Introduction to the Theory of Linear Differential Equations
INTRODUCTION TO THE THEORY OF LINEAR DIFFERENTIAL EQUATIONS BY E.
G. C. PODJLE Fellow of New College, Oxford OXFORD AT THE CLARENDON
PRESS 193d Oui, 1 oeuvre sort plus belle Dune forme au travail
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PRINTED IN GREAT BRITAIN PREFACE THE study of differential
equations began with Newton and Leibnitz, and most of the
elementary methods of solution were discovered in the course of the
eighteenth century. Where a problem could not be solved in finite
terms, expansions in power-series were tentatively em ployed by
Newton. But the theory was not placed on a satisfactory logical
basis until about a century ago, when Cauchy distinguished between
analytic and npn-analytic systems, and constructed rigorous
existence - theorems appropriate to each type. Ordinary linear
equations, with which this book deals, have always attracted
particular attention by their comparative tractability and their
numerous practical applications. Extensive monographs have been
devoted to many separate branches of the theory, such as spherical
and cylindrical harmonics, expansions in series of ortho gonal
functions, oscillation and comparison theorems, the Heaviside
calculus, polyhedral, elliptic modular and automorphic functions.
While some branches arose out of physical problems, others were
created by the progress of the theory of functions and of the
theory of groups. Many important ideas were first worked out in
connexion with the hypergeometric equation by Euler, Gauss, Kummer,
Rie mann, or Schwarz, and were then generalized by Fuchs, Klein,
Poincare, and many other writers of the highest distinction. The
present Introduction is based on lectures tosenior under graduates
at Oxford, and is designed for students who have already taken an
elementary course of differential equations, but have not yet
specialized in one of the more advanced branches. It is not a
compendium of this vast subject to which no single author could do
justice, but a selection of investigations of moderate length and
difficulty, illustrating those aspects of it which are most
familiar to myself. The first five chapters deal with properties
common to wide classes of equations, and the last five are devoted
to a more detailed examination of the hypergeometric equation,
Laplaces linear equation, and the equations of Lame and Mathieu. I
have not discussed systematically the equations of Legendre and
Bessel, as there are so many admirable accounts of them in English
suitable for students of every grade. On the other hand, I have
thought it well to devote a chapter to equations with constant
coefficients. I find vi PREFACE that candidates in university
examinations have great difficulty in constructing the solution of
such equations which takes assigned initial values, even when they
can write down the complete primitive. A very slight sketch of
Heavisides method should enable them to make short work of this
problem, which is of great practical impor tance. Again, the theory
of simultaneous equations with constant coefficients gives an
excellent opportunity of introducing in an easy context the notion
of invariant factors, which is of fundamental importance in the
Fuchsian theory. The short bibliography and the footnotes serve
both to acknow ledge my debt to the authorities and to guide the
more ambitious reader. Besides some of the great classical memoirs
and thesystematic treatises of Forsyth, Heffter, and Schlesinger,
the books from which I have learnt most are Kleins lectures on the
icosahedroii and on the hypergeometric function, the masterly
summaries of the general theory in the works of Goursat, Jordan,
and Picard, and the studies of particular equations in Whittaker
and Watsons Modern Analysis. Those vho wish to learn more about
existence-theorems should con sult the recent work of Kamke...
