Introduction 1
Chapter Ⅰ．First Order Equations:Some Integrable Cases 9
1．Explicit First Order Equations 9
2．The Linear Differential Equation.Related Equations 27
Supplement:The Generalized Logistic Equation 33
3．Differential Equations for Families of Curves.Exact Equations 36
4．Implicit First Order Differential Equations 46
Chapter Ⅱ:Theory of First Order Differential Equations 53
5．Tools from Functional Analysis 53
6．An Existence and Uniqueness Theorem 62
Supplement:Singular Initial Value Problems 70
7．The Peano Existence Theorem 73
Supplement:Methods of Functional Analysis 80
8．Complex Differential Equations.Power Series Expansions 83
9．Upper and Lower Solutions.Maximal and Minimal Integrals 89
Supplement:The Separatrix 98
Chapter Ⅲ:First Order Systems.Equations of Higher Order 105
10．The Initial Value Problem for a System of First Order 105
Supplement Ⅰ:Differential Inequalities and Invariance 111
Supplement Ⅱ:Differential Equations in the Sense of Carathéodory 121
11．Initial Value Problems for Equations of Higher Order 125
Supplement:Second Order Differential Inequalities 139
12．Continuous Dependence of Solutions 141
Supplement:General Uniqueness and Dependence Theorems 146
13．Dependence of Solutions on Initial Values and Parameters 148
Chapter Ⅳ:Linear Differential Equations 159
14．Linear Systems 159
15．Homogeneous Linear Systems 164
16．Inhomogeneous Systems 170
Supplement:L1-Estimation of C-Solutions 173
17．Systems with Constant Coefficients 175
18．Matrix Functions.Inhomogeneous Systems 190
Supplement:Floquet Theory 195
19．Linear Differential Equations of Order n 198
20．Linear Equations of Order n with Constant Coefficients 204
Supplement:Linear Differential Equations with Periodic Coefficients 210
Chapter Ⅴ:Complex Linear Systems 213
21．Homogeneous Linear Systems in the Regular Case 213
22．Isolated Singularities 216
23．Weakly Singular Points.Equations of Fuchsian Type 222
24．Series Expansion of Solutions 225
25．Second Order Linear Equations 236
Chapter Ⅵ:Boundary Value and Eigenvalue Problems 245
26．Boundary Value Problems 245
Supplement Ⅰ:Maximum and Minimum Principles 260
Supplement Ⅱ:Nonlinear Boundary Value Problems 262
27．The Sturm-Liouville Eigenvalue Problem 268
Supplement:Rotation-Symmetric Elliptic Problems 281
28．Compact Self-Adjoint Operators in Hilbert Space 286
Chapter Ⅶ:Stability and Asymptotic Behavior 305
29．Stability 305
30．The Method of Lyapunov 318
Appendix 333
A．Topology 333
B．Real Analysis 342
C．Complex Analysis 348
D．Functional Analysis 350
Solutions and Hints for Selected Exercises 357
Literature 367
Index 372
Notation 379