Ordinary Differential Equations

Ordinary Differential Equations is an outgrowth of courses taught for a number of years at Iowa State University in the mathematics and the electrical engineering departments. It is intended as a text for a first graduate course in differential equations for students in mathematics, engineering, and the sciences. Although differential equations is an old, traditional, and well-established subject, the diverse backgrounds and interests of the students in a typical modern-day course cause problems in the selection and method of presentation of material. In order to compensate for this diversity, prerequisites have been kept to a minimum and the material is covered in such a way as to be appealing to a wide audience. The book contains eight chapters and begins with an introduction the subject and a discussion of some important examples of differential equations that arise in science and engineering. Separate chapters follow on the fundamental theory of linear and nonlinear differential equations; linear boundary value problems; Lyapunov stability theory; and perturbations of linear systems. Subsequent chapters deal with the Poincare-Bendixson theory and with two-dimensional van der Pol type equations; and periodic solutions of general order systems.

Preface

Acknowledgments

1 Introduction

1.1 Initial Value Problems

1.2 Examples of Initial Value Problems

Problems

2 Fundamental Theory

2.1 Preliminaries

2.2 Existence of Solutions

2.3 Continuation of Solutions

2.4 Uniqueness of Solutions

2.5 Continuity of Solutions with Respect to Parameters

2.6 Systems of Equations

2.7 Differentiability with Respect to Parameters

2.8 Comparison Theory

2.9 Complex Valued Systems

Problems

3 Linear Systems

3.1 Preliminaries

3.2 Linear Homogeneous and Nonhomogeneous Systems

3.3 Linear Systems with Constant Coefficients

3.4 Linear Systems with Periodic Coefficients

3.5 Linear nth Order Ordinary Differential Equations

3.6 Oscillation Theory

Problems

4 Boundary Value Problems

4.1 Introduction

4.2 Separated Boundary Conditions

4.3 Asymptotic Behavior of Eigenvalues

4.4 Inhomogeneous Problems

4.5 General Boundary Value Problems

Problems

5 Stability

5.1 Notation

5.2 The Concept of an Equilibrium Point

5.3 Definitions of Stability and Boundedness

5.4 Some Basic Properties of Autonomous and Periodic Systems

5.5 Linear Systems

5.6 Second Order Linear Systems

5.7 Lyapunov Functions

5.8 Lyapunov Stability and Instability Results: Motivation

5.9 Principal Lyapunov Stability and Instability Theorems

5.10 Linear Systems Revisited

5.11 Invariance Theory

5.12 Domain of Attraction

5.13 Converse Theorems

5.14 Comparison Theorems

5.15 Applications: Absolute Stability of Regulator Systems

Problems

6 Perturbations of Linear Systems

6.1 Preliminaries

6.2 Stability of an Equilibrium Point

6.3 The Stable Manifold

6.4 Stability of Periodic Solutions

6.5 Asymptotic Equivalence

Problems

7 Periodic Solutions of Two-Dimensional Systems

7.1 Preliminaries

7.2 Poincaré-Bendixson Theory

7.3 The Levinson-Smith Theorem

Problems

8 Periodic Solutions of Systems

8.1 Preliminaries

8.2 Nonhomogeneous Linear Systems

8.3 Perturbations of Nonlinear Periodic Systems

8.4 Perturbations of Nonlinear Autonomous Systems

8.5 Perturbations of Critical Linear Systems

8.6 Stability of Systems with Linear Part Critical

8.7 Averaging

8.8 Hopf Bifurcation

8.9 A Nonexistence Result

Problems

Bibliography

Index