Book:J. David Logan/A First Course in Differential Equations

J. David Logan: A First Course in Differential Equations

Published $\text {2005}$, Springer

ISBN 978-1441975911

Subject Matter

• Differential Equations

Contents

Preface

To the Student

1. Differential Equations and Models

1.1 Differential Equations

1.1.1 Equations and Solutions

1.1.2 Geometrical Interpretation

1.2 Pure Time Equations

1.3 Mathematical Models

1.3.1 Particle Dynamics

1.3.2 Autonomous Differential Equations

1.3.3 Stability and Bifurcations

1.3.4 Heat Transfer

1.3.5 Chemical Reactors

1.3.6 Electric Circuits

2. Analytic Solutions and Approximations

2.1 Separation of Variables

2.2 First-Order Linear Equations

2.3 Approximation

2.3.1 Picard Iteration

2.3.2 Numerical Methods

2.3.3 Error Analysis

3. Second-Order Differential Equations

3.1 Particle Mechanics

3.2 Linear Equations with Constant Coefficients

3.3 The Nonhomogeneous Equation

3.3.1 Undetermined Coefficients

3.3.2 Resonance

3.4 Variable Coefficients

3.4.1 Cauchy–Euler Equation

3.4.2 Power Series Solutions

3.4.3 Reduction of Order

3.4.4 Variation of Parameters

3.5 Higher-Order Equations

3.6 Summary and Review

4. Laplace Transformations

4.1 Definition and Basic Properties

4.2 Initial Value Problems

4.3 The Convolution Property

4.4 Discontinuous Sources

4.5 Point Sources

4.6 Table of Laplace Transformations

5. Linear Systems

5.1 Introduction

5.2 Matrices

5.3 Two-Dimensional Systems

5.3.1 Solutions and Linear Orbits

5.3.2 The Eigenvalue Problems

5.3.3 Real Unequal Eigenvalues

5.3.4 Complex Eigenvalues

5.3.5 Real, Repeated Eigenvalues

5.3.6 Stability

5.4 Nonhomogeneous Systems

5.5 Three-Dimensional Systems

6. Nonlinear Systems

6.1 Nonlinear Models

6.1.1 Phase Plane Phenomena

6.1.2 The Lotka-Volterra Model

6.1.3 Holling Functional Responses

6.1.4 An Epidemic Model

6.2 Numerical Methods

6.3 Linearization and Stability

6.4 Periodic Solutions

6.4.1 The Poincaré-Bendixson Theorem

Appendix A. References

Appendix B. Computer Algebra Systems

B.1 Maple

B.2 MATLAB

Appendix C. Sample Examinations

Appendix D. Solutions and Hints to Selected Exercises

Index