导语

  

内容提要

  

    本书是在1996年第六版《常微分方程》（德文）一书的基础上编写而成的。本书主要介绍了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唯一性，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。此外，本书还增加了在一般相关教材中很少涉及但具有一定难度的内容，并对一些复杂基本定理给出了新的证明。阅读本书须具备一定的计算代数、线性代数及泛函分析的基础知识。

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. Differential 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 Invariance
  Supplement II: Differential Equations in the Senseof Caratheodory
  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 Sohltions 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 Principles
  Supplement II: Nonlinear Boundary Value Problems
  27. The Sturm-Liouville Eigenvalue Problem
  Supplement: Rotation-Symmetric Elliptic 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. Functional Analysis
Solutions and Hints for Selected Exercises
Literature
Index
Notation