常微分方程

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，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。 目次：一阶微分方程，一些可积的例子；一阶微分方程理论；一阶系统，离阶微分方程；线性微分方程；复线性系统；边值问题与特征值问题；稳定性与渐进稳定性。

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

W.沃尔特（Wolfgang Walter）是德国卡尔斯鲁厄大学(Universität Karlsruhe)教授。

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