本书是在1996年出版的《常微分方程》(德文)一书的基础上编写而成的,书中主要介绍常微分方程的基础理论。内容包括:可积一阶微分方程,微分方程解的存在性和**性,微分方程的初极值问题,边值问题和特征值问题,稳定性与渐进稳定性理论。阅读本书需要具备一定的计算代数、线性代数及泛函分析的基础知识。适用于高校数学专业、牧业专业和计算机科学等相关专业的本科生和研究生。

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

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.