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