内容

目次

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.- 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.- 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 Sense of Carathéodory.- § 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.- 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 nwith Constant Coefficients.- Supplement: Linear Differential Equations with Periodic Coefficients.- 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.- 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.- VII: Stability and Asymptotic Behavior.- § 29. Stability.- § 30. The Method of Lyapunov.- A. Topology.- B. Real Analysis.- C. C0111plex Analysis.- D. Functional Analysis.- Solutions and Hints for Selected Exercises.- Literature.- Notation.