestly impossible, since for f=xn this equation
reduces to 1=0.
Hence the n functions yv ..., yn-i, xn are independent,
and we may introduce them as n new independent
variables. By Art. 35, it will be easily seen that Yf then
assumes the form
It
dxn'
which is a mere translation.
Hence, we may remark, incidentally, that by a proper

INVARIANCE. 53
choice of variables, every infinitesimal transformation
may be brought to the form of a mere translation.
In the new variables the equation Q = 0 has the form
and xn can only occur formally in this equation. For if
xn be really present, we might solve and find xn in terms
of yv ..., yn-i, so that the invariant equation will have
the form
But for this equation to be invariant under Yf we
must have Y(F) zero, either identically or by means of
F = 0. Now
and hence we see that the variable xn cannot occur in
the function F.
If now we return to our original variables and desig-
designate the equation which is invariant under Iff by Q = 0,
it is clear that Q must he capable of being expressed as
a function of the (n — 1) independent solutions
of the linear partial differential equation Yf= 0, or of
its equivalent equation Uf=O.
This is a result of much importance for our subsequent
investigations.
For the special case of three variables, it follows that
to find the moat general equation which is invariant
under a given Qv
СГ/шЦх, у, *)%+ч(<°, 2/. *)%у+&х- У- *)%
it will be necessary to find two independent solutions of
the linear partial differential equation of the first order

54 ORDINARY DIFFERENTIAL EQUATIONS.
If these solutions be u(x, y, 2) and v(x, у, 2), the most
general invariant equation will have the form
F(it, «) = 0;
Or, written in a form solved for u,
u=f(v).
SECTION III.
The Lineal Element. The Extended Group of One
Parameter.
44. A lineal element is the aggregate of a point (x, y)
in the plane, and a direction through that point. If y'
represents the tangent of the angle which the direction
makes with the ж-axis, it is clear that -x, у, у' may be
regarded as the coordinates of the lineal element; and
by assigning to y\ which need not necessarily be con-
considered a differential coefficient, all possible numerical
values, we evidently obtain the x * lineal elements which
pass through the point (x, y).
An ordinary differential equation of the first order
in two variables, of the form
0(ж, у, У)=0,
may now be considered as an algebraic equation in the
three variables x, yt y', defining oo2 of the oo3 lineal
elements of the plane. The equation 0 = 0, as a differ-
differential equation, has oo1 integral curves; and the tangent
to an integral curve at any point (x, y) must be determined
by a value of у which satisfies the above equation. But
the same value of y' determines the lineal element
through the point (x, y); for when x and у are fixed, only
that value of у will satisfy Q = 0. Thus the x2 lineal
elements which are defined by the algebraic equation
in three variables, ?1 = 0, envelope the integral curves of

THE LINEAL ELEMENT.
55
the differential equation in two variables, ?2 = 0, as indi-
indicated in Fig. 1.
If the equation ii = 0 happens not to contain y' at all,
it still represents со2 lineal elements, although it can no
longer be considered a differential equation. These are
evidently the со2 lineal elements whose points He along
the curve ii = 0, as indicated in Fig. 2. Through each
point pass со1 lineal elements, since at that point x and у
are fixed, while у, being indeterminate, may have ao1
different values.
In the following, as we have only to do with differ-

56 ORDINARY DIFFERENTIAL EQUATIONS.
ential equations, we shall always consider that U actually
contains y'.
45. If a transformation be given by the equations
Vi = <t>(v>y)> У1 = Ф(х,у), A)
it is obvious that not only the points of the plane, but
also the со3 lineal elements are transformed by A)
according to a fixed law. For the value of the trans-
transformed y', which we shall call y\, and which determines
the direction of the transformed lineal element, is
determined by means of the equations,
Thus it is seen that the value of y\ depends merely
upon the transformation A) and the values assigned to
x, y, and y'. The transformation in the three variables
¦x, y, and ;/,
х1 = ф(х,у)) y^^x.y), У\ = ^, B)
which tells how the x3 lineal elements of the plane are
transformed, is called the extended transformation corre-
corresponding to the transformation A).
46. If a G1 be given by the equations
Ж1 = 0(ж. У, «X 2/1 = ^(ж> У' «). C)
each of its oo1 transformations may be extended in the
above manner. It is natural to expect that the oo1
transformations in the variables x, у, у' will themselves
form a (?!; and the proof that such is the case is very
simple. For let
be a second transformation of the 0г; and let the

THE LINEAL ELEMENT. 57
elimination of xx and y1 between C) and D) give a
transformation of the G1 of the form
хо_ = ф(х,у,Ъ), y2 = \J,(x,y,b), E)
b being a function of a and ax alone.
If each of the transformations C), D), and E) be
extended, it is easy to see that all the extended trans-
transformations form a Gv For C), when extended, becomes
х1 = ф(х,у,а), yi = i,(x,y,a), y\ = ^|j|' ^;.. .F)
and D) becomes
The successive performance of F) and G) upon the
lineal elements of the plane is equivalent to the per-
performance upon them of the transformation obtained by
eliminating xx, уг between F) and G). But by E), the
latter transformation must have the form
х2 = ф(х,у,Ь), уг = Мх,у,Ъ), г/'2 = аЦ^ (8)
where b is a function of a and аг alone. It is clear that
(8) is the transformation which would be obtained by
extending E); that is, the cc1 extended transformations,
corresponding to the G1 C), form themselves a Gv
47. It is also obvious that if a point transformation of
the form A) be given, not only will y1, but also y"t..., y<n\
be transformed by A) according to fixed laws.
The transformation in four variables,
is called the twice-extended transformation corresponding
to A). Each of the =ol transformations of the G2 A) may
be twice- extended in this manner; and it is very easy to
see that the x1 twice-extended transformations in the
four variables x, у, у', у" also form a Gv

58 ORDINARY DIFFERENTIAL EQUATIONS.
Similarly, it may be shown that the thrice-extended
transformations of a given G1 in the variables x, у, у, y",
y'" form a G-,, and so on to the «-times extended trans-
transformations 01 the Gv
48. We shall now give a method for finding the in-
Jinite&imal transformation of an extended Glt since the
conditions for the existence of such a transformation,
Art. 26, are obviously fulfilled.
If the finite transformations of the G± be given by the
equations
x1 = <t>(x,y,t), У1 = у№,у,Ь), (9)
and the infinitesimal transformation by
we know, Art. 29, that ? and у have the forms
where symbol 6 is equivalent to the symbol of differen-
differentiation. Thus, since
d
we have
„dy , Sdy i Sdx
St ~ St dxl
Since the order of the operations indicated by 6 and d
can be reversed,
a*.a%aya%
St
St dx2
or from A0),
8y'_di ffy dg_dt] , dg
St ~dx~dx dx~dx У 'dx
This is the increment assigned to y' by the transforma-

THE LINEAL ELEMENT. 59
tion (9); we shall usually indicate it by tf, so that the
infinitesimal transformation of the once-extended Gx has
the form
49. In an entirely analogous manner it may be shown
that the increment which y" receives under the trans-
transformation of the G1 (9) has the form
„ dr{ „ dg.
1 sdx~y Ясс'
and generally, the increment of y^ is
' dx y dx
Thus the infinitesimal transformation of the тг-times
extended G1 is
ё4Ф^
EXAMPLES.
In examples (l)-(9) below, it is required : (a) to find by Art. 30
the infinitesimal transformation, Uf, from the accompanying finite
transformation, t being the parameter of the Ог; (b) conversely, to
find the finite transformations of the Gp by Art. 36 or Art. 29,
considering the infinitesimal transformation as being given ; (c) to
find, by Art. 39, what points or lines, if any, are absolutely invariant
under each (У,; (d) to find, by Art. 37, an invariant of each G1;
(e) and, finally, by Arts. 29 and 38, to draw a figure representing
the path-curves along which the points of the plane are moved by
means of the transformations of each of the G1 respectively.
A) *•!=*
=i?.
This is the Gx of translations of all points of the plane, through
a distance t, in the direction of the ;r-axis.
The G-l of translations along the ^-axis.

60 ORDINARY DIFFERENTIAL EQUATIONS.
This is a Ог of so-called aff/ae transformations. The effect of
performing these transformations upon the points of the plane,
when t is positive, is equivalent to a stretching of the plane, as if it
were a homogeneous elastic plate, in the direction of the #-axis.
D) *,-*,„ = *; F/=Jj?+»|.
This is a O\ of so-called similitudinous transformations. All
coordinates are seen to Ъе increased, or diminished, from the origin
out, in the same ratio. Hence, a figure in the plane always remains
similar to itself under this G^.
By means of the finite transformations of this Gb all points are
moved along their radii vectores through the same distance t.
(в) *,-«,»,=!»! C9w!-y|.
G) xl=jccoBt-ysintfy1=xsint+ycoHt; Uf=-yJ^+x^
A0) Show that the family of all so2 conic sections whose axes
coincide with the coordinate axes,
is invariant under the Gl of affine transformations,
J?i = tt, уг=у.
Verify the result Ъу making use of the condition, Art. 40,
A1) Show that the family of oo1 concentric circles,
x2+y2 = r\
is invariant under the G1 of similitudinous transformations,
Xl = tx, yx = ty,
and verify the result by Art. 40.

EXAMPLES. 61
A2) Show that the family of «2 straight lines,
а о
is invariant under each of the Gt given in examples (l)-D)
and F)-(9) ; that is, that these G1 are protective. Verify the
results, as usual, by Art. 40.
A3) (a) Show that the family of «2 circles with radius 1,
(х-а? + {у-ЪУ = \,
is invariant under the &г of rotations given in example G).
(b) Show the same of the family of cc1 tangents to the circle,
[See Ex. A7), Chapter I.]
A4) Show that the family of w1 circles,
(x-af+f = l,
is invariant under the вг of translations,
A5) Show that the family of oo1 circles which touch both axes
of coordinates,
is invariant under the Glt x1 = tx, Ух=гу, verifying as
usual.

CHAPTER IV.
CONNECTION BETWEEN
EULER'S INTEGRATING FACTOR AND LIE'S
INFINITESIMAL TRANSFORMATION.
50. We are now prepared to show to what the develop-
developments of the preceding Chapters have been tending.
In the first section of this Chapter, we shall show how-
to integrate the exact differential equation of the first
order in two variables. In the second section, we
shall show that a differential equation of the first order
in two variables which is invariant under a known Gt
may always be integrated by a quadrature ; while in the
third section, we shall establish some of the most
important types of such invariant equations.
SECTION I.
Exact Equations of the First Order. Integrating
Factors.
51. A differential equation of the form
c№(x, 3,)=||<fa+|?<i2, = O, A)
since it is obtained by the complete differentiation of an
equation of the form
Ф(ж, у) = const,

EXACT EQUATIONS OF FIRST ORDER. 63
it is said to be an exact differential equation; and the first
member of A) is called a complete differential.
It is obvious that not every differential equation of the
first order,
X(x,y)dy-Y(x,y)dx=O B)
is exact; for, to be exact, it is necessary that the condition
ЭФ ЭФ
Ъу' У=~дх~
be fulfilled. But from this follows
ЭХ д?
C)
since each of these quantities must be an expression for
дудх
We shall see that this necessary condition that B) shall
be an exact equation is also sufficient. For the most
general function, Ф, which satisfies
is obtained from
the integration being performed as if у were a constant,
and Z being a function of у alone, which occupies the
place of the constant of integration. The only other
condition to be satisfied is that the partial differential of
Ф with respect to у shall be equal to X{x, y); that is,
X(x, j,)sl{-J7(x, y)dx+Z(y)}, D)
fy = X(x,y)+^\Y<,x,y)dx E)
Since Z is free of x, the second member of this identity

64 ORDINARY DIFFERENTIAL EQUATIONS.
must also be free of x; that is, its partial differential
with respect to x must be zero. Hence
Эж ду
which is exactly the condition C); that is, C) is a
necessary and a sufficient condition that the differential
equation B) shall be exact.
From the above it follows that the integral of the
exact equation B) may be found by quadrature in the
form
Ф
-\Y(X,
or, if more convenient, the equivalent formula,
Ф= -\x{x,y)dy+\(Y(X, y)+^X(-X^y)dy)dx = const.,
may be used. Here the integration with respect to у
is to be performed as if ж were a constant; and with
respect to x as if у were a constant.
It may be remarked that the equations of Chap. II.,
Sec. II., are a special class of exact equations.
Example 1. In the case of the differential equation
the condition C) is satisfied. For
Х~у*~Аху-2х\ 1'=-(з?
whence, as may be at once verified,
Ъх Ъу '
so that the differential equation is exact

EXACT EQUATIONS OF FIRST ORDER. 65
Using the first of the above formulae for Ф, we find
and hence
Ъ\Т(х,уLх
is the general integral sought.
It may Ъе readily verified that the use of the second formula for
Ф would lead to the same result.
We see that the condition C) is here satisfied. The second for-
formula for Ф may be used advantageously in this case. We have
thus
and the integral of the last expression with respect to x is therefore
x. Hence the general integral sought is
v3
— —+# = const,
(c = const.)

66 ORDINARY DIFFERENTIAL EQUATIONS.
52. It will usually not be the case that the functions
X and Y in B) satisfy the condition C). But since
every differential equation of the first order of the form
B) must have an integral of the general form
Q(x, j/) = const F)
the equation
«fe+d^O
must be equivalent to B). That is, there must always
exist a function M(x, y), such that we can write
^dx + ^dy = M(x, y)(Xdy-Ydx);
and since the left-hand member of this identity is a
complete differential, the right-hand member must be a
complete differential also. From C) we see that M, X,
and Y must satisfy the condition
911 W
дх ду " '
The factor M, which converts the equation B) into an
exact differential equation, is called, after its discoverer
Euler, an Euler's integrating factor of the differential
equation B).
Example. In the equation
the condition C) is not satisfied. Hence this equation ia not exact.
If the equation be multiplied, however, by
it will become exact; and the method of the preceding article gives
aa the general integral,

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 67
SECTION II
A Differential Equation of the First Order, which is
Invariant under a known Gv may be integrated
by a Quadrature.
53. Having seen in the last section that an exact
differential equation of the first order in two variables
may be integrated by a quadrature, and that the know-
knowledge of an integrating factor of a given differential
equation, which is not exact, enables us to put the
equation into an exact form, we shall show in this
section what it means for a differential equation of
the first order to be invariant under a given Ot; and
we shall see that such an invariant equation may be
integrated by a quadrature.
54. In order that an algebraic equation
w(x,y,y') = 0,
in the three variables x, у, у' may be invariant under a
given Gv in the same variables,
it is, by Art. 42, a necessary and sufficient condition
that the expression U'(o) shall be zero, either identically
or by means of w = 0. It was also shown, Art. 43, that
if и and v are two independent solutions of the linear
partial differential equation
the most general form of the invariant equation w = 0 is
Q(u,v) = 0, or u~F(v) = O.
55. If now y' be considered the differential coefficient
of у with respect to x, the equation
w(x,y,y') = 0
will be a differential equation of the first order; and if

68 ORDINARY DIFFERENTIAL EQUATIONS.
we consider U'f to be the once-extended 0г corresponding
to the (?! in two variables,
when the expression U'(w) is zero, either identically or
by means of the equation w~0,the differential equation
of the first order, w = 0,is said to be invariant under,
or to admit of, the Qv
Also we see that to find the most general differential
equation of the first order which shall be invariant
under a given Gv Vf, it is necessary to find two
independent solutions of the linear partial differential
equation of the first order,
that is to say, we must find two independent integral-
functions of the simultaneous system
One of theae integral-functions may be found from the
equation
dx _dy,
i n
and since ? and ц are free of y', this integral-function,
which we shall call u, will not contain y'. The second
integral-function, which we shall denote by v, and for
finding which one method has been indicated, Chap. II.,
Sec. 2, must contain y'. The most general invariant
differential equation will then have the form
v-F(u) = 0.
56. To find the integral-function и of the preceding

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 69
article, it is theoretically necessary to integrate a differ-
differential equation of the first order, namely,
But from the form of this equation, we know, Art. 38,
that u— const, must represent the path-curves of the Gv
Hence, if the path-curves of the Gv Uf, are known, of
course u is also known; and it will be remembered,
Chap. III., Examples, that the path-curves of a large
number of the most important G^'s in the plane can be
found by integrating differential equations of the first
order which are exact. Thus, in a large number of
the most important cases, и can be found by a quad-
quadrature.
We propose to show now that if и has been found, v can be found
by a quadrature.
We have already seen, Art. 43, tbat every infinitesimal transfor-
transformation in n variables can be brought, by a proper choice of variables,
to the form of a mere translation. If и be known, we shall first
show that in this case
can be brought to the form of a mere translation by a quadrature.
Let ua introduce into Uf the new variables хъ уг; and
demand that Uf assume the form of a translation. Thus Uf,
Art. 35, becomes
In order that UJ shall bave the form of the translation ?~
in the new variables, it is necessary to have
That is to say, xl must be a solution of the partial equation Uf=O ;

70 ORDINARY DIFFERENTIAL EQUATIONS.
and since и is a solution of this equation, being by hypothesia the
integral-function of the ordinary differential equation
v
we may assume xt =w.
Now yx must be a function of x and y, which satisfies the
equation
and we may assume that г/и x, and у are connected by an equation
of the general form
U{x, у,уг) = const.
By differentiating this equation with respect to x and with respect
to у successively, we find
за за Ъу,
oy oy1 ay
Multiplying the first equation by ? and the second by 77, aud
adding, we obtain
or, on account of the differential equation connecting x, y, and y1}
.an ъада .
But, by Art. 18, this linear partial differential equation is
equivalent to the simultaneous system
dx _dy _dyx
that is to say, y^ may be found as a function of x and у by in-
integrating this simultaneous system in the three variables x, y, yv
But we already know one integral-function of the system, namely,
xi or u. Hence it is obvious thatyj may be found by a quadrature ;
for we only need to eliminate, Art. 23, say x out of the equation
by means of w = const., when we have an ordinary differential
equation between у and уъ in which the variables are separate.

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 7l
Thus, by a quadrature, we have found the new variables which
make Uf take the form of a mere translation, ^-. But the
differential equations which are invariant under this translation
are easily found. The extended Gt in the variables xu yv evidently
haa the form
since. Art. 48, „', =A - ¦»', ^ = 0,
'1 dxy У1 dxl '
where ?j, t)b Tj'v and y, have the usual meaning. Hence to find
the invariant equations, we must find two integral functions of the
simultaneous system
dx1_dyi_d?/i
0 1 0 '
since ?j and г}\ are zero. But Xj and y\ are evidently two
independent integral-functions of this system. Hence the general
invariant differential equation in the variables xv yx will have the
form
n(*i,yi)=u
If, now, we return to our former variables, this equation must
take the form of a function of и and v equated to zero, say
F(«, v) -a
But since #, is identical with w, y\ must be a function of v; and
we can obviously assume y'l = v.
Hence when the path-curves, w=const, of a given Ox are
known, the most general differential equation of the first order
which is invariant under the given Gt may be found by
quadratures. Practically the calculations may usually be made
much shorter than indicated above, since in the most important
cases the variables in the simultaneous system to be integrated,
cto_cty\_dtf_
l~v~v'*
may be separated by inspection.
57. In Art. 37 the function u, which is a solution of
the linear partial differential equation
was called an invariant of the Ov Uf. Similarly, the

72 ORDINARY DIFFERENTIAL EQUATIONS.
function v, which we saw must always contain y', and
which is a solution of
4%
is called a differential invariant of the first order of
the Glt Uf
58. If x, y, and y' be considered the coordinates of a
lineal element in the plane, the equation
«>{x,y,y')=0 A)
represents, Art. 44, со2 of the со3 lineal elements; and to
demand that the equation w~0 shall be invariant under
the Gv U'f, is the same as to demand that the family
of со2 lineal elements shall, as a whole, be invariant
under U'f. For, the analytical criterion that A) shall
be invariant, means, interpreted geometrically, that the
transformed A) shall represent the same family of со2
lineal elements that A) itself does. But these со2 lineal
elements envelop the go1 integral curves of A), con-
considering this equation as an ordinary differential equation
of the first order; and since the family of lineal elements
is invariant, the family of со1 integral curves must also
be invariant under the Gv U'f.
Thus, if
Ф(ж, i/) = const B)
represent these integral curves, since B), which does not
contain y' at all, must be invariant under the extended
Gv U'f, this equation must also be invariant under the
Gv Uf; that is, by Art. 40, a condition of the form
C)
must hold, if the differential equation A) is invariant
under U'f.
Conversely, if a condition of the form C) holds, of
course the со1 integral curves B) are invariant—and
with them, the family of со2 lineal elements A)—or, as

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 73
we may say, the differential equation of the first order
A) is invariant under U'f, If, therefore, a (?j is known,
of which the integral curves of a given differential
equation of the first order admit, this equation, written
in the form A), always admits of the extended Gv
Hence, we may also define an ordinary differential
equation of the first order as being invariant under a
given Ov Uf, when an integral-function Ф of that
equation is transformed by means of Uf into a function
which is itself an integral-function of the differential
equation; that is, when a relation of the form C) exists.
59. We shall now show that a differential equation of
the first order in two variables, which is invariant under
a known Qv may be integrated by a quadrature.
Let the given differential equation be
п(х,у,у') = 0; D)
and suppose D) to admit of the Gv
We shall, for reasons explained in Art. 60, assume that
?3 = 0 is not the differential equation of the x1 path-
curves of the Ov Uf.
If D) be written in the solved form
Y(x,y)dx=Q, F)
and if its integral-function be designated by m(x, y), by
Art. 16, w must be a solution of the linear partial differ-
differential equation of the first order,
Moreover, since the family of integral curves w = const.
is invariant, it follows from Arts. 40 and 58 that
Я{<*)*(^ + ЧЦ = УГ(*(х,у)), (8)

74 ORDINARY DIFFERENTIAL EQUATIONS.
Now if Ф(ш) be a certain function of w alone, Ф will
also be an integral-function of F), and ?7(Ф) wu^ depend
upon Ф alone. For
and to may be removed from the right-hand member of
the last identity by means of
giving thus ?7(Ф) as a function of Ф alone.
Since we assumed above that the curves w = c were
not the always invariant path-curves of the Gx, Iff, the
function W (<o) in (8) cannot be zero; and we may easily
choose Ф as such a function of w that ?7"(Ф) = 1. For it
is only necessary to determine Ф so that
Since Ф = const, represents the same family of curves
that w = const, does, let us suppose w so chosen from the
beginning that U(w) = 1; that is, let us now designate
by a) the function which we have just called Ф. Then
we have
and
.... , ию, uw , Xdy — Ydx
that is, й— л- -1- -Л/ =
Since the first member of the last equation is necessarily

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 75
a complete differential, the same must be true of the
second member; that is, we have the
Theorem.* // a given differential equation of the
first order in two variables
Xdy-Ydx = O
admits of a known Gv
whose path-curves are not identical with the integral
curves of the differential equation, then
is an integrating factor of the differential equation:
and the general integral may be found by a quadrature
in the form
CXdy-Ydx ,
i— ./—^^ = const.
*This theorem was first published by Lie in the " Verhandlungen
der Gesellschaft der Wissenschaften zu Christiania," November,
1874.
By Art. 52 the equation
Xdy-Ydx=Q
always possesses an integrating factor, Af; and if M be known, it
follows from the developments in the text that it is only necessary
to choose ? or i\ in
in such manner that
when the given differential equation will be invariant under Uf.
Although it follows from this that every differential equation ol
the first order is invariant under an unlimited number of (?Д when
we speak of an invariant differential equation in this book, we
shall always mean one which is invariant under a known Gv

76 ORDINARY DIFFERENTIAL EQUATIONS.
Example 1. The differential equation
U = xy'-y+xs=O
admits of the &',, Uf = xJ-. For the extended transformation is
w J Ъу
found by forming, Art. 48,
* sdL-« ш
which in this case, since tj=x, ? = 0, is 1. Hence
By Art. 55, the expression U'(u) must be zero, either identically,
or by means of i2 = 0. We find
Hence the condition that 12 = 0 shall admit of Uf is satisfied. Now
write 12=0 in the solved form,
since this equation admits of x^~, the integrating factor
has in this case the value,
1
and the integral is found to be,
fxdy -{y — x2)dx
or, by Art. 51,
<i)=¦ ~= const.
X
We may at once verify that (i>=const. admits of the infinitesimal
transformation of the Gu x~ ; as well as-of the finite transforma-

DIFFERENTIAL EQUATIONS OF FIRST ORDER, 77
Example 2. The differential equation
п=*уУ-(дчУ)=о
admits of the already extended G1
For here П'(Щ has the form,
U'(u)
and the condition that 12 = 0 shall admit of Uf is satisfied. Now
write the differential equation in the form,
since it admits of
an integrating faetor must be given by
(xydy-{sc-ir-y%)dx _
Hence the integral is
¦'-—=const.,
or, Art. 51, ""У" conat
60. The method of integration of Art. 59 fails when
X4— FfsO.
For this case, we see
Y 4'
and since the first of these ratios gives the direction of
the tangent to the integral curves of О = 0 through the
point (x, y), and the second ratio gives the direction in
which the point (x, y) is moved by means of the Qv Uf,

78 ORDINARY DIFFERENTIAL EQUATIONS.
the above identity states that the point (x, y) always
moves on one of the integral curves of ?2 = 0. Hence
the invariant family of oo1 curves is none other than
the family of oo1 always invariant path-curves of the
Gj—each curve being separately invariant. In other
words, the Gv Uf tells us nothing new with regard to
the equation 12 = 0, and hence Uf is, in this case, said to
be trivial with respect to that differential equation. In
Art. 59 the case that Uf shall be trivial is always
excluded.
When Uf is trivial, since
Y_r,
we may write
(=p(x,y)X, ч = р(х,у)?;
so that Uf has the form
Thus it is seen that every transformation of the form
is trivial, with respect to the ordinary differential
equation
Xdy-Ydx = 0.
In future, we shall always disregard trivial infinitesimal
transformations.
SECTION III
Glasses of Differential Equations of the First Order
which admit of a given Gj in Two Variables.
61. Having shown that an ordinary differential equa-
equation of the first order in two variables can be integrated
by a quadrature when it admits of a known Gv the next

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 79
step will be to find the classes of differential equations
of the first order which admit of certain of the simpler
Gt in two variables.
From Art. 58 it is clearly immaterial whether we say
that the family of oo1 integral curves of the given
differential equation is invariant under a Gv Uf, or
whether we say that the differential equation itself is
invariant under the Glt Uf, or under the equivalent once-
extended Gv U'f
62. To find all differential equations of the first order
which admit o/ a translation along the x-axis.
This translation is represented, Example 1, Chapter III.,
b
We see at once, that since ? = 1 and r\ = 0,
that is, 0>/_?
To find the most general invariant differential equation,
we must, Art. 55, find two independent integral-functions
of the simultaneous system
dx dy dy'
It is evident that у and y' may be chosen as the functions
designated as и and v in Art. 55; and hence the most
general differential equation of the first order which
admits of a translation along the ж-axis has the form
or, if solved in terms of y',

80 ORDINARY DIFFERENTIAL EQUATIONS.
In this equation the variables are separate, so that the
integration may be accomplished by a quadrature.
Analogously, it is obvious that all differential equations
of the first order which admit of the Q1 of translations
along the y-axis
1 ъу
have the form
and are immediately integrable by quadrature.
63. To find all differential equations of the first order
which admit of the Ог of afftne transformations
Here, since ? = x and t\ = 0,
Hence the extended G^ is
and the simultaneous system to be integrated is
dx_dy _ dy'
One integral-function is evidently у; and, from
a second is found to be xy.
Hence the most general invariant differential equation
of the first order has the form
Q(xy',y) = 0;

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 81
Here again the variables are separate, so that the
equation may at once be integrated by a quadrature.
The general form of the differential equations which
are invariant under the corresponding G^ of affine trans-
transformations along the y-axis,
is readily seen to be
In this equation also the variables may be separated by
inspection.
64. To JiTid all differential equations of the first order
which admit of the Glf
J Ъх ^Ъу
Here, since ? = х and r\ = y, we find in the usual
manner r{ = 0. Hence the simultaneous system to be
integrated is
dx dy dy'
The integral-functions of this system, usually designated
by и and v, are obviously
Hence, the most general differential equation of the first
order which is invariant under the (rx
has the form

82 ORDINARY DIFFERENTIAL EQUATIONS.
or, when solved in terms of y',
This is the so-called general homogeneous equation of
the first order.
We may write the above equation in the form
and the method of Art. 59, gives
1
M=-
as an integrating factor. Hence the equation written in
the form
= 0
* w
is exact, and may be integrated, by the method of Art. 51,
by a quadrature.
Example. Given
Cx"-y + 2y3)*/ + xMx = 0.
This equation, being homogeneous, belongs to the class of the
present article.
Written in the form
it becomes, dy-\-—-^—-~ъ<1х=:0 ;
so that — У-—j- = 0,

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 83
must be an exact equation. It may easily be verified that such is
the case ; and the general integral is found by Art. 51 to be
7-=; = const.
65. It should be noticed that equations of the general
form
{ax-\-by + c)dx — (a'x+b'y + c')dy = 0, (a, ...,& = const.)
may usually be made homogeneous by a proper choice of
variables. For, let the new variables be
x=x — h, y = y — k, (h, /c = const.)
then the given equation becomes
If, now, h and к are determined from the equations
ah+bk+c =0,
a'h+b'k+c'^O,
the above equation in x and у will evidently become
homogeneous, and thus may be integrated by the
method of the preceding article.
This method fails when a:a' = h:h'. Let us assume
then
a = n ¦ a', h = n .b', (n = const.)
and the original equation becomes
Now introduce in place of у the new variable
z = ax-\-by,
and it is readily seen that the differential equation takes
the form
dz , z4-c
= a + b - „
dx nz+c
in which the variables may be separated by inspection.

4 ORDINARY DIFFERENTIAL EQUATIONS.
Example. Given Bt/-x-l)dt/+Bx-y+l)clx=0.
Here the equations ah + bk+ с =0,
have the forma ЗА-?+1 = 0,
-A + 2*-l = 0;
so that A= -J, i-J.
Introducing the new variables
the given differential equation becomes
The general integral of this homogeneous equation is found
to be
x2 — ,vy +y2=const.;
and if we now return to the original variables, the general integral
of the given differential equation is seen to be
X2 - хуЛ-у^Л-х - у = const.
66. To find all differential equations of the first
order which admit of the Gl of rotations.
A rotation around the origin is given, as will be
remembered, by the Gl3
m= -УЪ1+ХЖ
1 ~ УЗй:+ Эу
Неге ?=—у, ч = х\ and hence
It is necessary, therefore, to find two integral-functions
of the simultaneous system
dx =dy^ dy'
-У x 1+y'*
From the first equation, which may be written,

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 85
we see that one integral function is
By the method of Art. 23, we now write
х2+уг = с2, (c2 = const.)
whence x = s/ci — yi,
, dy dy' n
The variables are separate: hence by immediate in-
integration
sin ~1 - — tan '1y' = b; (b = const.)
or, sin'1 ,У— 9-ta1n~1y' = b;
But this may be written
tan- —tan-1^^?»;
or, taking the tangent of both sides, the second integral
is found to be
xy' — у
v = ~—-, = const
x+yy
Thus the most general invariant diiFerential equation of
the first order has the form
This equation may be written—when F is put for
(x-y?)dy-(y+x?)dx = O.
The method of Art. 59 gives, as an integrating factor
of all equations of this form,

86 ORDINARY DIFFERENTIAL EQUATIONS.
so that the above equation, written in the form
is exact, and may be integrated, by Art. 51, by a quad-
quadrature.
67. To find all differential equations of the first order
which admit of the Ox
тух Э/ Vf
Here rj' will be found to have the value — %J; so that
the simultaneous system to be integrated has the form
dx_dy__ dy'
Since the variables are here separate, it is seen at once
that two independent integral-functions are
v'
Уг
We may write the second integral-function in the form
xy.y'
and since xy is itself an integral-function, we see that —
must also be an integral-function. Thus we may assume
xy'
usxy, «3-2-;
so that the most general invariant differential equation
of the first order will have the form
y'.'^-?(xy) = 0.

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 87
We may assume F =¦} ~-t and write the last equation
symmetrically
fx{xy) . xdy~f2(xy). ydx= 0.
Of course all equations of this form may be integrated
by a quadrature; since the method of Art. 59 gives as
an integrating factor,
1
~ xy(fx{xy)+f2(xy))
Example, Given
This equation may be written
so that it is seen to belong to the class of the present article.
Hence an integrating factor is
so that the equation
is exact. The general integral is found in the usual way to be
low- =const.
V xy
68. To find all differential equations of the, first
order ivhich admit of the Gx
Proceeding as usual, we find for ц the value
so that the simultaneous system to be integrated has
the form
dx _ dy _ dy'

88 ORDINARY DIFFERENTIAL EQUATIONS.
One integral-function is evidently
u = x.
A second may be obtained from
т dy'
or dy = Y-Z-
Since x = const, is one integral of the simultaneous
system, ф(х) plays the r61e of a mere constant in the
last equation. Hence, by a quadrature, a second integral-
function is found to be
The most general invariant differential equation has,
therefore, the form
This is the so-caUed general linear differential equation
of the first order. Of course all equations of this form
may be integrated by a quadrature. For the above
equation may be written
and by Art. 59,
is an integrating factor, so that

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 89
must be an exact equation. By Art. 51 the general
integral is found in the form
In an analogous manner it may be shown that the
differential equations of the first order which are in-
invariant under
have the general form
This general equation, which, of course, may be in-
integrated by a quadrature by the usual method, is said to
be linear in x, у being chosen as the independent
variable.
Example. Given
In this linear equation the functions ф and ir have the form
_ x _ 1
хт
H
ence
so that -ф-(х)е ^
and
=(i+^)*'
f dx
Making these substitutions in the formula for the general
integral of the linear equation, we find
h>}' (c=co"at)
aa the general integral required.

90 ORDINARY DIFFERENTIAL EQUATIONS.
69. It should be noticed that equations of the form
may be easily brought to the linear form. For, dividing
by yn the equation becomes
y-n .y'~(j>(xy/-"-ijr(x) = 0.
If now we put z = 2/1"n,
,, , dy 1 dz
so that у".-/ = - -j-,
ax 1 — ndx
the above equation becomes
1-й dx r ' ry '
an equation which is linear in z.
Example. Given
The equation may be written
Assuming, now, y~x=z, we have
„ dy dz
J dx dx
so that the given equation takes the form
The general integral of this linear equation is found, Art. 68,
to be
2 = e# + log.r+l ; (c = const.)
so that the general integral of the given equation ia
y-1 = or 4- log x +1,

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 91
70. The types of differential equations which admit of
given groups of one parameter, like those which have
been discussed, Art. 62-68, may be multiplied indefinitely;
and in this fact may be recognized a part of the extra-
extraordinary fruitfulness of the Theory of Transformation
Groups as a foundation for the Theory of Differential
Equations. As we know, in order to find the differential
equations of the first order which are invariant under a Gx
it is only necessary to find the value of r( from the
equation
, dr] , d?
n ~dx У dec'
and then find two independent integral-functions of
the simultaneous system
? n if
Moreover, we have shown, Art. 56, that if an integral-
function и of the ordinary differential equation in two
variables
be known, the second integral-function v can always
be found by a quadrature.
Practically, therefore, it is only necessary to choose
? and rj so that the ordinary equation
dx dy
7=T
will have an integrable form, either in being an exact
equation or in assuming one of the forms discussed,
Arts. 62-69, when all differential equations of the first
order which are invariant under the given Gv> and which
are therefore immediately integrable, may be found by

92 ORDINARY DIFFERENTIAL EQUATIONS.
quadratures. For instance, if i=xfj(xy) and
the above differential equation has the form
f-^xy). xdy -f2(xy). ydx = 0,
which, by Art. 07, is integrable by a quadrature. This
gives us и; and, by Art. 56, v may be found by another
quadrature, so that all differential equations of the first
order which are invariant under the Ог
may be found by two quadratures.
This method will, in general, give rise to a new class
of integrable differential equations of the first order
in two variables. If desired, ? and n might now be so
chosen that the equation
will belong to this new class. Then, of course, two
quadratures will, in general, give us another new class of
integrable equations, etc.
71. It will be remembered that, Art. 55, the condition
that an ordinary differential equation of the first order
in two variables,
П(х,у,у') = 0,
shall admit of a Gv
J ьЭж 'ду
is that the expression
shall be zero, either identically, or by means of fi —0.
Here of course, U'f is put for the once-extended G±

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 93
From this condition it is often possible to find the Qx
of which a given differential equation of the first order
admits, especially whenever the form of the equation
suggests the Ov
For example, the equation
is homogeneous in all of its terms except one. Since, by Art. 64,
all homogeneous equations of the first order admit of the Gx
we are led to suspect that the above equation will admit of a GY
of the form
Uf=axZ+b*% (а, Ь соты.)
The corresponding extended Gx is
and the condition that the given equation ii = O shall be invariant
under this G, is that P'(ii) sball be zero identically, or by means
ofS2=0. That is,
66^2 + 3F — «)у3 - 4 (Ь — в)
must be zero identically, or by means of S}=0.
Comparing this expression with the expression
we must attempt to choose b and a in such manner aa to make
the first expression equivalent to the second multiplied by a
constant factor. It is only thus, in this case, that the condition
of invariance can be satisfied, since the first expression cannot be
zero identically unless a=b = O, in which case Uf would vanish.
If a represent any constant we see that we must have, in order
that

94 ORDINARY DIFFERENTIAL EQUATIONS.
the identities
166 = 8a,
4a + 46 + 4F_a) = 4a,
3F-a) = a.
These equations are not contradictory, and may evidently be
satisfied by assuming a = 2; 6=1; a = J. Hence the given differ-
differential equation admits of the Gl ;
If the equations for determining a and 6 had proved to be
contradictory, it would, of course, have meant that the given
differential equation did not admit of a G1 of the form
In a manner similar to the above, it may be readily seen that
the differential equation
admits of Uf=
when a = l, Ъ = -2, aa we should be led to expect from Art. 67.
That is, the above equation admits of
72. The types of integrable differential equations established,
Arts. 51-69 illustrate the fact, mentioned Art. 17, that every
ordinary differential equation of the first order has one general
integral. The rigid proof of this proposition is, as already stated,
Art. 17, a theorem pertaining to tne Theory of Functions; but a
further illustration is afforded by the fact that when the variables
x and у are connected by an equation of the form
Xdy-Ydx=O, A)
we may always, by means of an infinite series, express у as a
function of x and one arbitrary constant. We shall not investigate
the question as to whether this series always converges or not.
From A) we have
we may more conveniently write it,
У=/1(*.У) B)

DIFFERENTIAL EQUATIONS OF FIRST ORDER, 95
By differentiating B) we find
Let us, for brevity, write this
/'=«*, У) C)
By successive differentiations of C) we find, analogously,
<Г=/з(*, У),
Now let ф(х) be the general value of у; and when we assign
a particular numerical value л to x, let the corresponding value
of у be designated by у$. Here #„ and y0 are called the initial values
of x and y. Then, by Taylor's Theorem, we have
But ф(ха) is y0 ; ф'(х0) is the value of y' when x=x0 ¦ ф"(^) is the
value of y" when x = xa, etc. Hence, by B), C), etc.,
and this is an expression for the general integral of A).
Since x0 is a particular numerical value of x, it is seen that the
general integral E) contains only one arbitrary constant, #„.
In Chapter X. we shall see that the general integral of a differ-
differential equation of the mlb order may be similarly expressed by an
infinite series.
73. In the following examples of differential equations
of the first order to be integrated, the test for an exact
differential equation should first be applied. It will be
remembered that the equation
Xdy-Ydx = 0
is exact, if

96 ORDINARY DIFFERENTIAL EQUATIONS.
and the integral may be found by a quadrature in the
form
f Ydx - \{X + M Ydxj dy = const.
If the given differential equation is not exact, but
belongs to one of the types of Arts. 62-68, it may be
integrated, as already seen, by a quadrature.
In case the given differential equation does not belong
to one of the types established, Arts. 62-68, the method
of Art. 71 should be employed to find the Gl of which
the equation admits.
We give below a table of types of the most important
of the simpler G^ in the plane, with the corresponding
type of invariant differential equation. The reader will
do well to re-establish for himself those types given
below which were not established, Arts. 62-68.
Group of One Parameter. Type of Invariant Differential Equation.
A) ty=|j- A) У = F(y>
B) СУ=°р B) y = F(.c).
ЮиНш~1% C)/-F(a,+W
It is seen that C) includes types A) and B).
' ' -* ~ХЪх \ ) У — x
<5> ufsx4x^%- <6>y=F(!)-
Equatious of the form (o,'x-\-b'y-\-<?)dy — ^
usually be brought to the homogeneous form. See Ait. 65.
(в) и/- -,?+*?
G) ЩтхУ-уЩ
(8) tysj*<*№-|'. (8) •-*(*),-

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 97
The form у'-ф(х).у-у1/(х)у>>=0 may be reduced to this one.
See Art. 69.
A0) G/=2j
EXAMPLES.
A) (у1-**
xdx+yd^ ydx-xdy = 0
¦Jl+x>+y* я>+у*
(8) (%-fxy-x)dy+ydx=O.
(9) xdy-(y+4^+f)dx = 0.
A0) (ar+y)dy-(y-a:)<fe = O.
A1) л:соа-^г/—lycos-—^Jrfj-=O.
A2) (?jj, +
A3) .crf3,-
A4) (агу!
A5) (^-
A6) By-*-
A7) Gу-
A8) (x
A9) (x
B0) (

98 ORDINARY DIFFERENTIAL EQUATIONS.
B1) (x
() f)yfy
B3) (n+yJxy)dx+(x+x-fxy)dy=O.
B4) xy{l + <xHxy)}(xdy +ydx) + xdy-ydx=O.
B6) xdy-(ay + x+l)dx=O.
B6) A -x'fdy+yjl -x4x=(x+Vl -;c
B7) yfeify
B8) dy+
C0) ^*/ + (^->/!loga;)<fa=
C1) Zxydy+(x-y')dx=0.
C3) cosa;%
C4) (l-^Jt
C6) dy-2xt
C7) c%y(
C8) ydy + d/*-coBx)dx=O.
C9) жйу-^ Л
The following geometrical examples lead to ordinary differential
equations of the first order, which may be readily integrated by
Bome of the foregoing methods.
For convenience of reference hereafter, it may be noted that
for a plane curve referred to rectangular coordinates,
Subtangent s —, ; Subnormal =yy';
Length of the perpendicular from origin upon tangent = У- -^**-f ;
Length of the perpendicular from origin upon normal = ™ ;
Intercept of tangent upon х-юлъ=у-ху';
Intercept of tangent upon y-axis = x — —.;
У
Distance from the origin to the foot of the normal =x+yy'.
D0) Find the curve whose subtangent is proportional to the
absciesa (k times the abscissa) of the point of contact.
D1) Find the curve whose Bubtangent is constant, and equal to a.

DIFFERENTIAL EQUATIONS OF FIRST ORDER. 99
D2) Find the curve whose subnormal is constant, and equal to 2a.
D3) Find the curve in which the angle between the radius vector
and the tangent is one-half the vertical angle.
D4) Find the curve in which the subnormal is proportional to the
пл power of the abscissa.
D5) Find the curve in which the perpendicular from the origin
upon the tangent is equal to the abscissa,
D6) Find a curve such that the area included between the curve,
the axis of x, and an ordinate, is proportional to the
ordinate.
D7) Find the curve in which the subtangent is the arithmetical
mean between the abscissa and the ordinate.
D8) Find the curve in which the intercept of the normal upon the
.F-axis is proportional to the radius vector.
D9) Find the curve in which the intercept of the tangent upon the
y-axis is proportional to the radius vector.
E0) Find the curve in which the subtangent is equal to
E1) Find all differential equations of the first order which are
invariant under the GY
l
<52) Find all differential equations of the first order which are
invariant under the G,
4

CHAPTER V.
GEOMETRICAL APPLICATIONS OF THE INTEGRATING
FACTOR ORTHOGONAL TRAJECTORIES, AND
ISOTHERMAL SYSTEMS.
74. In this chapter we propose to give some of the
simpler geometrical applications of the integrating factor,
Art. 59, of a differential equation of the first order in
two variables.
75. In Art. 59 it was shown that if a differential
equation of the form
Xdy-rdx = 0 A)
admits of the Gv which is not trivial,
then U=X^Y?
is an integrating factor of A).
Suppose now that u>(x, i/)=const. represents the xl
integral curves of A); then by means of JJf each curve
(»> = c passes over into the position of the adjoining curve
w = c + Sc. At the same time every point of general
position (x, y) passes through an infinitesimal distance^
Art. 29, iJp + rfSt, of which the projections upon the
axes of coordinates are (St and 4M.

THE INTEGRATING FACTOR.
101
Now draw the tangent to the curve w = c at the
point {x, y); and lay off upon this tangent the distance
s/X2-\- Y2, of which the projections upon the axes are
X and Y respectively. The two distances */?'2+tJSt and
%/X2-\- Y2 determine a parallelogram of which the area,
by a proposition of Analytical Geometry, is
(X,- Y()lt,
But this parallelogram, if we neglect infinitesimals of
an order higher than the first, is equal in area to the
rectangle constructed upon the base s/X2+ Y2 with the
altitude Ss,—Ss being the distance from the curve w = с
to the curve w — с + Sc, measured at the point x, y. Hence
we have
J
st
Hence we see that if M is an integrating factor of a
given differential equation A), M is inversely pro-
proportional to the area of the rectangle, one side of which
is the perpendicular distance, measured at a point of
general position (x, y^, between the integral curve through
that point and the integral curve of the family at an

102 ORDINARY DIFFERENTIAL EQUATIONS.
infinitesimal distance from that one; while the other
side of the. rectangle is the distance %/X2 + Y2, measured
off upon the tangent to the curve through the point (x, y)r
and from that point.
76. Let us apply the above result to a simple example.
If equal distances, of length n, are laid off upon all
the normals of a given curve
the end points of the normals form a new curve. If,
now, n varies, we find a family of x1 curves which are
called the parallel curves of the curve \Js = 0. The
differential equation
Xdy~7dx=0
may always be integrated by a quadrature, if its integral
curves are a family of parallel curves. For in this case
the perpendicular distance between two adjoining integral
curves is constant: so that, by Art. 75,
must be an integrating factor.
Hence, if it is known that the differential equation
A) represents a family of x1 parallel curves,
an integrating factor of(l).
Conversely, it is easy to see that if
be an integrating factor of A), the distance between two
adjoining curves must be constant, and the curves are
parallel.

ORTHOGONAL TRAJECTORIES. ЮЗ
The oo1 involutes of a given curve form a family of
parallel curves; and hence their differential equation may
always be integrated by a quadrature.
For example the involutes of the parabola
are represented, as is shown in the Differential Calculus,
by the equation
2(# + Jx2-y)dy + dx = 0.
Hence M = - , т=
*/4,(x + Jx2-yf+l
must be an integrating factor of the above equation, as
may be at once verified.
77. An orthogonal trajectory is a curve which inter-
intersects at right angles each member of a given family
of x1 curves.
A family of x1 curves, represented by a differential
equation
jy-F=0 A)
will evidently have oo1 orthogonal trajectories; and
their differential equation is readily obtained from A).
For, at any point of general position (x, y), the integral
curve of A) through that point is perpendicular to the
orthogonal trajectory through the point; hence, if y' be
the tangential direction of the integral curve, , must
be that of the orthogonal trajectory. Thus if we
substitute in A) —; for «', we obtain the differential
У "
equation of the x1 orthogonal trajectories of the integral
curves of A), in the form
X+Yy' = 0 B)
Reciprocally, the integral curves of A) are the or-
orthogonal trajectories of B).

104 ORDINARY DIFFERENTIAL EQUATIONS.
Example. It is required to find the orthogonal trajectories of
the hyperbolas
scy = a. {a = parameter)
The differential equation of these со1 curves is obviously
xdy-\-ydx=Q,
or x/+.y=0.
Writing ——in place of y\ we find as the differential equation
of the orthogonal trajectories
' 0
The variables are here separate, so that we find at once the
integral curves
хг-уг=с\ («^parameter)
which is also a family of hyperbolas.
78. A family of x1 curves in the plane is said to be
isothermal when, together with their orthogonal trajec-
trajectories, they form a network of infinitesimal squares.
If Xdy-Ydx = O A)
represent a family of isothermal curves, their orthogonal
trajectories will, Art. 77, of course, be represented by
Xdx+Ydy = 0, B)

ISOTHERMAL SYSTEMS. 105
and both of the equations A) and B) may be integrated
by quadratures by means of the geometrical interpre-
interpretation of the integrating factor, Art. 75.
For, if we consider any two adjoining curves of each
of the families A) and B) which form a small square at
the point (ж, у), the breadth, Ss, of the strips enclosed by
both pairs of curves is the same, since Ss is the side of
the infinitesimal square.
XT
Hence
is an integrating factor of both B) and A).
But if two ordinary differential equations of the forms
B) and A) have a common integrating factor, this factor
may be determined by a quadrature. For, if M be the
common integrating factor, by Art. 52, M must satisfy
the equations
ЪМХ -dMY
+ by =
From the last two equations —58— and ~^§~ may
be determined as functions of X and у; and, if these
quantities satisfy the condition of integrability, Art. 51,
Э SlogJtf _ Э Blogif
by' at ~ъх ъу~' ('
we may find log M, or M itself, by a quadrature from the
exact equation
SlogJl/

106 ORDINARY DIFFERENTIAL EQUATIONS.
If the above condition of integrability were not
satisfied, A) and B) would have no common integrating
factor; and hence the families of integral curves would
not be isothermal. Thus, that C) shall give such values
siiw aiiw
tor —^— and —~— as satisty D) is a necessary
condition that A) shall represent an isothermal family.
This condition is also sufficient; for if D) is satisfied,
A) and B) have a common integrating factor—that is,
the quantity designated as Ss above must be the same
for both families of integral curves, and these integral
curves form a net-work of small squares, or are iso-
isothermal.
If A) represents an isothermal family of curves, there-
therefore, the Common integrating factor if, of A) and B),
may be found by a quadrature; and the equations A)
and B) may be integrated by another quadrature each.
Example 1. The differential equation
t-x')dx=0, F)
represents an isothermal family of curves. For the orthogonal
trajectories are represented by
so that the equations C) have the furm
and it may be immediately verified that the condition D) is
satisfied. Thus E) represents an isothermal family ; and the
integrating factor of E) and F) is obtained from

ISOTHERMAL SYSTEMS. 1()T
by a quadrature. We find
and hence M ==—-^tj—гту
It may be at once verified that this is an integrating factor of
E) and also of F). Hence, from E) and F) respectively, we find
by quadratures
T V
~z , = const., -,-— -5 = const.
as integrals.
The first isothermal family is that of all circles which touch the
y-axis at the origin ; the second is that of all circles which touch
the #-axis at the origin.
Of course equations E) and F) might also have been integrated
by the method of Art. 64.
79. In the following examples, such differential equa-
equations as represent curve-families consisting either of
isothermal or of parallel curves may be integrated by
the method of Arts. 76-78. It may usually be seen,
from the geometrical meaning of the equation given,
whether the orthogonal family will be isothermal or not.
Those differential equations which represent orthogonal
trajectories which are neither parallel nor isothermal
curves may be integrated by Art. 73.
EXAMPLES.
Find the orthogonal trajectories of the following curve-families :
A) The straight lines y = ax. (a^parameter.)
B) The parabolas y2 = ±ax.
C) The circles а*+у*=а\
D) The parabolas ys = 4a(.r + a).
x2 ifl
E) The ellipses ~2 + ga=1- (a = const., Ь = parameter.)
F) The circles л:2— 2ах+у2-2ау + а2=0.
G) The ellipses ~ъ+Къ=1П%- (и = parameter.)
(S) The circles x2+y2 + ax~l=Q

108 ORDINARY DIFFERENTIAL EQUATIONS.
(9) The confocal conies ^-—^+^=1. F = parameter.)
A0) The parallel curves у-а = ф(х),
which result from translating the curve у = ф(х) parallel to
itself along the #-axis.
A1) Apply the result of Ex. A0) to the case of the serai-cubical
parabola
щ
щ
A2) Show that the differential equation of the orthogonal trajectories
of the curves in polar coordinates
F(p, в, е)=0
is obtained by eliminating с between the above equation and
W dp_ 3F
39 dB рЪр ¦
A3) Find the orthogonal trajectories of the curves
p = log tan 6+a.
A4) Find the orthogonal trajectories of the curves
в-а = ф(р),
which result from rotating the curve
в = ф(р)
around a fixed point in the plane.
A5) Apply the result of Ex. A1) to find the orthogonal trajectories
of the circles which result from rotating the circle
p = b cos в.
around one end of its diameter.

CHAPTER VI.
DIFFERENTIAL EQUATIONS OF THE FIRST ORDER,
BUT NOT OF THE FIRST DEGREE. SINGULAR
SOLUTIONS.
80. We propose to give in Sec. I. of this Chapter
methods for integrating some of the simpler differential
equations in two variables which are of the first order,
but not of the first degree.
In Sec. II. we shall see that a differential equation of
a degree higher than the first is sometimes satisfied by
a function .which is not a function of the " integral-
function" of the equation. This peculiar function,
equated to zero, constitutes what is known as a " Singular
Solution" of the given equation. A simple method for
finding the Singular Solution—when one exists—of an
invariant differential equation of the first order will be
given.
SECTION I.
Differential Equations of a Degree Higher than the
First
81. In Art. 55 it was shown that the condition that
the differential equation of the first order

110 ORDINARY DIFFERENTIAL EQUATIONS.
shall admit of a given Gx
is that the expression JJ\U) shall be zero, either identi-
identically, or by means of Q = 0.
It is clear that this condition is independent of the
form of Q, and hence of the degree of the equation Q = 0.
It will be borne in mind, however, that in order to apply
the method of Art. 59 to integrate the equation О = 0 by
a quadrature, when it is known that this equation admits
¦of a given G,, U'f, it is necessary to solve ?2 = 0 in terms
of у, in the form,
лу-г=о.
The algebraic solution of Q = 0 in terms of y' is not
always simple; and in Cases II. and 1П. below we shall
indicate methods by means of which that work may
sometimes be simplified or avoided.
82. Case I. Suppose that the equation
u(x,y,y') = 0 A)
may be solved, algebraically, in terms of y' in such
manner that the resulting roots will be rational functions
of x and y. Thus, if A) is of the nth degree, it may be
written
(у'-ф^х, у))(у'-фг(х, у)) ...(у'-фп(х, у)) = 0, ...B)
where the фх> ...,фп are rational functions of x and y.
In this case, since A) can be resolved into the linear
factors in B), A) is called a "decomposable" equation.
But B) is satisfied by writing
and if the general integrals of the n equations C) have
been found, by the methods of Chap. IV., in the form

DIFFERENTIAL EQUATIONS OF A HIGHER DEGREE. Ш
the general integral of A) will have the form
But equation D) loses nothing of its generality if we
assume the arbitrary constants all equal to one constant,
say с For, in order to find any value of y, it is necessary
to equate to zero one of the n factors on the left-hand
side of D), which gives an equation of the form
у~Ф^,у,с) = 0 E)
Now since с is an arbitrary constant, by giving с all
possible values, the form E) may be made to contain all
the integrals which may be derived from the corre-
corresponding къь factor of B).
Hence we find as the general integral of the original
decomposable differential equation A) :
{y-$i(%, У> c)} •-• iy-$n(x,y, c)}=0.
Example. Suppose B) to be of the second degree, of the form
&*-(*+№+ яу-0. F)
This equation may be written
ty~x)(]/-y)=%
whence у'-л:=0, г/'-г/=О.
Integrating the last two equations we find
so that the general integral of F) is
83. Case II. If the given equation
п(х,у,у') = 0
can be readily solved with respect to y, in the form
У = Ф&У'1 G)

112 ORDINARY DIFFERENTIAL EQUATIONS.
it is sometimes best to differentiate G), regarding y' as
a variable as well as x and y, and substituting in the
result y'dx for dy. A differential equation of the first
degree between x and у must result. Integrate this
equation by the methods of Chapter IV., and eliminate
y' between its primitive and the equation O=0.
Example. Given y=xy'2 + 2/.
Hence dy =y"*dx + 2x/dy' + 2tf/,
or, putting y'dx for dy,
(y-i-y')dx + 2(xy' + \)dy' =0.
This is an example of Art 68; and we find as an integrating
factor
Thus we have to integrate the exact differential equation
By Art. 48, we find as the general integral of this equation,
{y' -1 Jx - 2 (log у' - у')=const.
The general integral of the first equation is to be found by
eliminating #' between that equation and the last one.
84. Case III. When the differential equation Q = 0
can be readily solved in terms of x, in the form,
Х-Ф(У,У) = О, (8)
it is sometimes best to differentiate (8), regarding y' as
variable as well as x and y, and substituting —% for dx.
A differential equation of the first degree in terms
of у and y' must result. Integrate this equation by
the methods of Chapter IV., and eliminate y' between
the resulting equation and U = 0.

Example
Hence
and
Putting for
. Give:
SINGULAR SOLUTIONS.
we find at once,
У У ~~
113
The general integral of this equation is yy' = c. Eliminating y'
from the first equation, the general integral required is found to Ъе
85. An important equation of the form
which is known as Clairaut's equation, and which may
be integrated in a manner analogous to that employed
in Case II., will be treated separately in the next
Chapter.
SECTION II.
Singular Solutions.
86. It will be remembered that in Art. 59 it was
shown that if a given differential equation of the first
order
admits of a known G1
the differential equation may always be integrated by a
quadrature, provided that the infinitesimal transforma-
transformation JJf is not trivial with regard to O = 0: that is,
provided that the path-curves of Uf do not coincide with
the integral curves of 0 = 0.

114 ORDINARY DIFFERENTIAL EQUATIONS.
But now the question arises, may not a limited number
of path-curves of the G1 coincide with particular integral
curves of Q = 0, when the G1 is not trivial ?
It will be found that such may be the case. For,
along curves which are at once path-curves of the Gj and
integral curves of fi = 0, the value of y' given by the Gx
must coincide with the value of y' given by Q = 0.
Hence, to find such curves we only need to substitute
in Q = 0, and the resulting equation
..B)
will give the path-curves of the Gx which are also integral
curves of A), if such exist.
But it is easy to see that we may also find in this
way the equation to a curve which is a path-curve of
the Gx and which satisfies the given differential equation
A), but which is not a particular integral-curve of A).

SINGULAR SOLUTIONS. Ш
For it may happen that the family of integral-curves A)
have an envelope, and if so, it is clear that the equation
to the envelope will satisfy the differential equation
A); for at any point on the envelope the direction of
the tangent to the envelope is the same as that of the
tangent to either of the two consecutive curves of the
family, which, from the Differential Calculus, we know
must coincide in that point. Hence the value of y' at
any point on the envelope will satisfy the differential
equation; and the equation to the envelope ia called a
" Singular Solution " of A). Since at any point on the
envelope two values of y' given by A) must coincide,
it is clear that equation A) must be of at least the
second degree in у in order that the integral-curves
may have an envelope.
But now the family of integral-curves of A) is in-
invariant under the transformation Uf; hence it is clear
that the envelope of the family, if one exists, ia an
invariant curve of which the points are interchanged
by means of the transformation Uf. In other words,
the envelope must be a path-curve ox the Qv Uf, of which
A) admits. To find this particular path-curve, we only
need to substitute -j for y' in A); and the resulting
curve, or curves, must, as we saw, be those curves in the
plane for which the values of y' given by the differential
equation A), and by the Gv are the same. Hence, we
find by this method the singular solution of A), if one
exists; and, occasionally, as indicated above, a limited
number of particular integral-curves of the differential
equation, which are, at the same time, path-curves of the
Gv
The particular integral-curves may be distinguished
from the singular solution by the fact that the equation
to a particular integral-curve may always be obtained
from the general integral of A) by assigning a special
value to the constant of integration, while the equation
to the singular solution cannot be so obtained.

116 ORDINARY DIFFERENTIAL EQUATIONS.
If the equation B) breaks up into factors, each factor
must be separately examined to see whether it is a
particular integral or a singular solution.
It may be remarked that ^ and r\ cannot both be zero
along the enveloping curve of an invariant family; for,
as we saw above, the points of the envelope are inter-
interchanged when the curves of the invariant family are
interchanged, whereas all points on curves along which
^=rj = O, are absolutely invariant.
Example 1. Given
y^2cos2a — 2у'д^ sin3a -\-y2 — .r3sin3a = 0.
This equation is homogeneous, and hence is invariant under the Glf
Thus, according to the above theory, we find the singular solution,
if one exists, by substituting in the above equation - for y'.
We obtain, after an obvious reduction, the two equations
The general integral is found by Art. 59 to be
a?+f-2cx + c2cos2a=0; (c=const.)
and hence we see that
which may be obtained from the general integral by assigning to
the arbitrary constant с the value zero, is a particular integral;
while
which may be written
must constitute a singular solution, since these equations satisfy
the given differential equation, and cannot be obtained by assigning
any special value to с in the general integral.

SINGULAR SOLUTIONS. Ц7
Example 2. We know, Art. 71, that the differential equation
уЗ-4адаЧ%2=0
admits of the Gv
To find the singular solution, if one exists, we substitute for y,
In the above equation, -A Hence
The general integral is
and hence we see that # = 0 is a particular integral, while
is the singular solution.
Example 3. The differential equation
being free of yt admits of the Ог
Since for this (?„ 8^ = 1, 8^ = 0, we have
Therefore we write the above equation
and, substituting for y' the value x , we find
This is a singular solution, since the differential equation possesses
the general integral
The geometrical meaning of the singular solution in connection
with the so 1 curves represented by the general integral is obvious.

118 ORDINARY DIFFERENTIAL EQUATIONS.
EXAMPLES.
Integrate the following differential equations, finding the singu-
singular solutions, when such exist, as well as the general integrals.
For types of invariant equations, see Art. 73.
A) У-5)' + в=0. B)У2-»У=0.
а D) У(У+*)=*(*+»)•
G) ,/
(9) а
A1) у- -*
A0) 3^у!-
A2) xy'O-
A3) у=а/ + 1п/г. (Art. 8a)
A5) /=^1 +У2)- 0в) У=
/=y. (Art. 84.) A8) *=/+log/.
B0) my-шУ=да
B3) аУ
B5) 4y(
B6) «V-fa/t»=0.
B4) y"
B7) jr

CHAPTER VII.
RICCATI'S EQUATION AND CLAIKAUTS EQUATION.
87. We propose in this Chapter to make brief mention
of two important historical differential equations of the
first order, which are known respectively as Riccati's
equation and Clairaut's equation. The treatment of
these equations sketched here will be the same as that
of the ordinary text-books : for, although both equations
may be treated most advantageously from the standpoint
of the Theory of Transformation Groups, that method
would require a more extensive knowledge of these
groups than it is advisable to give in an elementary
text-book.
SECTION I.
Riccati's Equation.
88. This equation takes its name from that of an
Italian mathematician, Kiccati, who was the first to
discuss it.
The general form of Riccati's equation is
but this equation can only be integrated in a few special
cases : and the particular form usually discussed is
^ cx", B)

120 ORDINARY DIFFERENTIAL EQUATIONS.
where a, b, c, n, are certain constants. By introducing
into B) the new variables z = xa, и = -^, that equation
takes the form
du b „ с "-2 .
-,—I—и2= г" C)
dz^a a K '
a special form of A), which is itself sometimes designated
as Riccati's equation, instead of the more general equa-
equation A).
The equation B) happens to be much more easy to
discuss than equation C); and it is easy to deduce from the
condition that B) shall be integrable the condition that
C) shall also be integrable. We shall first show that
equation B) is always integrable when n = 2a; then we
shall show that the integration of B) may always be
made to depend upon this case when —^— is a positive
integer.
89. Case I. The equation
x-d^-ay + by2 = cxn B)
is always integrable when n = 2a.
Let us assume y = 3fv ; then B) becomes
dx '
and if n = 2(t this equation becomes
dx
dv dx
In this equation the variables are separate, so that
it may be integrated by a quadrature. If we return to

RICCATFS EQUATION. 121
the original variables, we find the exact differential
equation
of which the general integral is given by
Ce « -1
according as & and с have the same or opposite signs—
С being the arbitrary constant of integration.
90. Case II. The equation
^ сх" B)
is always integrable when —^— is a positive integer.
zn
Let us assume
where A is a constant to be determined. The equation
B) is easily seen to take the form
9^ p cx....(b)
Ух Ух dx v l
We shall choose A so that the constant in this equation
shall be zero; thus we may choose A=r, or A=0, so
that there are two subdivisions for this case of the
problem.

122 ORDINARY DIFFERENTIAL EQUATIONS.
A) If, in the first place, А =т, the last equation, after
a slight reduction, takes the form,
It is seen that F) is of the same form as B), except
that b and с have changed places, and a has been
changed to a+n: and this change was brought about
by substituting in B)
a se"
in place of y.
Hence, if in F) we make the substitution
it is clear that F) will take the form
х^-(.а + 2п)уг + Ьу' = е^ G)
where b and с have again changed places, and a + n has
become a + 2n.
Thus, if X successive substitutions of the above forms
are made in the equation B), that equation will take
either the form
or the form
х^-(а + Хп)Ух + Ьу^ = сх- (9)
according as X is odd or even.
But by Case I., the equations (8) and (9) are in-
tegrable if

that is, if
RICCATrS EQUATION. 123
«-2a
B) Secondly, let us assume A=0. Then B), by
means of the substitution
is readily seen to take the form
х-^-{п-а)У1 + су* = Ъх\ A0)
an equation which is identical with F) except that a
has become — a. If the preceding series of substitutions
are now made, the final result will be found to be the
same except for the sign of a. Thus the equation B)
will be integrable when
Combining these results we see that the equation
is integrable whenever —^ -- is a positive i/nteger.
From the nature of the substitutions employed in
the above two cases, it is clear that the general integral
of Riccati's equation, when that equation is integrable
by the above method, will be given in the form of a
finite continued fraction, the last denominator of which
is to be found by a quadrature.
91. We found, in the last article, as a condition of
integrability, that —=— should be a positive integer,
say Л. "n

124 ORDINARY DIFFERENTIAL EQUATIONS.
If we assume —^ =X, from A) Art. 90, we have the
series of substitutions
a *"
where ц has the value Ъ or c, according as X is odd or
even.
From these equations we have
a | a*
0 a-±^+ (и)
the last denominator of the finite continued fraction
being
a+(\-l)n xn
/* У\
The value of y\ is to be determined by a quadrature
from one of the equations (8) or (9). These equations
may now be put into exact forms, analogous to D),
writing a + Xn for a :
and
X being supposed odd in A2) and even in A3).

RICO AITS EQUATION. 125
If now we assume
n + 2a
2« - = X'
it is easy to see that у will have the value
x"
A4)
where the last denominator is
(к — 1)п — а ж^
~~ /* " V\
Also, y\ is to be found from one of the following exact
equations, which result from A2) and A3) by changing
a into — a:
\ being orfrf in A5) and even in A6).
Thus, when -—¦— is a positive integer, A1) in con-
connection with either A2) or A3), according as \ is odd or
even, will represent the required general integral.
When —s-—¦ is a positive integer, A4) in connection
with either A5) or A6), according as \ is odd or even,
will represent the required general integral.
92. By making use of new constants, the equation C)
which is itself sometimes designated as Riccati's equation,
may obviously be written
i^+bu* = cz™ A7)

126 ORDINARY DIFFERENTIAL EQUATIONS.
If, in place of z and tt we introduce the new inde-
independent variables x, and у = их, A7) becomes
J[ x™J'1 A8)
But we know that A8) is integrable when
(m+2)±2
where Л is a positive integer. This is therefore the
condition that the special form A7), of Riccati's equation,
shall be integrable.
Making use of the negative and of the positive signs
in succession in the above condition, we find
*nd m = r*?=2) B0)
By changing, in B0), the integer X into \+l, which
is obviously allowable on condition that X may assume
the value zero, as well as any positive integral value,
{20) may be written
Here, if \ = 0, m = 0; and since for X = 0 in A9) we
also have m=0, it is clear that X may admit of the same
.series of values in A9) as in B1).
Combining these results, we see that Riccati's equation,
in the special form A7), is integrable whenever
X being zero, or some positive integer.
When the negative sign is used in the last equation,
the general integral is given by A1) in connection with
{12) or A3) according as X is odd or even. When the

RICCATFS EQUATION. 127
positive sign is used, the general integral is given by
A4) in connection with A5) or A6) according as X is
dd
()
odd or even.
Example. Given the equation
This is a case of equation A7). By substituting yjx for u, the
equation takes the form of A8),
The condition of integrability A9)
Hence, the integral of B3) is given by the equation A1) in
connection with A3). Here we have
a-1, A = 2, я=-?, a+wA=~i 6=-l, c = 2 ;
hence A3) becomes
This ia an equation of the form D), where the a, b, and с of that
equation have the respective values
-i> -!>*¦
Thus, since b and с have opposite signs, the integral of the last
equation is
This value of y2 substituted in
gives the general integral of B3); and if in that result we restore
to у its original value их, we find the general integral of B2).

128 ORDINARY DIFFERENTIAL EQUATIONS.
SECTION II.
Clairaut's Equation.
93. The equation of the form
у = ху'+ф(у'), A)
is generally known as Clairaut's equation. Although an
equation of the first order, it is not usually of the first
degree.
Differentiating A), regarding y1 aa a variable, as well
as x and y, we find
% B)
from which follows either
C)
From the former of these equations follows
y' = c, (c = const.)
so that, from A), the general integral must have the
form
y = cx + <j>{c); D)
and that this is correct may be immediately verified by
differentiation.
But now, if we eliminate y' between C) and A) we
obtain a relation between x and у which satisfies A)
also, and which is therefore a solution of the equation.
But this solution will obviously not contain an arbitrary
constant, nor will it be included in the general integral D).
It is easy to see that this is what was designated,
Art. 86, as a singular solution of A); for the equation
у = сх + ф{с) D)
represents a family of со1 straight lines. If they have
an envelope, it will be found, as is well known, by

CLAIRAUT8 EQUATION. 129
eliminating с between the above equation and the
equation obtained from the above by differentiating it
with respect to c,
О = ж + 0'(с) E)
But the result of eliminating с between D) and E)
must be the same as the result of eliminating y' between
the two equations
У=ху'+ф(у'\
and hence the curve represented by the resulting equation
will be the envelope of the family D)—or a singular
solution of the differential equation A).
Of course the method of finding singular solutions by
differentiation as here indicated might be employed in
all cases, instead of the method of Art. 86; but by this
method we usually find too much—not only the envelope
of the curve-family, if one exists—but various other loci
which may exist, composed of multiple points, cusps, etc.
The method of differentiation is generally the most
simple for finding the singular solutions in the case of
Clairaut's equation; since, of course, there can be no
multiple points or cusps on one of the straight lines C).
It is clear that a complete primitive representing a
system of со * straight lines will always give rise to a
differential equation of Clairaut's form—the singular
solution of the equation being the curve to which the
straight lines are tangent.
94. A more general equation of a form similar to A) is
F)
Differentiate, regarding y' as a variable as well as x
and y\ hence

130 ORDINARY DIFFERENTIAL EQUATIONS.
But this equation is linear in x, and may be treated
by the method of Art. 68. If the general integral of G)
has been found in the form
П(х,у',с) = 0,
the elimination of y' between this equation and F) will
give the general integral of F).
Example 1. Given
y~^/+j, (8)
Thia is an example of Clairaut's equation. Differentiating, we
find
From aM-=
dx
the general integral ie seen to be
From
we have У = '\— >
and this value of у1, substituted in (8) gives as a singular solution
Hence we see that the singular solution is the equation to a
parabola, while the general integral represents its со' tangents.
Example 2. Given
*+да-=ау" (9)
This is seen to be an equation of the form F). Differentiating,
we find, after eliminating y,
or, corresponding to G),
dx x at/

CLAIRAUT'S EQUATION. 131
The general integral of A0) is, by Art. 68,
and the elimination of y' between A1) and (9) will give the general
integral required.
EXAMPLES.
Integrate the following equations (l)-E) by Sec. I.:—
A) x^-ay + f^x-". B) x^-ay+f^x'".
C) x%-y+tf=x\ D) ?+»« = «-'
E) ^ + (w> = «-«.
Integrate the following equations F)-A5) by Sec. II., giving also
the singular solutions :—
F) y^xtf+jr G) y = xy'
(8) у=xyl +г/ -y*. (9) f - 2^
A0) y=(x-\)tf~tf*. (И) ^У2-
A2) (y-
The singular solutions, as well aa the general integrals of the
differential equations to which the following geometrical problems
give rise, are to be found and interpreted. For geometrical for-
formulae, see Chap. IV., Examples.
A3) Determine a curve such that the sum of the intercepts made
by the tangents on the axes of coordinates shall be constant
and equal to a.
A4) Determine a curve such that the portion of its tangent inter-
intercepted between the axes of x and у shall be constant and
equal to a.
A5) Determine a curve such that the area of the right triangle
formed by its tangent with the axes shall be constant, and
equal to aa.
A6) Determine a curve such that the projection upon the axis of у
of the perpendicular from the origin upon a tangent is
constant, and equal to a.

CHAPTER VIII.
TOTAL DIFFEEENTIAL EQUATIONS OF THE FIRST
OEDEE AND DEGEEE IN THEEE VAEIABLES,
WHICH ABE DEEIVABLE FKOM A SINGLE PRIMI-
PRIMITIVE.
95. In this chapter we shall indicate how the integra-
integration of an ordinary diiferential equation in three variables
of the form
P(x, y, z)dx+Q(x, y, z)dy+B(x, y, z)dz = 0
may, under certain conditions which we shall develop,
be made to depend upon the methods given in Chap. IV.
for the ordinary differential equation of the first order
and degree in two variables.
SECTION I.
The Genesis of the Total Differential Equation m
Three Variables.
96. In Chap. I. it was seen that an equation of the
form
О (x, у, с) = 0, (с = const.)
when treated as a complete primitive always gives rise
to an ordinary diiferential equation of the form
X(x, y)dy-Y(x, y)dx = 0.

DIFFERENTIAL EQUATION IN THREE VARIABLES. 133
Similarly, if the equation in three variables,
Y(x, y, z, c) = 0,
is given, we find by differentiation
3F, 3F, ,3F, .
-dx + -dy+?izdz = 0,
which, when с has been eliminated by means of F = 0,
may be written
P(x, y, z)dx + Q(x, y, z)dy + B(x, y, z)dz = 0. ...A)
Equation A) is called a total differential equation—
the word toted being used in contradistinction to the
word partial in the definition of a partial differential
equation in three variables, Art. 15.
If the equation
V(x, y, z, c) = 0
be solved in terms of с ш the form
Q(x, y, z)=c, B)
we find, by differentiating B),
The equation C), which, for obvious reasons, is called
an exact differential equation, must of course be equi-
equivalent to A), since Q — c is equivalent to F = 0. Hence,
when an equation of the form A) is given, if it is to
be reducible to the exact form C), certain conditions
must hold. That is, there must obviously exist a
function ^(х,у, z) such that,
шЛ 81nce »SS=SSw ete.

134 ORDINARY DIFFERENTIAL EQUATIONS.
we obtain from D),
дП ЭР
ultiplying the first of these equations by R, the
second by P, and the third by Q, we find as the
condition that A) may have a general integral of the
form B), У ё
seco
p
SECTION II.
Integration of the Total Differential Equation in
Three Variables.
97. We saw in the last section that the condition E)
must be satisfied, in order that the total equation A)
Pdx+Qdy+Rdz = O A)
may be derivable from a primitive of the form
Q(x, y, s) = const.
It is clear that that condition will be satisfied if the
variables in A) can be separated in such manner that
P, Q, and R contain only the variables x, y, and z,
respectively; and the general integral of A) will, in
that case, obviously be given in the form

TOTAL EQUATION IN THREE VARIABLES. Ш
Example. Given
yz dx+xzdy+xy dz=0.
The variables may here be separated Ъу dividing by xyz; and
we find
x^ y^ г •
The general integral is, therefore,
or xys=c. (c=const.)
98. The equation A) may sometimes be rendered exact
by an integrating factor which suggests itself. For
instance, the equation
zydx — zxdy — y2dz = 0,
for which the condition E), Sec. 1, is satisfied, becomes
exact when multiplied by the factor —g- ^e then have
ydx — xdy dz_n
—^2 — v-u-
of which the general integral is obviously
ОС ,
— log г = const.
99. If, however, the variables in A) cannot be separated
by inspection, and no integrating factor suggests itself,
the usual method for integrating A)—supposing the
condition of integrability to be satisfied—is as follows:—
Since the condition E), Sec. I., is satisfied, a general
integral of A) of the form
Q(ic, y, 2) = const B)
exists, which, when totally differentiated, must give rise
to an equation equivalent to A). Suppose that one of
the variables in B), say z, is temporarily regarded as
a constant; then the equation resulting from totally
differentiating B) will have the form
F)

136 ORDINARY DIFFERENTIAL EQUATIONS.
Now the general integral of the ordinary differential
equation in two variables F) will clearly include the
general integral of A), if in place of the arbitrary-
constant of integration an arbitrary function of z ib
introduced. In order to determine the form of this
function of z, it is necessary to differentiate the general
integral of F), considering z also as variable, and com-
compare the total differential equation thus resulting with
A). This will give rise to a second ordinary differential
equation for determining the arbitrary function of z.
The choice as to which of the three variables is to be
regarded as a constant will be determined by the form
of the resulting equation F). It may sometimes be
easier to integrate, Dy the methods of Chap. IV., one of
the two equations
s = 0, Qdy+Rdz^O,
which result from considering у and x respectively as
constants, than to integrate F).
Example. Integrate the equation
Here the condition E), Sec I., is found to be satisfied. If we
assume s to be constant, as in this case is most convenient, the
term xydz vanishes, so that the given equation, after dividing
by F - s), becomes
ydx+xdy=O.
The integral of this ordinary equation in two variables is
xy = cons,t. = ф(г).
Differentiating, we find
In order that this equation may be equivalent to the first one,
we must have
dz z — b

TOTAL EQUATION IN THREE VARIABLES. 137
Here the variables are separate, so that an immediate integration
gives
#)SD(|-Si (C = COnst.)
The required general integral is, therefore,
100. We shall give a method, based upon geometrical
considerations, by means of which the total equation A)
Pdx + Qdy+Bdz = 0 A)
may, when the condition E) of the last section is
satisfied, be integrated by integrating one ordinary
differential equation of the first order in two variables.
Since E) is satisfied, A) has a general integral of
the form
Q(x,y,z) = e; B)
and B) represents so1 surfaces in space, called the
integral surfaces of A). If we cut these surfaces by a
family of so1 planes, say,
z=x + ay; (a = const.) C)
then for each value of a we obtain со1 curves of inter-
intersection of one of the planes with the so1 surfaces B);
and these curves are represented by a differential equa-
equation in x and y. To find this differential equation, we
only need to eliminate z and dz from A) by means of
C) and of
dz = dx+ady;
giving the differential equation in the form
ф(х, у, a)dx + \jr(x, у, a)dy = 0 D)
If, now, D) has been integrated, by the methods of
Chap. IV.—a being an arbitrary constant—we may
easily find the so1 surfaces B), since we know their
cos curves of intersection with the planes C). For the
ool curves of intersection which pass through one point
on the axis of the family of planes C) will in general
form one of the integral surfaces B).

138 ORDINARY DIFFERENTIAL EQUATIONS.
Now a point on the axis of the planes B) is evidently
determined by
у = 0, x = к; (к = const.)
and if the general integral of D) be
W(x, y, a) = const., E)
in order for the curves E) to pass through the point
y = 0, х = к we must have
W(x, y, a)= W(k, 0, a) F)
When a varies, F) represents the со1 curves through
the point у = 0, x = к; and if к also varies we obtain
successively the x1 curves through each point on the
axis of C). That is, if by means of C) we eliminate
a from F), we obtain the integral surfaces required in
the form
Thus the complete integration of A) has been accom-
accomplished by integrating one ordinary differential equation,
D), in two variables. If the constant a happens to
factor out of D), some other family of planes must be
used in place of C).
This method is theoretically better than that of Art.
99, since only one differential equation in two variables
has to be integrated; but the differential equation D) is
often more difficult to integrate than are equations F)
and (9) of Art. 99.
Example. Given
{y + z)dx + dy + dz = 0.
This equation evidently satisfies E), Sec. I.; and if we write
z=x + ay,
the equation D) becomes

TOTAL EQUATION IN THREE VARIABLES. 139
This is a linear equation, with the general integral
W{xyy, «) = «*^+i^)=C (c=const)
Hence equation F) has the form
or, writing for a,
... ,=0
y+г-х y + i-z
that is, e%+z) = const.
EXAMPLES.
Integrate the following ordinary differential equations in three
variables, after verifying that the condition E), Sec. L, is satisfied:
A)
B)
C) (г-3^-г
D) ayWdic+UWdy + apy'dz^O.
E)
F)

CHAPTER IX.
ORDINARY DIFFERENTIAL EQUATIONS OF THE
SECOND ORDER IN TWO VARIABLES.
101. In this chapter we propose to develop a theory
of integration for ordinary differential equations of the
second order in two variables analogous to that developed
in Chapter IV. for ordinary differential equations of the
first order.
The linear differential equation of the second and
higher orders will be treated separately in Chapter XI.
. SECTION I.
Exact Differential Equations of the Second Order.
102. If an ordinary differential equation of the second
order of the form
Q(x,y,y',y") = 0
be given, we know that the complete primitive, or
general integral, is an equation involving two inde-
independent arbitrary constants, e^ c2, of the form
%(x, y, clt c2) = 0.
It may be shown by means of the Theory of Functions
that if the complete primitive of O = 0 be written in the
form

EXACT EQUATIONS OF SECOND ORDER. 141
and if
y-w(x,alt cr2) = 0
be a second equation, which, when treated as a complete
primitive, gives rise to the same differential equation of
the second order, Q = 0, then it must always be possible
to choose the av a% as such functions of cv c2, say
ai = Mci>c2)> a2=X2(ci'c2).
that W(x, clt c2) = w(x, \v Xg).
If the above complete primitive Ч^ = 0 be differentiated,
an equation of the form
*2(#>2Л2Лс1(с2) = О
will result, from which, by means of 4if1 = 0, cx and c2
may be successively eliminated, giving rise to two inde-
independent differential equations of the first order of the
form
The elimination of #'between Ф1 = 0 and Ф2 = 0 would,
of course, give Фх = О again.
Now let Ф]^ = 0 be again differentiated, and c, be
eliminated from the resulting differential equation of the
second order by means of Фх=0. By this means we
must obtain the given differential equation of the second
order, Q = 0. This last equation must also, of course, be
obtained by proceeding similarly with Ф2 = О.
The differential equations of the first order, #! = 0,
Ф2 = 0, may, on the other hand, be considered as having
been derived by an integration from Q = 0; and for this
reason, Фг — 0, Ф2 = 0 are said to be first integrals of the
given differential equation of the second order, fi = 0.
From the manner of their genesis it is seen that there
can be only two independent first integrals of fi = 0,
although the whole number of first integrals is infinite.
That is, the constant of integration of any third first
integral of ?2 = 0, of the form

142 ORDINARY DIFFERENTIAL EQUATIONS.
can always be expressed as a function of the constants of
any two independent first integrals; otherwise, by
eliminating y' from Ф1 = 0 and Ф2 = 0 by means of Ф3 = 0,
two different forms of the complete primitive, ^ = 0 and
?" = 0, would be found, which is impossible.
For a method of expressing the general integral in the
form of an infinite series, see Art. 122.
Example. It may be readily verified that the ordinary differ-
differential equation of the second order,
has the first integrals
-JL =y cot(# + a) ; (a3, b, c, a eonsts.)
but only two of them are independent. For, squaring and adding
the second and third, and comparing with the first we find that a
relation of the form
63+<? = аа
must exist Also, developing cot(#+a) in the fourth equation, and
comparing the result with the sum of the second and third, we find
c + 6tana=0.
Hence, 6=acoso, c=-asina;
and only two of the four constants are independent.
The general integral of the original differential equation may be
found either by integrating any one of the first integrals again by
the methoda of Chap. IV., or by eliminating -У- between any two
independent first integrals. Thus the result of eliminating -У-
between the second and the third of the above first integrals will be
which is the general integral of the given differential equation of
the second order.

EXACT EQUATIONS OF SECOND ORDER. 143
Also, the first of the four first integrals may be written
which integrated gives
or sin у = a sin (x + /?).
If now we put m = acos C, n = a sm.fi,
the last equation reduces to
y = mshix+ncosx,
which is evidently the general integral as found above. The same
result may, of course, be found by eliminating -^ between the first
and fourth of the first integrals.
103. A differential equation of the second order
is said to be exact, when the expression Qdx is the exact
differential of a function of the form F(a;, у, у') Since
the equation Q = 0 is derived from F(a;, y, y') = c by
direct differentiation, y" can only occur in the first
degree. Otherwise п = 0 will not be an exact differential
equation.
Thus the exact differential equation of the second
order may be written
In order to integrate this equation completely, we
notice, first, that we can determine how the variable y'
enters the function F by integrating the term —, y"
with respect to if. If the expression thus found be
differentiated totally with respect to x, and the result be
subtracted from the first member of C), the remainder
must be a differential expression of an order not higher
than the first. Also since this remainder is the difference

144 ORDINARY DIFFERENTIAL EQUATIONS.
of two exact differentials, it must itself be an exact
differential, and yr can occur in it only to the first
degree. Ite integral, together with the terms already
found by integrating with respect to yr> will be the
integral of the whole equation C).
Example. The equation
xyf+xtf*-ytf=O, D)
is exact. For the term involving У is aW', and this being integrated
with respect to У gives xyy1. The last expression when differ-
differentiated totally with respect to x gives
Subtract this result from the first member of D), and the remainder,
- 2yy', will also be exact, having for its integral -f. Hence equa-
equation D) is exact, and a first integral is
If now we divide D) by a?, it will be seen that a second first
integral ia
Hence the complete primitive is
SECTION II.
Lie's Differential Equations of ike Second Order.
104. Of course not every differential equation of the
second order is exact; and there is no general method
for integrating all such equations when not exact. It
will be our object, however, to show in this paragraph,
how the knowledge that the given differential equation
of the second order admits of or is invariant wider, a
given (?j can be used to reduce the problem of integration
in a number of the most important classes of differential
equations of the second order. These invariant differ-
differential equations of the second order are sometimes called
" Lie's Equations of the Second Order."

LIE'S EQUATIONS OF SECOND ORDER. 145
105. Suppose that the infinitesimal transformation
of a given Gz in two variables be twice extended by
Art. 47; in the four variables x, у, у', у" the twice-
extended infinitesimal transformation will have the form
, , dy ,df „ dy' (
where «'еД-«г4| »"з-f- — y~X
1 dx " dx' ' dx " dx
The necessary and sufficient condition that an equation
in the four variables x, у, у\ у"', of the form
П(я>У.ЗЛу") = °.
may be invariant under the G^ U"f is, by Art. 42, that
the expression U"(Q) shall be zero, either identically, or
by means of Q = 0. It was also shown in Art. 43 that if
u, v, and w be three independent solutions of the linear
partial differential equation
the most general form of the invariant equation, Q — 0,
is obtained when Q, is expressed as an arbitrary function
of u, v, and to, say
fl(a> У> У'> У") = FO> v> w) = 0.
Or, if we choose, we may solve F = 0 in terms of one
of the three quantities u, v, or w, say in terms of w,
and thus put the most general invariant equation in
the form
ги — Ф(и, v)=0.
But x and у may be interpreted as point coordinates
in a plane; and y' and y" as the differentials
dx' dx2>

146 ORDINARY DIFFERENTIAL EQUATIONS.
respectively. In this case the equation Q = 0 is a differ-
differential equation of the second order in the variables x and
у; and if the expression U"(Q) is zero, either identically
or by means of Q = 0, the differential equation 12 = 0 is
said to be invariant under, or to admit of, the Ог U"f
Now the differential equation Q = 0 represents a doubly
infinite system of curves in the plane; and that the
equation 0 = 0 shall be invariant under XI"f means that
the system of curves must be invariant under XI"f also.
For, if we designated the new variables, as usual, by
xi> Vv Vi' У\> and the transformed equation by 121 = 0,
then, by hypothesis, Qt must have the same form in the
new variables that О had in the variables x, у, у', у":
that is, 0a=0 must represent the same family of curves
that О = 0 represented.
But this family of curves will also be transformed by
and at any point of general position {x, у), у' and y" will
receive the same increments, Art. 47, by means of Uf,
that they receive at that point by means of JJ"f. Hence,
a point P, in the plane which satisfies the system of
values x, у, у', у", will always be transformed by means
of Uf to the point Px which satisfies the system of
transformed values xv yv y{ y{'; and it is clear that
since P passes to the same position Px under the trans-
transformation Uf that it does under U"f and since y' and y"
are transformed by Uf exactly as they are by U"f, the
family of curves, represented by 0 = 0, must also be
invariant under Uf. Thus it is clear that the condition
that a differential equation of the second order
Q(x,y,y',y") = 0
shall be invariant under the twice-extended Gx U"f is
the same as that the family of oo2 integral-curves of
0 = 0 shall be invariant under Uf.

LIE'S EQUATIONS OF SECOND ORDER. 147
When the condition that U"(u) is zero, eith er identically
or by means of Q = 0, is satisfied, we sometimes say, for
brevity, that 0 = 0 is invariant under the Ot Uf, instead
of under the twice-extended G1 U"f.
106. We have a general method, Art. 23, for deter-
determining the quantities w, v and w of the preceding article,
and we have seen, Art. 56, that when и has been found
from the differential equation
?-?¦ : (l>
then v can be found as the second integral-function of
the system
dddtf
by means of a quadrature. Unless the path-curves of
the Gv which are represented by
u = const.,
are known, it win be necessary to perform an integration
to find и from the above differential equation of the first
order A); but we shall show that when и and v have
been found, w can be found by mere processes of differ-
differentiation. Also w, as will be seen, must necessarily
contain y"; and as v(x, у, у') was called, Art. 37, a
differential invariant, of the given Gv of the first order,
so w(x, у, у', у"), for reasons which are obvious, is called
a differential invariant of the given Qv of the second
order.
Ю7. We shall now show that when и and v have the
meaning of the last article assigned to them, and are
known, that w, the third independent integral-function
of the system
J:=^=Y==Y C)
can be found by differentiation.

148 ORDINARY DIFFERENTIAL EQUATIONS.
To this end let a and 6 be any two constants. Then
v — au-b = 0 D)
is a differential equation of the first order which is
invariant under U"f, since
U"(v-au-b)sO.
If now a is supposed to retain a fixed value, while b
varies, D) represents ool differential equations of the first
order which are invariant under U"f. Each of these
differential equations represents oo1 integral curves in the
plane, so that there are oo1 families, each of oo1 curves,
which as a system are invariant. This system of x2 curves
must be represented by a differential equation of the
second order, which must also be invariant under U"f.
We obtain this differential equation of the second order
by differentiating
v(x> y.y')-aw(%,y) = b
totally with respect to x. Hence, we find,
d/v dm л ,_.
si-as=° <5>
э» э»
or, briefly, w(x,y,y',y")—a = 0 F)
Since F) is invariant under the Gv we must have
U"(w-a) = 0,
by meana of w — a = 0. But, since a is a constant,
U"(w-a)=U"(w),
that is, U°(w)sO;
and w is a solution of the linear partial differential
equation U"f= 0.

LIE'S EQUATIONS OF SECOND ORDER. 149
From E') we see that since v must contain y', w must
also contain y"; hence we can take this function to be
the third solution of U"f=0, that is, the third integral of
the system C). It is seen further from E) that
dv
w
so that the moat general invariant differential equation
of the second order may always Ъе written in the form
G)
108. The complete integration of G) may be accom-
accomplished by the integration of an ordinary differential
equation of the first order in two variables, together with
one quadrature. For G), as is seen by its form, may be
considered a differential equation of the first order in the
variables и and v, and if G) has been integrated, say in
the form
ъ = ф{ща), (8)
this will be a differential equation of the first order in
x and y, since v contains y'. But since и and v are
solutions of
it is clear that the equation (8) must admit of the G-JJ'f.
Hence, as we know, Art. 59, (8) may be integrated by a
quadrature.
Summing up the above results it is seen that the com-
complete integration of a given differential equation of the
second order, which admits of a known Gv may be
performed as follows:
Л differential equation of the first order must be
integrated to find и—unless the path-curves и = const
of the Ox happen to be known—then a quadrature is
necessary to find v; then the integration of a differential

150 ORDINARY DIFFERENTIAL EQUATIONS.
equation of the first order between uandv; and finally
a quadrature.
A method which is theoretically an improvement upon
this one is given Art. 163.
SECTION III.
Classes of Lie's Differential Equations of the Second
Order.
109. We shall now apply the results of the last para-
paragraph to find the classes of differential equations of the
second order, which are invariant under some of the most
common G^s in the plane, and which are therefore
integrable by the above methods.
110. To find all differential equations of the second
order which are invariant under the G1 of translations
along the x-axis.
Here the O1 has the form
and it ia seen at once that tf = y" = 0, that is, the twice
extended Gl has the form
Thus it is necessary to find three independent integral-
functions, u, v, w, of the simultaneous system
dx _dy_dy'__dy"
1 ~0~ 0 ~ 0 '
It is obvious that we may assume
dv y"
"' "' du у
. ., dv
У tf V fll

GLASSES OF LIE'S EQUATIONS OF SECOND ORDER. 161
The most general invariant differential equation of the
second order has therefore the form
where, of course, since y1 is itself an integral-function of
the above simultaneous system, y" might be written in
place of ^7; that is, any differential equation of the second
order which does not contain the variable x is invariant
under the translations along the ж-axis.
If F = 0 be written in the form
or, Art. 107, ^-ф(м' ") = °-
it is seen that we obtain a differential equation of the
first order in и and v. Its integration will give an
equation of the form
v = \lr(u, ct), (c^const.)
This equation must also admit of the given (¦?,, Art. 108;
hence by Art. 59 a quadrature gives the general integral
sought in the form
Jv&-B=<- (°2=const-)
If F = 0 has the special form
/-Ф(у) = 0,
it is only necessary to assume
. dv
and the equation may be integrated by two quadratures.

152 ORDINARY DIFFERENTIAL EQUATIONS.
Example. Given yy" — у'2 = 0.
Here, according to the above method, we put
in order to obtain the differential equation of the first order
between и and v.
uv —г>2=0,
du
or *=«.
аи и
A quadrature now gives -=cx,
or t=c .
У
whence, by a second quadrature,
111. In a manner entirely analogous to that of the
laat article we may find the differential equations of the
second order which are invariant under the G1 of
translations along the i/-axis. Since
it is seen at once that we may write
, dv ,,
u = x, v = y, w = ^ = y .
Thus all differential equations of the second order,
F(x,y',y") = 0
which do not contain the variable y, are invariant under
the translations along the i/-axis.
If F = 0 be written in the form

CLASSES OF LIE'S EQUATIONS OF SECOND ORDER. 153
it is seen that thia is a differential equation of the first
order in и and v. Its integration will give an equation
of the form
so that a quadrature will give the general integral
required in the form
Example, Given A-js2)/- ay-2=0.
Thia equation, not containing the variable y, admits of
*-%
Hence we must substitute
, , dv
* = "• * = v-fsdi>
so that there results A -w2)-J^-wv-2=O,
This differential equation of the first order in и and v is linear;
its general integral is given by Art. 6S in the form
This equation iu x and i/ must also admit of Uf, Art. 10S, so that
another quadrature gives the required general integral in the form
у=(sin ~ lxJ+2c1sin~I^+ca.
112. It may be remarked that the equation
У" = +(У') A)
that is, the general differential equation of the second
order in which neither of the variables is present, admits
of both of the Gj's

154 ORDINARY DIFFERENTIAL EQUATIONS.
We find from A) by a quadrature
or, say y' = w(x+c1);
whence, by a second quadrature,
113. To find аи differential equations of the second
order which are invariant under the Qx of ajjine trans-
transformations
Uf^xX
1 dx
The twice-extended Ot is
while the simultaneous system to be integrated is
It is evident that we may assume, Art. 63,
u = y, v=xy';
,,, dv xdy'+y'dx ot"+i/'
so that w = ^- = —e-J-2—= v ' g_.
du dy у
The required differential equation of the second order
has therefore the form
By integrating this differential equation of the first
order, a result of the form
xy' — w{y, сх)=0 (c1 = const.)

CLASSES OF LIE'S EQUATIONS OF SECOND ORDER. 155
is found; and since this last differential equation of the
first order is known, Art. 108, to be invariant under
a quadrature will give the general integral required in
the form
у = п (x, cv c2) = 0. (c2 = const.)
114. In an analogous manner the most general differ-
differential equation of the second order which is invariant
under the (?х
since и s x. v = -, will be found to have the form
У
It should be noticed that the so-called abridged linear
equation of the second order of the form
where Хг and X are functions of x alone, is л particular
case of the general differential equation of the second
order which is invariant under
For the above invariant equation may obviously be
written

156 ORDINARY DIFFERENTIAL EQUATIONS.
when we assume
If now we suppose "ir to have the special form
the invariant equation will assume the form of the
abridged linear equation.
A further discussion of this equation will be given in
Chapter XI.
115. To find, all differential equations of the second
order which are invariant under the (?x
Here it is readily seen that
where ф' and ф" are written for
йф сРф
dx' dx*
respectively. Thus the most general invariant differential
equation is
фу"-ф"у~^(",фу'-ф'у)=о-
If ф is an integral-function of the abridged linear
equation of the second order
that is, if ф satisfies the identity
ф"+Х^х)ф'+Хг(х)фш0;
then the general linear equation of the second order

CLASSES OF LIE'S EQUATIONS OF SECOND ORDER. Ш
will be a particular саяе of the above invariant differential
equation.
For we only need assume Ф in the form
when the invariant equation becomes
that is, since ф satisfies the abridged equation
уГ+Xlx) у'+Х
116. To find aU differential equations of the second
order which are invariant under the в1 of perspective
transformations
&4
It is seen from Art. 64 that we may here assume
u=ff, U = y.
x
, , dv xy"
so that -j- = —2_.
Thus the most general invariant differential equation of
the second order is
yl
or, as it may obviously be written,

158 ORDINARY DIFFERENTIAL EQUATIONS.
The integration of the differential equation of the first
order in»s° and v = y',
ОС
will give a result of the form
y'-*(|-o)=°-
This equation must, of course, admit of Uf, and hence
its general integral, and thus the general integral of
F = 0, may now be found by a quadrature.
Example, Given the differential equation
xyf-xy-*+yy'=Q.
This is obviously an equation of the form F=0. Hence it may be
written in the form ,, , ,
and, in fact, we have JHOL -5^-=
Henee *?-Л ¦
du и
that is, v=cxu,
A quadrature will now give the general integral required in the
form y=c,.f.
117. The values of и and v for the groups
3/ , 3/ Э/ Э/ Э/ Э/ ,Э/ Э/
уЪаГ Ъу1 дх УЪу' Ъу' Удх> Х Ъх^ Уду'
are given Arts. 66-73. It will be a valuable exercise for
the reader to find the corresponding invariant differential
equations, or Lie's equations, of the second order.

CLASSES OF LIB'S EQUATIONS OF SECOND ORDER. 159
118. The criterion that a given differential equation of
the second order,
Я(я. у, tf, y") = 0,
shall admit of a given twice-extended Qv
is that the expression
«3fi , ЭП , ,3Q , „ЭП
shall be zero, either identically or by means of Q = 0.
It is often possible to find, from this condition, the G1 of
which the given equation admits, as is best illustrated by
an example.
Example. The condition that the equation
shall admit of Uf, is that the expression
ehall be zero identically, or by means of ii=O. On comparing this
expression with ?2 =0, we see that for ^sO(>| =y, and thus rj' =tf,
The given equation must therefore admit of ут4-, and hence be one
of the type УЧ У-— Ф[х. —)=0 :
уЯ \. ' у) '
and, in fact, it may be written
-"''*-'--"'' a
Now, Art. 114, assume u=x, v = -,
and the last equation takes the form
dv yt(Z + uv)
du 1 — uv '

160 ORDINARY DIFFERENTIAL EQUATIONS.
an equation of the first order which may be integrated, Art. 67, by
a quadrature. Since the resulting equation will admit of the Gv
a second quadrature will give ua the general integral of the given
differential equation of the second order, ?2=0.
119. We shall apply the theory of integration of this
paragraph to an example involving geometrical con-
considerations.
A family of curves in the plane is often defined by an
equation expressing a relation between such magnitudes
as the subtangents, the radii vectores, the perpendiculars
from the origin on the tangents, etc., giving rise by that
means to a differential equation of a certain order.
For example, suppose it is required to find all curves
which are denned by a relation between the line r
connecting the point (x, y) with the origin; the angle ^
between this line and the radius of curvature, p; and
tlf Sh lti i i
the radius of curvature itself.
by an equation of the form
p;
Such a relation is given
It is geometrically evident that such a curve will pass,
by means of a rotation, into a congruent one. In other

CLASSES OF LIE'S EQUATIONS OF SECOND ORDER. 161
words, the family of curves represented by Ф = 0 admit
of the в1 of rotations
where, of course, the values of r, \Js, and p, in ф = 0, are
to be expressed in Cartesian coordinates.
Since, in the variables x and у, Ф = 0 will be a differ-
differential equation of the second order, and since we know
the path-curves of Uf, it is clear that, by Art. 108, the
above family of oo2 curves may be determined by the
integration of a differential equation of the first order,
together with a quadrature.
120. In integrating the following differential equations
of the second order, the equations should first be examined
to see whether they are exact or not, Art. 103. If an
equation is exact, ita integration may, Art. 103, be
reduced to the integration of a differential equation of
the first order.
If the differential equation of the second order is not
exact, it should be compared with the types of invariant
equations, Arts. 110-117 ; and if the equation belongs to
one of those types, its integration is to be accomplished
by the method indicated for that type.
If the equation is not exact, and does not belong to
one of the given types, the Gx of which it admits is to be
sought by the method of Art. 118.
. In examples (l)-B5) it should be verified geometrically,
whenever practicable, that the family of oo2 integral
curves found admit of the Gx which was used in
integrating the differential equation. Thus, it is geo-
geometrically evident that the integral curves
of Ex. 26, which are the oo2 circles through the origin,
admit of the G1 of rotations.
We give, for convenience of reference, a table of some
of the more important invariant differential equations of
the second order.

162 ORDINARY DIFFERENTIAL EQUATIONS.
Group of One Parameter. Type of Invariant Deferential Equation.
A) Щ=% D*4* •./)=<>.
This invariant equation includes, of course, the types
1\2лУ)=о, F(y,/)=o.
B) Щ=Щ B)Г^,у',/) = 0.
This equation includes
EXAMPLES.
• B) (l-
C) y>-««". D) y-
=1+У!. A0)jf*=jf"+l.
B/'»+l). A2) яу
A3) аУ+/=О. A4) У-Я/" =/(/).
A5) (l+*V+l+ya=O. A6) A -^уч^
A7) .y'+yW. A8) (»!-^J/--^
A9) У'+да'=0. B0) y(l-logy)y
B1) /=J. B2) jy-;K,''+y
B3) !?-!}¦-Зу-О. B4) xif'+xyf-
B5) xy-xi/+y = 0.

CLASSES OF LIE'S EQUATIONS OF SECOND ORDER. 163
The following geometrical examples lead to ordinary differential
equations of the second order which admit of a Gv The formulae
for a plane curve given, Art. 73, together with the two given below,
are convenient for reference.
Differential of an arc s, measured from a fixed point on the curve
up to a point (x, y),
da
Radius of Curvature, T
B7) Determine a curve such that the length of the arc measured
from a fixed point, on it is equal to the intercept of the
taugent on the axis of x.
B8) Show that the curve whose radius of curvature is proportional
to the cube of the normal is a conic section.
B9) Required the family of curves in which the radius of curvature
is constant and equal to a.
C0) Determine the family of curves in which the radius of curva-
curvature is equal to the normal (a) when the two have the same
direction, F) when the two have opposite directions.
C1) Determine the family of curves in which the radius of curva-
curvature is equal to twice the normal (a) when the two have the
same direction, F) when the two have opposite directions.
C2) Show that if т is the angle which the tangent at any point
(x, y) on a given curve makes with the ж-ахда (see figure,
Art. 119), while p is the radiua of curvature at that point,
all curves defined by a relation of the form
may be determined by the integration of a differential
equation of the first order and a quadrature.
C3) If the angle between the дг-axisand the line joining tiie centre
of curvature with the origin be designated by $, and if ф
have the meaning of Art. 119, show that all curves defined
by a relation of the form
?2(ф,т, в)=0
between the three angles <A, т, в may Ъе determined by the
integration of a differential equation of the first order and a
quadrature.

CHAPTER X.
THE DIFFERENTIAL EQUATION OF THE m'h ORDER
IN TWO VARIABLES.
121. In this chapter we propose to indicate briefly
how the methods of Chapters IV. and IX. for integrating
invariant differential equations of the first and second
orders may be extended to equations of an order higher
than the second.
SECTION I.
Lie's Differential Equations of the mtK Order.
122. A differential equation of the m1*1 order in two
variables has the general form
where Q must, of course, actually contain ^<; and the
complete primitive, or general integral, (Art. 5), of this
equation has the general form
Щъу.ъ, ...,cm) = 0,
where съ...,ст are m independent arbitrary constants.
We saw in Art. 72 that the general integral of an ordinary
differential equation of the first order in two variables may always
be expressed in the form of an infinite series involving oue arbitrary
constant. We shall now show that the general integral of the
ordinary differential equation of the ma order in two variables
may, similarly, be expressed as an infinite series involving m

LIE'S EQUATIONS OF THE тл QBB&R. 165
arbitrary constants—although, as in Art. 72, we shall not investi-
investigate the question of whether this series always converges or not.
It may be remarked that thia method for obtaining the general
integral is of little practical value for such equations as are neither
linear (Chap. XI.) nor reducible to a linear form. If
y-=F1(«,3,,y,...y~-") A)
be the given differential equation, by differentiation we find ym+1)
as a function of x and y, and the differential coefficients up to the
wi, if F2 actually contains yfn-i>. Substituting, in the result, for У"
its value as given by A), we find for ym+l),
y(-+1) = Fl(tf)y)y',...y—'>) B)
Differentiating B), and reducing as before, we find
y-+*=.F3te у, у', ...У™') ; C)
and proceeding in this way we see that all differential coefficients of
у of an order higher tban the mlb are expressible in terms of
xt У-i У\ •••> У<т~ь> by means of A).
Now when we assign to x the initial value л*0, let the correspond-
corresponding initial values of у, у1, ...,У™~1)Ье represented by Уо>Уо'г ,..,W""",
while the second members of (l), B), C),... become FJ°\ Fa@), F3@>,...
respectively ; then, by Taylor's theorem, we find, as in Art. 72,
'*-- w
In this expression for the general integral of A), x0 is a special
numerical value of x; and, as the Ft@> are functions of the m
arbitrary constants y^ /0, ...уо'', D) really contains only m inde-
independent arbitrary constauta.
123. We shall now give a method for finding all
fil i f b
are
123. We shall now give a method for finding
differential equations of the mtb order, Q = 0, which
invariant under a given G1# Such equations are some-
sometimes designated as " Lie's equations o/ the mlk order."
In order to find all differential equations of the second
order which, are invariant under a given Gx

166 ORDINARY DIFFERENTIAL EQUATIONS.
we saw that, if
!4
represent the twice-extended Gv it is necessary to find
three independent solutions of the linear partial equation
in four variables
U"f=0.
If these solutions are represented by
«(re, y), v(x, y, y'), ^ = w{x, y, y', y"),
then the most general invariant difFerential equation of
the second order is
F(u,v>w) = 0.
In an entirely analogous manner we may see that
to find the most general differential equation of the third
order which is invariant under Uf, it is necessary to find
four independent solutions of the linear partial differ-
differential equation in five variables
where JJ'"f is the thrice-extended Gv so that
Just as we saw in Art. 107 that we could assume
dv
du
so now, if u, v, w be three solutions of U'"f=Q, the
function
dw _ d2v
du ~ dv?
may be seen to be a fourth solution of U"'f=0, which
must contain y'". Hence the most general invariant

LIE'S EQUATIONS OF THE тл ORDER 167
differential equation of the third order has the form
-,/ dv d2v\ .
\ du' duV
Unless the path-curves of the Gx
и = const.
are known, it will be necessary to integrate a differential
equation of the first order in two variables
dx _dy
to find n. Then v. Art. 56, may be found by a quadrature,
and of course the other two solutions
d^v_ <Pv_
Ш? dv?
by mere differentiations.
It is obvious now that the most general invariant
differential equation of the mth order will have the form
™/- dv d2v dm-4\ л
and it is clear that, in the most unfavourable case, this
differential equation may be established by the integra-
integration of a differential equation of the first order in two
variables, a quadrature, and (то — 1) differentiations.
Since и and v are given, Arts. 62-68, for the Gl of those
Articles, of course the corresponding invariant differential
equations of the mth order may be found, as indicated
above, by mere differentiations.
In accordance with the definitions of differential in-
invariants of the first and second orders, given Arts. 57
and 106, we now define the function
d*v
du1
as a differential invariant of the (i + l)th order of the
exu/.

168 ORDINARY DIFFERENTIAL EQUATIONS.
124. By Arts. 42 and 43 it is clear that the necessary
and sufficient condition that a given differential equation
of the mth order,
shall be invariant under a given G1 Uf, is that the
expression
shall be zero, either identically, or by means of Й = 0,—
where
t/ i™Vf — > —*- + и~^^*т~ и —^Ч-f" ... -г Я — —
J о% ay oy oym
is the m-times extended 0v
In order to reduce the problem of integrating 12 = 0,
we must first find u, by integrating, if necessary,
dx_dy
T~T'
Then v may be found by a quadrature; and the above
differential equation of the mth order may be put into
the form
t,/ dv dtn~1v\ л
or, if we choose,
dm~4 ,/ dv dm~4\ л
, .— Ф(гь, v, -т~, ... -j -I —0.
This is a differential equation of the (m— l)th order in и
and v. If its general integral has been found in the
form
v~f(u, cu c2> ... cm.1) = 0,
the last equation will be a differential equation of the
first order in x and y, which, by Art. 42, must admit of
the given Ог Uf; and hence the general integral of
fi — 0 may be finally found by a quadrature.

CLASSES OF LIE'S EQUATIONS OF m* ORDER. 169
SECTION II.
Glasses of Lie's Differential Equations of the m(A Order.
125. We shall now illustrate the method given in the
last section for finding classes of invariant differential
equations of the mtb order by some of the simplest
possible examples.
126. To find all differential equations of the mth
order which are invariant under the Gx of translations
along the x-axie,
By Art. 62, we have
. dv y"
"' "' du у
dw_d*v_y'"y'-y"*
ш=дмг ^ ¦etc-
Thus the most general invariant equation of the
order may obviously be written
V' а'>у" y3
or, in the equivalent form,
Hence, an equation of the тш order, ?2 = 0, which ia
free of x, admits of the Gv TJf\ and may always be
written as an equation of the (m — l)th order in the
variables u = y,v = y', in the form
_,/ dv dm~1v\ л
Flit, v, -j-, ... ,-;йгт) = 0.
\ du du /

170 ORDINARY DIFFERENTIAL EQUATIONS.
127. To find all differential equations of the mth order
which are invariant under the G1 of all translations
along the y-axis,
Here, by Art. 62, we have
, dv „
u = x, v = y ; w = -,- =y .
13 ' du У *
. dw d% ,„
s5ss» ¦etc-
Hence, in this case, the most general invariant differential
equation of the mth order haa the form
and it is obvious that it may be written as an equation
of the (m — l)tt order in the form
F[u, v, -j-, ... t т =0.
V ' ' аи' dum-'J
128. To fond all differential equations of the mth order
which are invariant under the G1
Here we have, Art. 63,
у dv, у у \у /
dw d2v у'" оу' у" n/y'\s
du~du2 = Y~ У У ^У' '
Hence the most general invariant differential equation
of the mth order may be written
{Х>У'~У~Ч''~У~ У^У+ \y)"")~ '

CLASSES OF LIE'S EQUATIONS OF ma ORDER. Ш
or in the equivalent form
¦\ у у у vs
Thus every equation of the form Q = 0, of the mth
order in x and y, may be written in the form of an
equation of the (m —1)°* order in и and v, by making
the substitutions indicated above.
It is seen that the so-called abridged linear equation
of the 77ith order, of the form
is a particular case of the equation Q = 0.
129. To find all differential equations of the mth order
which are invariant under the GY
Here, by Art. 115, we have for и and v the forms
, . dv ,
u = x, У=фу-фу; ^=ФУ ~ФУ,
а^^ФУ ~Ф У + ФУ -ФУ, etc.
Hence, the most general invariant differential equation,
of the tn}h order may be written
F (ж, фу' - ф'у, фу" —ф"у, фу'" - ф'"у + ф'у" - ф"у\ ...) = 0.
Any differential equation of the mth order of this
form may be written as an equation of the (m — l)th
order in the variables и and v by making the substitu-
substitutions indicated above.
It may be noticed that the general linear equation of
the mth order

172 ORDINARY DIFFERENTIAL EQUATIONS.
is a particular case of the equation F*=0, if d> is an
integral-function of the corresponding abridged linear
equation. The proof is precisely analogous to that of
Art. 115 for equations of the second order.
130. It will be very valuable exercise for the reader
to find the differential equations of the 3rd, 4th,... orders,
which are invariant under the simple G/s given in
Art. 117.
131. It may be noticed that the simplest form of
differential equation of the mth order that is invariant
under the Q,
is y№ = X{$), A)
where X is a function of x alone. It is obvious that the
general integral of this diiferential equation of the mth
order may be found by w successive quadratures.
132. It is clear that when the equation of the type of
Art. 126 has the special form
its integration may be facilitated by assuming jfi ~ z.
The equation ?2 = 0 is then of only the (m — i)®1 order in
the variables x and z,
~/ dz dm~iz\_t\
\ ' dx' '" dxm~ / '
and is of the type of Art. 126 still.
Its integration will give a result of the form
so that, by the preceding article, the general integral
required may now be found by i successive quadratures.

CLASSES OF LIE'S EQUATIONS OF m" ORDER, 173
This method is particularly applicable when Я = 0 has
one of the simple forms
O^»)-/^—4)=0, B)
or Пэ^-Д^е-4) = 0 C)
The first equation, when we put jfm"^ = z, becomes
dz ,, .
and may be integrated by a quadrature giving z as a
function of x and one constant.
The second equation, when the same substitution is
made, becomes
/72,.
for the integration of which a method is given in Art.
110, by means of which z is found as a function of x and
two constants.
Example I. Given y"y"'
his ia an exam
у" = 2, and we fin
This ia an example of equation B), Art. 132. Hence, assume
" d fid
whence x=c + */l+ г2. (с = const.)
Thus, solving for 2, we have
By two successive quadratures the general integral may now be
found.
Example 2. Given aye=/'.
Tliis is au example of equation C), Art. 132. If we write г for/',
the equation becomes аъ^' = г
By Art. 110, we find from this

174 ORDINARY DIFFERENTIAL EQUATIONS.
Hence,-from ?/"—с1ея + й2е ",
by two successive quadratures, we find the general integral
у = cxa4a + c2a2e а-\-с^с-\-с^
EXAMPLES.
Integrate the following differential equations :
A> чГ=1 B) af=y'. C) f=^±-^
D>у=жсоял E) ж=у=2у". F) e^" + 4cos^ =
G) У*=xf. (8) f=eiiAe. (9) /" = 1 + cos x
A0) У"=УA+У). A1) Щ"У=у"*-а>. A2) /"/'=!.
A3) д;/ч-ЗУ"=О. A4) УУ=A -

CHAPTER XL
THE GENERAL LINEAR DIFFERENTIAL EQUATION
IN TWO VARIABLES.
133. In the first section of the present chapter we shall
give a method for finding, by mere algebraic operations,
the general integral of the ordinary linear differential
equation of the mth order with constant coefficients and
the second member zero.
In the second section we shall give methods for the
same equation, when the second member is not zero.
In the third section we shall show how the knowledge
of the fact that the general linear differential equation is
always invariant under a known Gx may be used to lower
the order of the equation.
SECTION I.
The Abridged Linear Equation of the mth Order; with
Constant Coefficients.
134. The differential equation of the mth order of
the form
y^+Xm^{x)y^-^+ ...+X1(x)y = X(x) A)
is known as the general linear differential equation of
the 771th order. We reserve for the third section a
treatment of this equation when the Xi are functions of

176 ORDINARY DIFFERENTIAL EQUATIONS.
x; for the present we assume that the Xi are all
constants, so that A) may be written
y™+Am-iyb»-V+:.+Aiy = X(x) B)
The last equation is known as the general linear
equation of the mth order with constant coefficients. If,
in particular,
X(x) = 0,
the equation B) becomes
yfw+Am.iy«»-»+... + Aiy'+A1y = O C)
which is called the abridged linear equation corresponding
to B). We shall discuss equation C) in this section.
135. Although, as we know from Art. 128, C) is
invariant under the вг
%
so that the integration of C) may be reduced by the
method of that article, a more expeditious way of finding
the general integral, and one not involving any processes
of integration, will now be explained.
To this end, substitute in C) for у the value
where a is a constant to be determined. It is seen tnat
each of the terms of C) wUl be multiplied by the factor
e°*, which may therefore be discarded, so that we have
This is an algebraic equation of the mtt degree in
terms of a; and for each root, сн, of this equation, it
is clear that we have a corresponding particular integral
of C) of the form
Thus, if av av ... am be the m roots of D), the equation
E)

ABRIDGED LINEAR EQUATION OF m* ORDER. 177
will be the general integral of C), Art. 122. For this
equation contains m independent arbitrary constants,
and the value of у given by E), when substituted in C),
satisfies C) identically.
Example. Given the abridged linear equation of the second
order> y
The corresponding algebraic equation of the second degree is
аг-5я + 6 = 0
with the roots 2 and 3. The general integral of the differential
equation is therefore, by E),
That this is correct may be immediately verified.
136. In the case when D) has a double root, say
the equation E) no longer represents the general integral
of C). For in that case the first two terms in E)
reduce to the form
where c1+f2 may obviously be replaced by a single
arbitrary constant c; and since E) now only contains
(m —1) independent arbitrary constants, it is no longer
the general integral of C).
In order to obtain the general integral, let us suppose
that the above-mentioned two roots are not exactly
equal, but that they differ by a quantity к, which will
ultimately be made to vanish. The part of E) depending
upon the roots a^ and a2 will then have the form
F)

178 ORDINARY DIFFERENTIAL EQUATIONS.
Since ct and c2 are arbitrary, we may assume them to
be infinite in such manner that, as к approaches zero,
C# approaches a finite quantity Вг, while Cj and c2 are
taken with opposite signs, in such manner that f^+Cg is
finite and equal to Br Thus the sum F) has the form
so that, when л: = 0, F) becomes
Thus we see that in the case when D) has a double
root ax = a2, the arbitrary constant (Ci + C^) must be
replaced in E) by a binomial expression of the form
In an entirely analogous manner it may be shown
that if D) has an T'-fold root, the т terms coalescing in
E) must be replaced by a polynomial of the form
Example. The algebraic equation corresponding to
yy4y
has the double root 3. Hence the general integral of the differ-
differential equation is B B)
137. When D) has a pair of imaginary roots, the
corresponding constants of integration are to be assumed
imaginary in order that the pair of terms in E) may be
reduced to a real form. Thus, if
Ье а pair of imaginary roots, the corresponding terms in
E) are
= eax [(Л+c2) cos /Зж + s/^1. (cj - c2) sin

LINEAR EQUATION OF m* ORDER. 179
К now cl and c2 be considered imaginary, and if we
assume
the real form sought will be
ea*(^cos/3;c+.Bsin/&<;) G)
It is readily seen, as in Arfc. 136, that if a pair of r-fold
imaginary roots occurs in D), each of the arbitrary con-
constants in G) must be replaced by a polynomial of the
(r_ l)th degree of the form
Example. Given y
The corresponding algebraic equation,
а2-6а+13=0,
has the pair of imaginary roots,
Hence, by G), the general integral is seen to be
у = <?%A cos 2x + В sin 2x).
SECTION II.
The Linear Equation of the ma Order, with Constant
Coefficients and the Second Member a Function of x.
138. The problem of finding the general integral of
the equation
is intimately connected with that of finding the general
integral of the corresponding abridged equation
0 B)
For suppose that the general integral of B) has been
found in the form
C)

180 ORDINARY DIFFERENTIAL EQUATIONS.
and that ф(х) is any function of x which satisfies A),—
then it is clear that
у^с1еЛ'+с2е'^+... + с„е^'+ф(х) D)
is the general integral of A). For, if we denote the
second member of C) by F, the result of substituting
D)
in A) will be equal to the sum of the results of substituting
y у ф()
successively. The first of these results will be zero,
because Y satisfies equation B); the second result will
be X(x), because ф(х) satisfies A). Hence D), which
contains m independent arbitrary constants, must, Art.
122, be the general integral of A).
We shall call the function ф(х), which does not contain
any arbitrary constant, a particular integral-function of
A); the function Y is usually called the complementary
function.
Thus, when a particular integral-function of A) is
known, the general integral of A) may be found by
adding this function to the general integral of B).
139. There are numerous methods for finding the
general integral D); and probably the most elegant is
that based upon the theory of Transformation Groups.
But this method will have to be reserved for a later
occasion, since its application presupposes a knowledge
of groups of more than one parameter.
The method which most naturally suggests itself is, by
differentiation and (if necessary) elimination, to derive
from A) a differential equation of the (m+7i)th order in
which the second member is zero. The general integral
of this equation, found by Art. 135, will contain (m-\-n)
arbitrary constants, and will be of the form

LINEAR EQUATION OF m" ORDER. 181
where the B% are arbitrary constants. But this general
integral of the differential equation of the (m-\-n)th order
must necessarily include the general integral of A);
and since the complementary function of A) has the
form C), m of the bi above must coincide with the m a,i
in C). Thus the algebraic equation of the (m+n)
degree of which the bi are the roots, Art. 135, has m
roots in common with the algebraic equation corre-
corresponding to the abridged equation B). Hence, when the
Щ have been found, the n bK in E), which do not coincide
with the ait may be found by solving an algebraic
equation of the n™ degree only.
Suppose now that b1 = av ... bm=am; then the n arbi-
arbitrary constants Bm+i, ...Bm+n in E) are superfluous for
the general integral of A); and these n constants may
be so determined, by substituting the value of у from E)
in A), that A) will be satisfied. Since the value of у
given by E) must satisfy A), no matter what values the
Si, ... Bm have, we may facilitate the determination of
the Дп+i,... Bm+n by assuming Вг = 0г... Bm = 0.
Example 1. Given y"-nfy^z+l F)
By differentiating twice we find
^_wy=0; G)
and the algebraic equation corresponding to G) is
*-nW=0 (8)
The algebraic equation eorreaponding to the abridged equation
of which the roots are a = n, a=~-n. Hence—as in this case is
evident anyhow, a = n and a= —n are two roots of the equation
of the fourth degree (8). The other roots are obviously a = 0
repeated twice. Hence, Art. 136, the general integral of G) is
while the complementary function of F) ia

182 ORDINARY DIFFERENTIAL EQUATIONS.
ThuB we must be able to determine S3 and Bt in such manner
that В3+В&
will be a particular integral-function of F). Substituting
in F), we find
and hence Bs s Bt = - i.
Thus the general integral of F) is
У
Example 2. Given
y" + Syl + Wy=4ez-ea (9)
By differentiating twice we may eliminate e* and e2*, and find
у"+by1" - 6У' - 32y-+32y=0.
The corresponding algebraic equation is
а4 + 5а3-6а!-32а + 32=О; A0)
and this equation has two roots in common with
of which the roots are —4 and —4. Hence the remaining two roots
of the algebraic equation of the fourth degree may be determined
from the equation which results when A0) has been divided by the
factor (a+4f, that is, from
a!-3a + 2=0.
The other two roots of A0) are therefore a=l, a=2.
The general integral of (9) has therefore the form
so that it must be possible to determine U3 and BA in such manner
is a particular integral-function of (9).
By substituting y=B&+B^
in (9), we find
tbatis,

LINEAR EQUATION OF m* ORDER. 183
Thus we find for (9) the general integral
140. A second method for finding a particular integral-
function of A) is that which is commonly known as the
" Variation of Parameters." To find ф{х) (Art. 138), by
this method, we first find the general integral of B) in
the form C); then, considering the m arbitrary constants
as variable parameters, by substituting the value of у
given by C) in A), we determine the parameters in such
manner that A) is satisfied.
The m parameters may evidently be subjected to
(m—1) arbitrary conditions; and the system of con-
conditions which produces the simplest result is that which
demands that all the derivatives of у of an order lower
than the mth shall have the same values when the
parameters are considered as variables that they have
when the parameters are considered as constants.
Example. Given y" + n*y=X(x) A1)
The general integral of the abridged equation corresponding to A1)
18 у =¦ cx coa nx + e2 sin nx. A2)
Now, supposing Cj and c2 to be variables, we wish to determine
these quantities in the simplest manner possible, so that A2) will
be the general integral of A1). Differentiating A2), we have
y'= ~ ncx Bin 7ix+nc2coa7ix+сов nx-~+smnx-^ ;
and thus, in order that y' may have the same value as if cx and c2
were constants, we muat have
^ A3)
Also, differentiating the equation
у = — nct sin nx+nc2 coa nx
again, we find
y"= — п\сг cos nx + c2 sin nx) — n sin nx -^ + я coa nx -j^-
In order that у ahall satisfy A1) we muat have, therefore,
A4)

184 ORDINARY DIFFERENTIAL EQUATIONS.
and from A3) and A4) we find
-n^ = Xsinnx, n^ = Xcosnx.
dx dx
Hence, by two quadratures,
Ci=—IXsinnxdx + fti, c2 = - lXcoanxd% + as ;
and the general integral of A1) is
y= --cosTixl Xsin.nxdx + -ainnxl Xcosnxdx + alcosnx + a2sinnx.
It will be seen that the same result may be obtained directly by
Art. 146.
141. It should be noticed that all equations of the
form
) = О A5)
may be transformed into linear equations with constant
coefficients by the simple substitution
а+Ъх = е\
t being the new independent variable.
If, in equation A5), the constant a happens to be
zero, A5) is called the general homogeneous linear
equation of the mth order.
Example. The equation
when we assume a+f)X=ee
becomes linear with constant coefficients. For we have
so that the above equation becomes
an equation of the form B),

GENERAL LINEAR EQUATION OF m" ORDER 185
SECTION III.
The General Linear Equation of the mih Order in which
the Coefficients are Functions of x.
142. It was shown, Art. 129, that the linear equation
of the mtb order
3/»+Xm-I(%(»'-4+...+X1(%+Xo(:c) = 0, ...A)
where the Xi are functions of x alone, is invariant under
the Gj
if ф is any function of x which satisfies the abridged
linear equation corresponding to A)
у = 0 B)
We shall call ф(х), under these circumstances, a particular
integral-function of B).
143. In order to depress the order of equation A), we
know, Art. 129, that we must make the substitutions
, v , ф' „ 1 dv , ф"
~ i^_«^ ф «Г te
y ~~j>dv? фЫи+~4?У+ ф у' etCl
Now all the terms in A) which will contain у explicitly
when the above substitutions are made, are evidently
included in the expression
у{фт+Хт_1(и)ф«п-1'>+...+Х1(и)ф};
and, since ф is a particular integral-function of B), the
above expression is identically zero. That is, the equation
of the (m— 1)ш order, which corresponds to the equation
A) in the variables и and v, is also linear. Hence, if a
particular integral-function, ф(х), of the abridged linear

186 ORDINARY DIFFERENTIAL EQUATIONS.
equation B) is known, the general linear equation A)
will admit of the Q^
and will be linear, of the (m —1)"> order, when written in
the variables и and v by the method of Art. 129. Thus
the complete integration of A) may, in this case, be
accomplished by the integration of a linear differential
equation of the (m — 1)" order and a quadrature.
144. If, in particular, m = 2 in equation A), we see that
the general linear differential equation of the second
order
= O A')
may Ье completely integrated by two quadratures, when
a particular integral-function, ф(х), of the corresponding
abridged equation
з/Чадг/+вд^о B')
is known.
Written in the variables и and v, Art. 129, (Г) becomes
or, since ф satisfies B'),
X2, Xo, and ф being expressed as functions of u.
The general integral of this linear equation of the
first order is, Art. 68,
or, restoring the variables x and y,

GENERAL LINEAR EQUATION OF mih ORDER 187
The general integral of this linear equation of the first
order, that is, the general integral of (!') is, Art. 68,
145. The abridged linear differential equation of the
mth order B) admits of the Gt
so that the order of B) may always be depressed by
unity by the method of Art. 128; but the resulting
differential equation of the (m —1)№ order in the vari-
variables и and v is usually not linear.
146. If, in particular, we assume that B) has constant
coefficients, and is only of the second order, of the form
y"+Ay'+By = 0, (А, В const.)
written in и and v, by Art. 128, it becomes
an equation of the first order which may obviously be
integrated by a quadrature, when another quadrature
will give the general integral of the differential equation
of the second order. In this manner the particular
integral-function ф(х) may be found,—that is, by ascrib-
ascribing any numerical values desired to the two arbitrary
constants in the value of у found. Also, ф(х) may be
found by Art. 135.
When <p(x) is known, the general linear equation of
the second order, with constant coefficients
y"+Ay'+By+X(w) = 0,
may, by Art. 129, be written as a linear differential
equation of the first order in и and v. The last equation
may be integrated by a quadrature, Art. 68; when a

188 ORDINARY DIFFERENTIAL EQUATIONS.
second quadrature will give the general integral of the
general linear equation of the second order with constant
coefficients.
Hence, since ф{х) may always be found, by Art. 135, by
algebraic operations, we see that the linear differential
equation of the second order with constant coefficients
may always be integrated by two quadratures.
For practical work, however, the method of Art. 139
will usually be found more advantageous.
Example. Given the differential equation
y" + n2y=cos nx. C)
It is seen that sin nx is a particular iutegral-f uuctiou of the abridged
equation у" + я8#=0.
Heuce the above differential equation of the second order admits
of 7,.
Uf-smmxSf;
Oy
and to depress the order of the equation we have, Art. 129, to
substitute in C),
, v ,, 1 dv „
x = u, у = — \-nuotnxy, у — — j—vry.
' 3 sniTiic 3i э вшил? du 3
We find ~Л— ~=соъш,
siu пи du
or dv=si n nu cos nu du.
Hence V=^T + Clt ^Cl =const^
or, since v = sin пхч/ - n cos nxy, u = x,
, . , sin2n^
we have^ siu nxy — n cos nxy =—~ 1- cv
The integral of this linear differential equation of the first order
inay, by Art. 68, be found by a quadrature in the form

GENERAL LINEAR EQUATION OF m" ORDER. 189
EXAMPLES.
Integrate the following abridged linear equations with constant
coefficients :
A) y"-ty + 12y = 0. B) 3f- 10y43j>=O.
C) /' - 4y' = 0. D) f - Чу' + 6j> = 0.
E) y"-12y" + 2Vy=O. F) у-4/
G) у"-4аЪ</+(а> + Ъ*)'у=0. (8) </"+
(9) y" + 2y"-8s.=O. A0) /"-з
A1) у» + 2»У + я'у=0. A2) У-ЗУ"+3«("-У=О.
Integrate the following linear equations with constant coefficients
and the second members functions of x :
A3) f-y + liy=x. A4) У'-2
A5) f-W+y'~e-. A6) f+nb
. A8) у"-з
л B0) y'"-2
Integrate the following equations by the method of Art. 141 :
B1) яУ-аУ-%=0. B2) (ж + а)у ~4(tf + a)/ +6y=O.
B3) «У'-^Ч2у-х1о8ж. B4) Bа;-1)У"+Bж-1)у'-2у=0.
Integrate the following equations by the method of Sec III. :
B5) (\-a?)f + xy'-y=x(l-x*Y. B6) f -xrf +(Ж-1)у=Л
B7) а^"+4аУ+2у=е-. B8) яУ'+2у=*.
B9) A-^)У-^'-з/=0. C0) У'-~у'+^-з.=я-1.
Other examples for practice may be found in the Examples at
the ends of Chapters IX. and X.

CHAPTER XII.
METHODS FOE THE INTEGRATION OF THE
SIMULTANEOUS SYSTEM.
147. In the first section of this chapter we shall give
briefly the simplest of the methods which do not involve
transformation groups for integrating certain forms of
simultaneous systems of ordinary differential equations.
In the second section we shall give a general method
of integration for a simultaneous system in three vari-
variables, when the equivalent linear partial differential
equation of the first order in three variables admits of a
known Gx; while in the third section we shall give
an application of the theory developed in the second
section to ordinary differential equations of the second
order in two variables.
SECTION I.
Special Methods for Integrating Certain Forms of
Simultaneous Systems.
148. In Art. 23, Chap. II, we gave a method for inte-
integrating a simultaneous system of the form
dx ___ dy _ dz
that is, we saw that when the first equation had been
integrated,—by the methods of Chapter IV.,—either x or

INTEGRATION OF SIMULTANEOUS SYSTEMS. 191
у might be eliminated from Z, so that a second integral
of the simultaneous system might be found by integrating
a second differential equation of the first order in two
variables. It is obvious, therefore, that a simultaneous
system of the above form may be completely integrated
by integrating two ordinary differential equations of the
first order in two variables.
149. The general form of the simultaneous system in
three variables is
dx Ay dz ...
T = T=Z A)
where X, Y, and Z are usually functions of all three
variables x, y, z.
We may write the ratios A)
dx_dy _dz _
X~ Z~
where X, fx, v may be either constants or functions of the
variables. If it is possible to choose X, ,«, v in such
manner that
then also \dx+/idy + vdz = O; C)
and the integral-function of the total differential equa-
equation C), if it may be found by the methods of Chap. VIII.,
will obviously also be one of the integral-functions of A).
That is, if n(a, y_ 0) = c
is the integral of C), it is also an integral of A).
The second integral of A) may then be found by inte-
integrating an ordinary differential equation in two variables
from, say,
dx dy
when z has been eliminated from X and Y by means of

192 ORDINARY DIFFERENTIAL EQUATIONS.
Example, Given the simultaneous system
dx dy dz
mx ~ny nx-h ly — mx'
The method of the present article may be applied twice. If we
choose A, ft, v equal to I, m,n respectively, we find
ldx + mdy + ndz=O.
If we choose A, ft, v equal to x, y, z respectively, we find
xdx+ydy +zds = Q.
The integrals of these equations are obviously
lx+my+m=cv
and these two eqnations are the general integrals of the given
system. For the geometrical meaning of the integrals of a simul-
simultaneous system in three variables, see Art. 19.
150. The general simultaneous system in the (n + 1)
variables xh ..., xn, t, has, as we know, the form
dxi dx2 dxn dt , .
~х1='х~2 = -=ж=т1' ( }
where the Хъ ..., Xn, T, are usually functions of all the
variables. If we choose any one of the variables, say t,
as the independent variable, it will always be possible,
by differentiating these equations a sufficient number of
times, to eliminate all but one of the dependent variables
and their differential coefficients. In fact, if no method
for abbreviating the work suggests itself, we may always
obtain, by differentiating each of the given equations
(ti—1) times, exactly n2 equations, which are just suffi-
sufficient to eliminate (n — 1) variables with their n(n— 1)
differential coefficients. The resulting differential equa-
equation of the 7ith order in two variables must then be
integrated; and from its general integral, and the system
D), the values of the other dependent variables may be
round, giving a system of general integrals consisting,
Art, 20, of n equations involving n arbitrary constants.

INTEGRATION OF SIMULTANEOUS SYSTEMS. 193
Of course this method is most appropriate for the
integration of systems of linear equations with constant
coefficients, since we then have a definite method, Art. 135,
for the integration of the system.
151. As an illustration of the preceding article, suppose
that a system of two differential equations of the first
order is given, connecting the variables x, у, and t, t being
chosen as the independent variable. To find the equa-
equation connecting x and t, we differentiate, if necessary,
both of the given equations with respect to t; thus
obtaining four equations connecting the quantities
dx dy d2x d2y
X> % Ш' W oW' ~Ш%
from which we can eliminate y, h|, -?Д. The resulting
equation will, of course, be a differential equation of the
second order in x and t.
The general integral of this equation will give x in
terms of t and two arbitrary constants; and by substitut-
substituting this value of x in one of the equations of the given
system, у may be found.
Example 1. Given the simultaneous system of linear equations,
Jf-=^uJi E)
3# — у x+y 1
These equations may Ъе written
and by differentiating the first we find,
d'x dx dy
Hence ^_3^+j;+y
or, from the first equation,

194 ORDINARY DIFFERENTIAL EQUATIONS.
The general integral of this linear differential equation of the
second order, with constant coefficients, ie found by Art. 136 to be
F)
Substituting this value of x in
wefindfory, y=(S1-S2 + S20«a 0)
Thus the equations F) and G) represent the system of general
integrals of E).
Example 2. Given the system of linear equations,
&-*+**=*, (8)
in which the independent variable t occurs explicitly. Differentiat-
Differentiating the first equation, we have
By means of the equations (8) we may eliminate у and ч*- from the
last equation, giving
By the method of Art. 139 the general integral of this equation
ie found to be
and this value being substituted into the first of the equations (8)
gives us at овсе,
^1
152. It is clear that the differential equation of the
second order,
may be regarded as equivalent to a simultaneous system

INTEGRATION OF SIMULTANEOUS SYSTEMS. 195
of equations of the first order in three variables. For we
have
,. , dx dy dy'
so that — = -? = —,—2—
1 У Ф, У, У)
In an analogous manner it is clear that a differential
equation of the mth order in two variables is equivalent
to a simultaneous system of m differential equations of
the first order in (m+1) variables.
Similarly, a simultaneous system of differential equa-
equations of an order higher than the first may always be
written as a simultaneous system of differential equatious
of the first order in the proper number of variables.
For example, if in the simultaneous system of the
second order
dP~ ' dp~ ' dP~ '
where X, Y, Z are certain functions of x, y, z, t,—we
designate by x', y', z' the differential coefficients, with
respect to t, of x, y, and 2 respectively,—the above simul-
simultaneous system may obviously be written,
dx , dy , dz ,
Ж=*- ТГУ- dl^'
dx! „ dy' v dz'
n=x' m=Y' ~m=z-
Thus the simultaneous system of equations of the second
order in four variables may be replaced by the simul-
simultaneous system of equations of the first order in seven
variables.
If the six general integrals of this system, involving
six arbitrary constants, have been found, Art. 150, the
elimination of x', -if, z' between these integrals will give
the three general integrals, involving six arbitrary con-

196 ORDINARY DIFFERENTIAL EQUATIONS.
stants, of the simultaneous system of equations of the
second order.
153. A method of integrating a simultaneous system
of linear equations with constant coefficients and of an
order higher than the first, analogous to that of Art. 150,
will be sufficiently illustrated by the following example:
Example. Given the system
lx+*»(9)
By differentiating the first equation twice we find
and from thie equation, by means of the equations (9), we find
The general integral of this equation is, by Art. 136,
and by substituting this value of x in the first of equations (9),
we find
SECTION II.
Theory of Integration of a Simultaneous System in Three
Variables which is Invariant wider a known Gt.
154. It was shown in Chapter II. that the general
simultaneous system in three variables, of the form
dx _dy _dz

INTEGRATION OF SIMULTANEOUS SYSTEMS. 197
is equivalent to the linear partial differential equation of
the first order in the same variables,
and that two independent solutions of the latter were
always two independent integral-functions of the former,
and vice versa.
Thus we may consider the linear partial equation A)
as taking the place of the above simultaneous system;
and when we speak of the equation A) admitting of,
or being mvariant under a given 0г—an expression
which we shall immediately explain—we may also, if
we choose, say that the simultaneous system admits of,
or is invariant under the given Gv The theory of
integration of this section will be developed, therefore,
for the linear partial differential equation A), using that
equation as the representative of the corresponding
simultaneous system.
155. A (?L in three variables has the general form
where f, y, f are functions of the variables x, y, z. We
say that the linear equation
—where X, Y, Z are, of course, certain functions of
x, y, z—is invariant under, or admits of the Q1 (//when,
by means of the G± Iff, each solution of A) is trans-
transformed (compare Art. 58) into a solution of A). Thus,
if w^x, y, z), w2(x, y, z) be two independent solutions of
A), using the customary symbolic method for expressing
the fact that the transformation Iff is performed upon
the function wj, the condition that A) shall be invariant
under Iff is
V(oh) = a<«i. «a) i = l, 2 C)

198 ORDINARY DIFFERENTIAL EQUATIONS.
This condition for the invariance of A) can, of course,
only be applied when the solutions <av w2 are known;
but we shall in the next article develop a condition
which is practicable when w: and w2 are unknown.
156. The expression
U(Af)-A(Uf)
has a definite meaning: it means, for the first term,
put Af in place of f, in Uf; and, in the second term,
put iff in place off, in Af. Thus it is seen
If the differentiations here indicated are carried out,
the terms involving differential coefficients of / of the
second order will cancel out. For instance ^K~% will
occur in the first term with a positive sign, and in the
fourth term with a negative sign, etc.
Thus we find the noteworthy symbolic expression
U(Af)-A(Uf)

INTEGRATION OF SIMULTANEOUS SYSTEMS. 199
But when it is remembered that
and that similar expressions hold for U(Y), A(i}\ etc,
the above identity may be written (putting for brevity
XJX for ЩХ), etc.,)
U(Af)-A(Uf)
s(UX-A()dl
Now if <av w2 be the (unknown) solutions of Af=0,
we must have
Hence also
Further, from C), if 4/=0 admits of the вг Uf,
= 1,2.
But since Wj, w2 are solutions of Af=0, this last expres-
expression must be zero identically. Thus the whole expression
U(Af)~ A(Uf) becomes zero if eoi is put in place of /,
and D) becomes
?.=1, 2 E)
Also we know that
so that from E) and F) must follow the identities,
UX-A( ITY-Aq VZ-Al
X S-~Y—S—Z—-

200 ORDINARY DIFFERENTIAL EQUATIONS.
Let the value of the ratios G) be represented by
\(x, y, z); then we may write
CX-A(=\.X, JJY-A^ = \.Y, UZ-A^ = \.Z.
Hence from D)
This then is the condition that a linear partial differ-
differential equation of the first order shall admit of a given
Qx: and it is clear that the condition may at once be
extended to n variables (including n = 2). It is customary
to write (8) in the brief form
(U,A) = \.Af, (9)
where the left-hand member of (9) is merely an abbrevia-
abbreviation for the left-hand member of (8).
It is easy to see that the necessary condition (8) is
also sufficient. For if щ is put in place of / in (8), we
obtain
A(U(a>i))^0, A0)
since the other terms in (8) vanish identically. But A0)
means that U(wi) is a solution of Af=0; that is,if (8) is
a true equation, the solutions щ must admit of the trans-
transformation Iff—that is, the differential equation Af=0
itself must admit of IT/. Hence, the necessary condition
(8) is also sufficient.
157. It is evident from (9) and (8) that every expres-
expression of the form {A, A) or (IT, IT) is identically zero, and
hence the condition (9) that the equation Af=0 shall
admit of the 0г p . Af, where p is an arbitrary multiplier,
is satisfied. But this transformation p . Af, which tells
us nothing new concerning the equation Af=0, and
which has therefore no value in the problem of integra-
integration, is said to be trivial with respect to Af=0. This
accords with the definition of Art. 60 for trivial trans-
transformation in two variables. Such transformations are
always to be disregarded in our investigations.

INTEGRATION OF SIMULTANEOUS SYSTEMS. 201
158. We shall now for the moment write Iff equal to
zero, and consider the equation Uf= 0 as a second linear
partial differential equation, and we shall show that if
the condition (9) exists, that is, if
(U,A) = \.Af, (9)
then Uf= 0 and Af= 0 may be put into forms for which
(U, A) = 0, and ultimately that these equations have one
solution in common.
When the condition (9) holds, and X is not zero, the two
linear partial equations Uf=0, Af=0 are said to form a
complete system of two members. When, in (9), X = 0,
the two equations are said to form a Jacobian system, of
two members.
For the sake of symmetry we shall assume the two
linear partial differential equations of the first order in
the forma
and shall merely assume that they fulfil a relation of
which (9) is a particular case, that is, we shall assume
that the relation
(Av A^Pl(x, y, z)AJ+plx, y, z)AJ A1)
exists.
If pl = 0, the condition A1) is identical with (9).
As far as the maintenance of the condition A1) is
concerned, we shall see that a condition of the form A1)
must still hold when the equations Alf=0, A^f—0 are
replaced by any equations which are consequences of
these two, as
where Xi, цк are arbitrary multipliers, whose determinant,
however,
must evidently not be zero.

202 ORDINARY DIFFERENTIAL EQUATIONS.
Hence
(AJ, Аз/ХХ^ + Х^, ^
111 аЛI/11^42^ч/
Since we know that (Аг, А1)=(А2, A2)=0; and since
(as is easily verified) (A2, Аг)= ~(AV A2), while the four
last terms are affected by coefficients which are functions
of x, y, z, it is clear that (A J, AJ) is an expression which
is linear in_terms of AJ and AJ', that is, by_means of
A2), (AJf A2f) is linear in terms of AJ and AJ,
Thus, as far as the relation A1) is concerned, it is
certain that the equations A1f=0, A2f=O may be re-
replaced by any equations of the form AJ=0, A%f=0, as
given in A2).
Let us therefore take the two linear partial equations
in the forms,
^/-g-^w^O, AJ^-^(x,y,z)g=0. A3)
Here (Aj/p Azf) must still be capable of being written
as a linear expression in terms of A1f and A2f; but
when the operation indicated by (Av A2) is carried out,
it will be found by A3) that the result is free of
—¦ and j-: that is, in the expression
дх ду ' r _
(AvA2)STi.AJ+t2.AJ,
when A^f and A2f are chosen in the form A3), we must
have тх = т2 = 0: or,
(AJ, Z2/) = 0. A4)
Hence, if two linear partial differential equations of
the first order satisfy a condition of the form A1), they
may always be chosen in such a manner as to satisfy
the condition A4).

INTEGRATION OF SIMULTANEOUS SYSTEMS. 203
159. It now remains for us to show that if two given
linear partial differential equations of the first order
satisfy a condition of the form A4), they must have a
common solution.
If и and v be the solutions oi^A1f=0, it is known
that the most general solution of A1f=0 must be some
function of и and v of the form Q(u, v). We now wish
to determine Q in such manner that it shall also be a
solution of A2/=0. We have
and by means of the relation A4), putting и and v respec-
respectively for / in __ _ _ __
A&fl-AjA^O A4)
it is easy to see that, since A^(u) = A^v) = 0,
1А) 1&)
That is to say, A^u) and A2(v) are solutions of Ах/=0,
and are therefore functions of и and v, say,
ЛО) = <t>(u: «). 22(i;) s ^(«,, v).
Hence
AlQiu, v)) = ф(и, v)^ + ^(u, у)Щ.
The condition, therefore, that Q(u, v), which is a
solution of ^^=0, shall also be a solution of A2f=0,
takes the form
O. A5)
This is a linear partial differential equation of the
first order in и and v; and it is always satisfied by the
integral function of the corresponding system
du _ dv

204 ORDINARY DIFFERENTIAL EQUATIONS.
If this integral function be W(u, v), then W is the
common solution of Ax/=0 and A2f=0, the existence
of which was to be proved. Of course W is a function
of x,y,z\ so that
W(x, y, z) = const.
represents a family of surfaces in space.
160. We shall now return to our original equations of
Art. 158,
having proved that the existence of the condition (9),
that the equation Af~ 0 shall admit of the Qx Uf, means
that the equations A6) form a complete system—that
is, that they have one solution in common.
If W be the common solution of A6), at any point
ж, у, z on one of the surfaces TT=con8t., two tangential
directions are assigned to the point by means of XJf and
Af; and the direction cosines of these tangential direc-
directions are proportional respectively to ? y, ? and X, Y, Z,
Art. 19.
If a, jS, у be three quantities proportional to the
direction cosines of a line perpendicular to the above
two tangential directions at the point x, y, z, we have
whence,
If now dxt dy, dz represent the differential coefficients
of the variables x, y, z on the surface W= const., it
follows that the relation
must be satisfied by the coordinates of all the points on
those surfaces.

INTEGRATION OF SIMULTANEOUS SYSTEMS. 205
In other words, the common integral surfaces of the
complete system Af= 0, Uf= 0 satisfy the total differential
equation A7).
A method for integrating equations of the form A7)
has been given in Art. 100, Chapter VIII. If
W(x, y, s) = const.
¦be the integral required, we know that W will be one of
the solutions of the given invariant linear partial
differential equation.
161. Considering one of the solutions of the linear
partial differential equation of the first order in three
variables which admits of a known G, as having been
obtained, we shall now show that the other solution may
be found by a mere quadrature.
To this end let us suppose that W(x, y, z), the solution
already found, actually contains the variable z—for, of
course, it must contain one at least of the three vari-
variables—and in place of x, y, z, let x, y, W be introduced
as new variables. In these variables A/, by Art. 35,
will have the form
or, since by hypothesis. A(W)
Now eliminate z from Ax and Ay, and Af will have
the form
Analogously, we find for the transformed Uf,
UfSy(x,y, W)? + S(x,y,W)?.

206 ORDINARY DIFFERENTIAL EQUATIONS.
We see that in Af=0 no differential coefficient with
respect to W occurs at all; and Uf does not transform
this variable; hence, W plays the r6le of a mere constant,
and x and у are the only variables.
The problem has now been reduced to the integration
ofthe linear partial differential equation in two variables
Af=0, which admits of the known QV in the same two
variables, Uf But as this partial differential equation is
equivalent to the ordinary differential equation, Art. 16,
which admits of Uf the solution can be found, by the
methods of Chapter IV.. by a mere quadrature, in the
form
F_ f o(a, V, W)dy-{3(x, У>
-)a(x, y, W).S(x,y, W)-$(x,y,
W)dx
, W).y(x)y, W)'
The integration here is to be performed as if W were a
constant, and afterward the value of W as a function of
x, y, z is to be introduced.
We may sum up the results of this section as
follows: If a Imear partial differential equation of the
first order in three variables admits of a known infini-
infinitesimal transformation which is not tnvial,it8integration
may be accomplished by the integration of cm ordmary
differential equation of the first order in two variables,
together with one quadrature.
Example. The linear partial differential equation of the first
order
admits of the
since the application of the criterion (9) gives in this case
(Uf,Af) = Af.

INTEGRATION OF SIMULTANEOUS SYSTEMS. 207
Thus Uf—d and Af=0 form a complete system with a solution
which is the integral of the ordinary differential equation A7).
The latter equation will be found to reduce itself in this case to the
form
xzdx+ysdy -(a?+y^)dz = Q, B1)
when the substitutions Х=%2+у2+уг, ? = #, etc., are made.
By the method of Chapter VIII. we find at once aa the integral
of B1),
W{x, y, z) = ^+У2 = const.;
and it may be readily verified that W is really a solution of both
C/=0 and Af=0, since it is found that
Now we shall introduce x, yt and W aa new variables, eliminating
s by means of
Hence, the second solution of Af=0 is
V=
If, now, in place of W, its value—in terms of z, у, г—be put,
there results finally for the second solution,
162. A theory of integration of an invariant linear
partial differential equation in n variables—-that is, of an
invariant simultaneous system in n variables—analogous
to the theory of this paragraph for the invariant linear
partial equationin three variables, might now be developed.

208 ORDINARY DIFFERENTIAL EQUATIONS.
But a discussion, both of that theory and of the method
of integration to be employed when a linear partial equa-
equation is invariant under more than one known Gv must be
reserved for a later occasion.
SECTION III.
Second Method for Ordinary Differential Equations of
the Second Order in Two Variables.
163. The theory of integration of the last section
may be readily applied to ordinary differential equations
of the second order in two variables which admit of a
known Gv
For, Art. 152, it was seen that the differential equation
of the second order
y"-w(x, y, i/) = 0 A)
is equivalent to the simultaneous systemin three variables,
dx dy dy
which, in turn, is equivalent to the linear partial differ-
differential equation of the first order in three variables,
Thus, if the differential equation A) admits of a known
twice-extended Qx U"f, the partial differential equation
C) must admit of the once-extended Olt
that is, we must have
(V,A) = \.Af. D)

SECOND ORDER IN TWO VARIABLES. 209
Thus the condition (9) of the foregoing section is
satisfied, and the partial differential equation C) may be
integrated by the methods of that section. That is to
say, Art. 100, if an ordinary differential equation of the
second order i/n two variables,
admits of a known Glt the differential equation of the
second order may be completely integrated by the inte-
integration of an ordinary differential equation of the first
order in two variables, and a quadrature.
Example. Given
This equation may be written
and hence C) has the form
It may be at once verified that E) admits of the Gj
to which corresponds the once-extsnded Gx
so that the condition (9), Sec. 2, is satisfied, and the equations
Af=% U'f=0 F)
form a complete system.
The common solution of the equations F) must be the integral-
function of the total equation corresponding to A7), See. 2,
yrfdx-Qy-xtf)dy+xydtf=<i G)
By Art. 99, or Art. 100, the integral-function of G) is found to be
Щх,У,?/)=У2-яУ1/- (8)
P. С, О

210 ORDINARY DIFFERENTIAL EQUATIONS.
Now introduce x, «, and W—the common solution of the equations
F)—as new variables ; thus,
_
Eliminating y' from Л/ by means of (8), we find
Now Af=0, a linear partial equation in the variables x and y,
admits of Uf in the same variables: hence the second solution of
Af—d is found by a quadrature in the form
Thus, eliminating / between
^=Уя-ЧУ-в1 and ^=^.=c3;
we find
'+*
or as it may be written
mxi+ny2 = l) (m, и consts.)
which is the form of the complete integral of E).
Thus we see that E) represents the °os conic sections whose axes
coincide with the axes of coordinates ; and it is clear, geometrically,
that this family of oos curves is invariant under the Gx of affine
transformations
I
164. The simultaneous systems given in the following
Examples are simple; and, for the most part, they may
be integrated by the methods of both Sec. I. and Sec. II.
Examples A) to D) are, however, intended to illustrate
Arts. 148, 149; Examples E) to A4), Arts. 150-153;
while the remaining Examples are intended to be treated
by the method of Sec. II., after it has been verified in
each case, Art 156, that the given simultaneous system
is invariant under the accompanying Qv Examples illus-
illustrating Sec. III. may be found at the end of Chapter IX.

EXAMPLES.
EXAMPLES.
dx dy _ <&
Idx _ mdy _ ndz
m{y — z) tU(z~x) lm(x — y)'
Idx mdy ndz
211
D)
. ¦ dx _ dy _ dz
v ' -1 4 2
A0) 4^+
(П) 4^+91+44^+493,-t, 3^+7^
A2) ^+»V =
A5)- Л—
^ ; з;-з/-г+2~2

212 ORDINARY DIFFERENTIAL EQUATIONS.
B1) Verify that the linear partial differential equation correspond-
corresponding to the simultaneous system,
dxcfydz
aiX+b1y+c1z+dl
admits of a Gx of the form,
where a, ft, у are certain constants, and that therefore the
above simultaneous ayatem may usually be integrated by the
method of Sec. II.
B2) The method of Sec. II, fails for the preceding example only in
the case of Uf being trivial. For what values of the con-
constants O[, bv ..., d3 is Uf trivial ? "What are then the integrals
of the simultaneous system ?
B3) Verify that the linear partial differential equation correspond-
corresponding to the simultaneous system,
admits of the Ol
if X, Y, Z are homogeneous functions of x, y, z; and that
therefore, when Ufis not trivial, the integration of the above
system may be reduced by the method of Sec. II.

EXAMPLES. 213
B4) Verify that the linear partial differential equations correspond-
corresponding to the simultaneous systems in Ex. A6) and A8) admit,
respectively, of the GVe,
B5) Verify that the linear partial differential equation,
admits of the &v

ANSWERS.
CHAPTER I.
E) (l+^)y+3/-tan->«. (в)
G) y = vi/+y'-y". (8)
(9) «y-2ay+2y=o. (lo)
(II) ;г
A3) У" = 7з/'-вз,. A4) у»-
A5) 3^A +У») = Л A6) гУ>
A7) l+y'-to-ayj'-O. A8) ^'¦
A9) «
CHAPTER II.
A) Bufy+cy+2=0. B)A+ж)A+у)=о.
C) l*±^
E)
(в) i
G) тпг

ANSWERS. 215
A0) x2+y2 = c12; tan ^-tan^z^ const, or taking the tangent of
bot
A1) «•-3
both sides, -=c2.
CHAPTER III.
N.B.—Only such invariant points and lines as are within a finite
distance from the origin will he taken into consideration.
A) No invariant point. An Invariant is I2(,y).
B) An Invariant is 12(д?).
C) All points on the y-axis invariant. An Invariant is J2(y).
D) The origin is an invariant point. An Invariant is 12 (•-).
E) The origin is an invariant point. An Invariant ia Ш-)•
F) The origin ia an invariant point. An Invariant is Q.{xy\
G) The origin is an invariant point. An Invariant is п(х*+уг).
(8) All points on the y-axis are invariant. An Invariant is 12 (^).
(9) All pointa on the #-axis are invariant. An Invariant is 12 (У- J.
CHAPTER IT.
B) x*-

216 ORDINARY DIFFERENTIAL EQUATIONS.
A7) (y-x+lfty+x-iy-c.
A9) xy>=c(x+2t/).
B1) jr=M".
B3) ay-u
B5) y=fiz°+- .
B7) m-tan-'y-l+e.e-1"-1'
B8) t =»+! + «*.
C0) i=logtf + l+ra.
C1) Admitaof Uf=%M+y
C2) yVr+Jl-log^kt^i
C4) ,.|^Г7-»|-|.
C5) 3f={cc^ + ^ + J(-*
sinjr+c
C9) Admitaof Uf=-W-+y
D2) y>a>4oz + e (parabola).
D4)j.J = K.ic»« + o.
D6) y"=cf.
'2 л;
B0) у=ет.
B2) xy =log<?y2.
B4) ,x^ + logsin(^)=log—.
У
B6)^ = -^ + ^.
B9)i = cVb^-l.
Э/ л , /
7~f-t Ans. log^+ —= c.
+ с{33)у(зесх + Ша;)=х+с.
C6) y={c^+Iog.r+l^'.
C8) 5ya = 2sinj; + 4co8^+ce-21.
^. Ans. ^-——= = c.
D3) /) = c.(l-cos6).
D5) я^+у2 = 2аг.
D7) (#-y)a-2^=0 (parabola).
= K(x-cy (a conic). D9) Jf = |{(|)"- (*)"}•
E0) с . у" = (я- l)* + *jl E1)
E2) ,rf-y = F(x).
CHAPTER V.
A) »>+Jf»=A B)
C) у=ет. D) The system is self-orthogonal.

ANSWERS. 217
F) The differential equation is homogeneous; hence the ortho-
orthogonal trajectories are,
'©
(8) laothermal. x?+y2 + cy+l=§.
(9) The system is self-orthogonal.
A0) y=J^+eonst. A1) jr= -
A3)? = «n"fl + u A4)fl=-
A5) S = {cos-1
CHAPTBE VI.
A) (у-2« + в)(у-8» + с)=0. B) y=.o.«", y = c.e—.
C) (ay + c
E) (y
General integral, о
Singular solution, ^=0.
(8) уа=2сд7+са. Singular solution, .r3+y2=0.
(9) ж2+о(а;-Ззг)+(!!=О. Singular sralution, (х+Ьц)(!с-у)=0.
A0) д:г+у2~4сг+Зс3=0. Singular solution, Xs-3y2 = 0.
A1) y=.2+cs. Singular solution, 1+4ж"у=0.
Admits of и/=х^-2уЩ.
A2) 2у=сэ? + ~. Singular solution, y*=aa?.
A3) x±Ja'+4by=alog(a±-Ja?+4by)+c. Singulareolution,y=0.

218 ORDINARY DIFFERENTIAL EQUATIONS.
A4) Admits of и/зх^+ъЖ.
General integral,
Singular solutio
A5)
Singular solution, #=0.
A6) 4(ж+бK + (а;+сJ-18у(а;+
Singular solution, y=0.
A7) y! = 2oj; + A Singular solution, д
A8) a;+l = ±V2y+c"+log(±V2y+c-l).
A9) е!»+2ам»+<!>=0. Singular solution, x- ±1.
B0) 0{i!^
B1) (*>-y
B2) (y-^-c
B3) x* = c(y- c). Singular solution, у = ± 2л:.
B4) Admits of UfaaS^ + iyW. General integral, y = <?(i:-cf.
Singular solution, a?—16^=0-
B5) {у + с)*°=х{х-а)(х~b). Singular solution, х(х-а){х-Ъ) = 0.
B6) <? + 2oiC«y - to2) - зАу+ay=0. No singular solution.
B7) Admits of Ufsx^+iyJ. Singular solution, y=^j—
OHAPTBE VII.
A) The answer is given by A1) in connection with A2), Art. 91,
since A=l.
B) The answer is given by A1) in connection with A3), since Л=2.
C) Use A4) and A6). General integral is :
D) and E). See Art. 92.

AJfSWXRS. 219
F) y=cx+™. Singular solution, уг=тх.
G) у = ся + Vo2 + a2c2. Singular solution, —-j.3L = l.
(8) y=cr+c-A Singular solution, 27у2=4A+яM.
(9) (у-етJ=1+с2. Singular solution, x'+y'=l.
A0) y=c(x- l)-c2. Singular solution, %=(ж-1J.
A1) (у -ел;J + 4с=0. Singular solution, яу=1.
A2) (у-м)(ас-Ь)=о6с. Singular solution, I -J ±^j) =1.
A3) ж*+у*=а*. A parabola.
A4) ^+/=al
A5)^ = *^. Equilateral hyperbola.
A6) а!» = 4а(»-у). A parabola.
CHAPTEE VIII.
(i) у,+ет+ед=й B) ?+г.'=с:
F) у(*+г)=с(у+2). G) e"
CHAPTER IX.
B) о,2-2
G) (c^

220 ORDINARY DIFFERENTIAL EQUATIONS.
(9) Sj^c^+of^-i+e,. A0) с1е>=соа{х+с1).
A1) ^logajxtx-cj+cr A2) (аг+
A3) j^clogz+c, (H) у~
A5) y=o1a; + (c1!+l)log(^-Ci) + i;11.
A6) у=д;+с1{31п-1
A8) yg !!
B0) logj,,.i+_L-, B1) у-ДЧс^
B2) y= -log^-e.log^ B3) y=e,»1+
B4),
B6) а? + 2с1я+у* + 2с21/=0. ( Admits of U/^ -#
B7) ctf'-logy'-icfc+c,). B8) c,/-^(tf +
B9) (S-e,
C0) (a) f-
(b) x=c,+ci\og{y+Jy'-c11} (a catenary).
C1) (a) x+a,=o, vera-1^ - */2cty -у* (a cycloid).
F) (kx+e^f=3e1jf-<sl' (a parabola).
C2) It is geometrically evident that the family of curves admit of
the translations Vf = *J-.
C3) It is geometrically evident that the family of curves admits of
the group of similitudinoua transformations
CHAPTER X.
A) yal^ + c^ + c^+^logx. B) y=c,e°+c&: + c
C) y=c, + c^+c^+c^-fx+ay log Jx+a.

ANSWERS. 221
CHAPTER XI.
A) Jf-o^ + ojr". B) ц
C) зг-^ + ^е-^+с,. D) y
E) jf=oI«" + c^-1" + <y"*'+ll,«-<v>:
F) У = «"fa + V + c#? + 0,1s).
G) у=n«i sin (a! - 6>+», cos (a» - V)x).
(8) yeI=c1.
(9) y=c,. e
A0) y=c1e-
A1) у=(с1 +
A2) Jf=(o1 + <vr+(i3a^)«4-c4. A3) 3,
A4) y=c,sina;+CjCosa;+(»3+c,a;)e"+l.
A6) У = Сг 8^

222 ORDINARY DIFFERENTIAL EQUATIONS.
(IV) »-( ?)
A9) »=(«¦-?) »Ь2*+(<*-^<=°»**+!
B0) ^^«-«Ч-^-^)«¦«**+(<%+!?)«¦ sin*
B1) у-с
B3) y=
B5) A particular integral-function is x.
General integral:
B6) A particular integral-function is ef.
General integral:
y = e'{\e~*~(\xe~nz~dx+cl)c
B7) A particular integral-function is -.
General integral: y= +-^-^~-
B8) A particular integral-function is -.
л?2
General integral: у=с1 + с%\щх+ 4 .
B9) A particular integral-function is e'ln~4
General integral: iy=c1eBio^l*+c2ee0B~1:
C0) A particular integral-function is e*.
General integral: у = с1ех + й2.* — (nfi +

ANSWXBS. 223
CHAPTEE XII.
B) Рх+т^+пЬ^с^, рл;!+яУ + п!г! = A,.
C) b?+mt/1+Ks*=c1; Pi<+my+»?=jr
D) x+y + z^^; хуг = сг.
E) у=-2с,е—+Oje-'«i г = С1е- + сае-'-.
Зг г2 г 2г2
F) ж=с1г-4+2с!г-3+^+^ ; ^=-^2 -Vs-%)+lb'
(8) ^=(«1+V)e-*-^+g; y=-^'
A0)
.,,, , « 29e" , 19< 56
A1) tf=c1e-+c2<re'-nj-+-3---9-;
24e" lit 56
+ +
A2)
A3) ж=4
c^ + c4«-1>/? +
-- ж Й
A4) y = (c1 + cie)e"+3»3e 9-j; 2=2(Зс2-с1-сг»;)<!"-Сзе"'! - J.
A5) y + 22 = oi; 2*+y-21og(a;-y-2+l) = C2.
A6) «*+#=<!, j a;+y-log2=cs.
(IV) ar-yso,; ак+зга + ^у = С8. A8) y+j!2 = o,; ж+зв = с2.
1; у=0,6--0,2-'-|.

224 ORDINARY DIFFERENTIAL EQUATIONS.
) The G^ is trivial for
with the rest of the constants zero. The integrals in this case are