Theory of Ordinary Differential Equations

This book is written for second- and third-year honours students, and indeed for any mathematically-minded person who has had a first elementary course on differential equations and wishes to extend his knowledge. The main requirement on the reader is that he should possess a thorough knowledge of real and complex analysis up to the usual second-year level of an honours degree course. After the basic theory in the first two chapters, the remaining three chapters contain topics which, while fully dealt with in advanced books, are not normally given a connected or completely rigorous account at this level. It is hoped therefore that the book will prepare the reader to continue his studies, if he so desires, in more comprehensive and advanced works, and suggestions for further reading are made in the bibliography. S Title List of Publications of The New University Mathematics Series THEORY OF ORDINARY DIFFERENTIAL EQUATIONS COPYRIGHT © M. S. P. EASTHAM 1970 Library of Congress Catalog Card No. 70—105345 Dedicated To Heather and Stephen Preface Contents Notation CHAPTER 1 Existence Theorems 1.1 Differential equations and three basic questions 1.2 Systems of differential equations 1.3 The method of successive approximations 1.4 First-Order systems 1.5 Remarks on the above theorems 1.6 Differential equations of order n 1.7 Dependence of solutions on parameters Problems CHAPTER 2 Linear Differential Equations 2.1 Homogeneous linear differential equations 2.2 The construction of fundamental sets 2.3 The Wronskian 2.4 Enhomogeneous linear differential equations 2.5 Extension of the variation of constants method 2.6 Linear differential operators and their adjoints 2.7 Self-Adjoint differential operators Problems CHAPTER 3 Asymptotic Formulae for Solutions 3.1 Introduction 3.2 An integral inequality 3.3 Bounded solutions 3.4 L2(0, \infinity) solutions 3.5 Asymptotic formulae for solutions 3.6 The case k = 0 3.7 The case k > 0 3.8 The condition r(x) —> 0 as x —> \infinity 3.9 The Liouville transformation 3.10 Application of the Liouville transformation Problems CHAPTER 4 Zeros of Solutions 4.1 Introduction 4.2 Comparison and separation theorems 4.3 The Prufer transform 4.4 The number of zeros in an interval 4.5 Further estimates for the number of zeros in an interval 4.6 Oscillatory and non-oscillatory equations Problems CHAPTER 5 Eigenvalue Problems 5.1 Introduction 5.2 An equation for the eigenvalues 5.3 Self-adjoint eigenvalue problems 5.4 The existence of eigenvalues 5.5 The behaviour of \lambda_n and \psi_n as n----> \infinity 5.6 The Green's function 5.7 Properties of G(x,\zeta, \lambda) as a function of \lambda 5.8 The eigenfunction expansion formula 5.9 Mean square convergence and the Parseval formula 510 Use of the Prüfer transformation 5.11 Periodic boundary conditions Problems Bibliography Index