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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