| The author's book on GewShnliche Differentialgleichungen (Ordinary Differential Equations) was published in 1972. The present book is based on a translation of the latest, 6th, edition, which appeared in 1996, but it also treats some important subjects that are not found there. The German book is widely used as a textbook for a first course in ordinary differential equations. This is a rigorous course, and it contains some material that is more difficult than that usually found in a first course textbook; such as, for example, Peano's existence theorem. It is addressed to students of mathematics, physics, and computer science and is usually taken in the third semester. Let me remark here that in the German system the student learns calculus of one variable at the gymnasium1 and begins at the university with a two-semester course on real analysis which is usually followed by ordinary differential equations. |
| Preface Note to the Reader Introduction Chapter I. First Order Equations: Some Integrable Cases 1. Explicit First Order Equations 2. The Linear Differential Equation. Related Equations Supplement: The Generalized Logistic Equation 3. Dif ferential Equations for Families of Curves. Exact Equations 4. Implicit First Order Differential Equations Chapter II: Theory of First Order Differential Equations 5. Tools from Functional Analysis 6. An Existence and Uniqueness Theorem Supplement: Singular Initial Value Problems 7. The Peano Existence Theorem Supplement: Methods of Functional Analysis 8. Complex Differential Equations. Power Series Expansions 9. Upper and Lower Solutions. Maximal and Minimal Integrals Supplement: The Separatrix Chapter III: First Order Systems. Equations of Higher Order 10. The Initial Value Problem for a System of First Order Supplement I: Differential Inequalities and Invarian e Supplement II: Differential Equations in the Sense of aratheodory 11. Initial Value Problems for Equations of Higher Order Supplement: Second Order Differential Inequalities 12. Continuous Dependence of Solutions Supplement: General Uniqueness and Dependence Theorems 13. Dependence of Solutions on Initial Values and Parameters Chapter IV: Linear Differential Equations 14. Linear Systems 15. Homogeneous Linear Systems 16. Inhomogeneous Systems Supplement: L1-Estimation of C-Solutions 17. Systems with Constant Coefficients 18. Matrix Functions. Inhomogeneous Systems Supplement: Floquet Theory 19. Linear Differential Equations of Order n 20. Linear Equations of Order n with Constant Coefficients Supplement: Linear Differential Equations with Periodic Coefficients Chapter V: Complex Linear Systems 21. Homogeneous Linear Systems in the Regular Case 22. Isolated Singularities 23. Weakly Singular Points. Equations of Fuchsian Type 24. Series Expansion of Solutions 25. Second Order Linear Equations Chapter VI: Boundary Value and Eigenvalue Problems 26. Boundary Value Problems Supplement I: Maximum and Minimum Prin iples Supplement II: Nonlinear Boundary Value Problems 27. The Sturm-Liouville Eigenvalue Problem Supplement: Rotation-Symmetri Ellipti Problems 28. Compact Self-Adjoint Operators in Hilbert Space Chapter VII: Stability and Asymptotic Behavior 29. Stability 30. The Method of Lyapunov Appendix A. Topology B. Real Analysis C. Complex Analysis D. Fun tional Analysis Solutions and Hints for Selected Exercises Literature Index Notation |
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