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