Based on a translation of the 6th edition of Gewöhnliche Differentialgleichungen by Wolfgang Walter, this edition includes additional treatments of important subjects not found in the German text as well as material that is seldom found in textbooks, such as new proofs for basic theorems. This unique feature of the book calls for a closer look at contents and methods with an emphasis on subjects outside the mainstream. Exercises, which range from routine to demanding, are dispersed throughout the text and some include an outline of the solution. Applications from mechanics to mathematical biology are included and solutions of selected exercises are found at the end of the book. It is suitable for mathematics, physics, and computer science graduate students to be used as collateral reading and as a reference source for mathematicians. Readers should have a sound knowledge of infinitesimal calculus and be familiar with basic notions from linear algebra; functional analysis is developed in the text when needed.
Inhaltsverzeichnis
I. First Order Equations: Some Integrable Cases. - § 1. Explicit First Order Equations. - § 2. The Linear Differential Equation. Related Equations. - § 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. - § 7. The Peano Existence Theorem. - § 8. Complex Differential Equations. Power Series Expansions. - § 9. Upper and Lower Solutions. Maximal and Minimal Integrals. - III: First Order Systems. Equations of Higher Order. - § 10. The Initial Value Problem for a System of First Order. - § 11. Initial Value Problems for Equations of Higher Order. - § 12. Continuous Dependence of Solutions. - § 13. Dependence of Solutions on Initial Values and Parameters. - IV: Linear Differential Equations. - § 14. Linear Systems. - § 15. Homogeneous Linear Systems. - § 16. Inhomogeneous Systems. - § 17. Systems with Constant Coefficients. - § 18. Matrix Functions. Inhomogeneous Systems. - § 19. Linear Differential Equations of Order n. - § 20. Linear Equations of Order nwith Constant 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. - § 27. The Sturm Liouville Eigenvalue Problem. - § 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. FunctionalAnalysis. - Solutions and Hints for Selected Exercises. - Literature. - Notation.
