Ordinary differential equations

Preface v
Note to the Reader xi

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
2.1. 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
6.1. Supplement: Singular Initial Value Problems 70
7. The Peano Existence Theorem 73
7.1. Supplement: Methods of Functional Analysis 80
8. Complex Differential Equations. Power Series Expansions 83
9. Upper and Lower Solutions. Maximal and Minimal Integrals 89
9.1. 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
10.1. Supplement I: Differential Inequalities and Invariance 111
10.2. Supplement II: Differential Equations in the Sense of Caratheodory 121
11. Initial Value Problems for Equations of Higher Order 125
11.1. Supplement: Second Order Differential Inequalities 139
12. Continuous Dependence of Solutions 141
12.1. 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
16.1. Supplement: $L^1$-Estimation of $C$-Solutions 173
17. Systems with Constant Coefficients 175
18. Matrix Functions. Inhomogeneous Systems 190
18.1. Supplement: Floquet Theory 195
19. Linear Differential Equations of Order $n$ 198
20. Linear Equations of Order n with Constant Coefficients 204
20.1. 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
26.1. Supplement I: Maximum and Minimum Principles 260 
26.2. Supplement II: Nonlinear Boundary Value Problems 262
27. The Sturm-Liouville Eigenvalue Problem 268
27.1. 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