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.
Author(s): M. S. P. Eastham
Series: The New university mathematics series
Publisher: Van Nostrand Reinhold
Year: 1970
Language: English
Pages: 128
Preface
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 approxitnations
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 Inhomogeneous 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(0,00) 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 -> 00
3.9 The Liouville transformation
3.10 Application of the Liouville transformation
Problems
CHAPTER 4 ZEROS OF SOLUTIONS
4.1 Introduction 66
4.2 Comparison and separation theorerns 68
4.3 The Priifer transformation 69
4.4 The number of zeros in an interval 73
4.5 Further estimates for the number of zeros in an interval 76
4.6 Oscillatory and non-oscillatory equations 77
Problems 80
CHAPTER 5 EIGENVALUE PROBLEMS
5.1 Introduction 84
5.2 An equation for the eigenvalues 86
5.3 Self-adjoint eigenvalue problems 87
5.4 The existence of eigenvalues 91
5.5 The behaviour of An and 1fJn(x) as n -+ 00 95
5.6 The Green's function 97
5.7 Properties of G(X,A) as a function of A 100
5.8 The eigenfunction expansion formula 102
5.9 Mean square convergence and the Parseval formula 105
5.10 Use of the Priifer transformation 108
5.11 Periodic boundary conditions 110
Problems 111
Bibliography 115
Index 116
