06200 - Ordinary Differential Equations, with an Introduction to Lie's Theory of the Group of One Parameter [Page, 1897].pdf

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ORDINARY DIFFERENTIAL EQ!)ATIONS.
WITH AN
INTRODUCTION
TO
LIE"S
THEORY OF THE
GROUP
OF ONE
PARAMETER
ORDINARY
DIFFERENTIAL EQUATIONS
AN ELEMENTARY TEXT-BOOK
WITH AN
INTRODUCTIOl'~
TO
LIE'S THEORY OF THE GROUP OF ONE PARAMETER
BY
JAMES MORRIS PAGE
PH,D,, UNIVJi:RSil'Y OF LEIPZIO: J.I'ELLOW BY COURTESY JOHNS HOPKINS
UNIVERSITY; ADJUNCT PROF!;SSQR OF PURB MATHEMATICS
UNIVERSITY OF VIRGINIA
:ll[:anbon
MACMILLAN AND CO.,
LIMITED
NEW YORK : THE MACMILLAN COMP Al'<I'"Y
1897
AU
right•
re.served
GLASGOW: PRINTII::O
AT TRE
UNIVERRlTV PRESS
BY ROBERT MACLi-;SOSE AND CO,
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. Ever
since the discovery of the Infinitesimal Calculus, the
integration of differential equations ha.s 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 wa.s
found possible to give theories of integration for certain
classes of these eq nations ; for instance, for the
lwmo-
genwus
or for the
linear,
differential equation of the
first order. Also, important theories for the linear differ-
ential equations of the second or higher orders, have
vi
ORDINARY DIFFERENTIAL EQUATIONS.
been developed. But all these special theories of in-
tegration 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 differ,
entia! equations.
Only a very small part of Lie's extensive and im-
portant 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," 18 7 0-7 4 ; in the
Mathematische Annalen,
Vol. II., 24 and 25; and in
his
Vorlesungen
uber
Differentialgleichungen mit
Be-
lannten Infinitesimalen Transformationen,
edited by Dr.
G. Scheffers, Teubner, 18 91. 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-
formation groups, employed in Chapters III.-V., and
IX.-XII. of this book, are due
ea:cl1UJively
to Prof. Lie.
Lie has also developed elegant theories of integration
for Clairaut's and Riccati's equations, as well as for the

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