Ordinary Differential Equations and Stability Theory: An Introduction (Dover Books on Mathematics)

Sanchez, David A.

ISBN 10: 0486638286 / ISBN 13: 9780486638287

Editore: Dover Publications, 2012

Codice articolo 0486638286-2-20775757

Riassunto:

This is a brief, modern introduction to the subject of ordinary differential equations, with an emphasis on stability theory. Concisely and lucidly expressed, it is intended as a supplementary text for the advanced undergraduate or beginning graduate student who has had a first course in ordinary differential equations.
The author begins with a general discussion of the linear equation and develops the notions of a fundamental system of solutions, the Wronskian, and the corresponding fundamental matrix. He then introduces the nonhomogeneous linear equation and the important variation of parameters formula, by which, following a consideration of the nth-order linear equation, the solution of the nonhomogeneous nth-order linear equation is obtained. A chapter is then devoted to the linear equation with constant coefficients.
The two following chapters are introductory discussions of stability theory for autonomous and nonautonomous systems. Included here are two results for nonlinear systems, Liapunov’s direct method, and some results for the second-order linear equation. The final chapter takes up the problems of the existence and uniqueness of solutions and related topics. Two appendixes — “Series Solutions of Second-Order Linear Equations” and “Linear Systems with Periodic Coefficients” — are also provided, and there are problems at the end of each chapter.

Contenuti: Chapter I. Introduction 1.1 Preliminary Notation 1.2 The Ordinary Differential Equation 1.3 An Existence and Uniqueness Theorem 1.4 The Maximum Interval of Existence Problems Chapter 2. The Linear Equation: General Discussion 2.1 Introduction 2.2 Fundamental Solutions 2.3 The Wronskian 2.4 The Nonhomogeneous Linear Equation 2.5 The nth-Order Linear Equation 2.6 The Nonhomogeneous nth-Order Linear Equation Problems Chapter 3. The Linear Equation with Constant Coefficients 3.1 The nth-Order Linear Equation 3.2 The Nonhomogeneous nth-Order Linear Equation 3.3 The Behavior of Solutions 3.4 The First-Order Linear System Problems Chapter 4. Autonomous Systems and Phase Space 4.1 Introduction 4.2 Linear Systems--Constant Coefficients 4.3 A General Discussion 4.4 Nonlinear Systems Problems Chapter 5. Stability for Nonautonomous Equations 5.1 Introduction 5.2 Stability for Linear Systems 5.3 Two Results for Nonlinear Systems 5.4 Liapunov's Direct Method 5.5 Some Results for the Second-Order Linear Equation Problems Chapter 6. Existence, Uniqueness, and Related Topics 6.1 Proof of the Existence and Uniqueness of Solutions 6.2 Continuation of Solutions and the Maximum Interval of Existence 6.3 The Dependence of Solutions on Parameters and Approximate Solutions Problems Appendix A. Series Solutions of Second-Order Linear Equations Appendix B. Linear Systems with Periodic Coefficients References; Index

Dati bibliografici

Titolo: Ordinary Differential Equations and ...
Casa editrice: Dover Publications
Data di pubblicazione: 2012
Legatura: Paperback
Condizione: Very Good