Table of Contents

1 Introduction

1.1 Mathematical Models and Solutions...1

1.2 Qualitative Methods: Phase Lines and Direction Fields...2

1.3 Definitions, Classification, and Terminology...8

2 First Order Differential Equations

2.1 Separable Equations...11

2.2 Linear Equations: Method of Integrating Factors...16

2.3 Modeling with First Order Equations...23

2.4 Differences between Linear and Nonlinear Equations...30

2.5 Autonomous Equations and Population Dynamics...33

2.6 Exact Equations and Integrating Factors...36

2.7 Substitution Methods...42

3 Systems of Two First Order Equations

3.1 Systems of Two Linear Algebraic Equations...49

3.2 Systems of Two First Order Linear Differential Equations...56

3.3 Homogeneous Linear Systems with Constant Coefficients...62

3.4 Complex Eigenvalues...79

3.5 Repeated Eigenvalues...87

3.6 A Brief Introduction to Nonlinear Systems...94

4 Second Order Linear Equations

4.1 Definitions and Examples...103

4.2 Theory of Second Order Linear Homogeneous Equations...106

4.3 Linear Homogeneous Equations with Constant Coefficients...108

4.4 Mechanical and Electrical Vibrations...122

4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients...128

4.6 Forced Vibrations, Frequency Response, and Resonance...134

4.7 Variation of Parameters...139

5 The Laplace Transform

5.1 Definition of the Laplace Transform...149

5.2 Properties of the Laplace Transform...154

5.3 The Inverse Laplace Transform...159

5.4 Solving Differential Equations with Laplace Transforms...163

5.5 Discontinuous Functions and Periodic Functions...170

5.6 Differential Equations with Discontinuous Forcing Functions...174

5.7 Impulse Functions...185

5.8 Convolution Integrals and Their Applications...193

5.9 Linear Systems and Feedback Control...201

6 Systems of First Order Linear Equations

6.1 Definitions and Examples...205

6.2 Basic Theory of First Order Linear Systems...209

6.3 Homogeneous Linear Systems with Constant Coefficients...211

6.4 Nondefective Matrices with Complex Eigenvalues...228

6.5 Fundamental Matrices and the Exponential of a Matrix...240

6.6 Nonhomogeneous Linear Systems...249

6.7 Defective Matrices...255

7 Nonlinear Differential Equations and Stability

7.1 Autonomous Systems and Stability...263

7.2 Almost Linear Systems...269

7.3 Competing Species...283

7.4 Predator-Prey Equations...293

7.5 Periodic Solutions and Limit Cycles...302

7.6 Chaos and Strange Attractors: The Lorenz Equations...310

8 Numerical Methods

8.1 Numerical Approximations: Euler's Method...315

8.2 Accuracy of Numerical Methods...317

8.3 Improved Euler and Runge-Kutta Methods...321

8.4 Numerical Methods for Systems of First Order Equations...326

9 Series Solutions of Second Order Linear Equations

9.1 Review of Power Series...331

9.2 Series Solutions Near an Ordinary Point, Part I...334

9.3 Series Solutions Near an Ordinary Point, Part II...349

9.4 Regular Singular Points...355

9.5 Series Solutions Near a Regular Singular Point, Part I...361

9.6 Series Solutions Near a Regular Singular Point, Part II...368

9.7 Bessel's Equation...377

10 Orthogonal Functions, Fourier Series, and Boundary Value Problems

10.1 Orthogonal Systems in the Space PC[a,b]...383

10.2 Fourier Series...385

10.3 Elementary Two-Point Boundary Value Problems...394

10.4 General Sturm-Liouville Boundary Value Problems...398

10.5 Generalized Fourier Series and Eigenfunction Expansions...407

10.6 Singular Sturm-Liouville Boundary Value Problems...415

10.7 Convergence Issues...418

11 Elementary Partial Differential Equations

11.1 Heat Conduction in a Rod: Homogeneous Case...431

11.2 Heat Conduction in a Rod: Nonhomogeneous Case...443

11.3 The Wave Equation: Vibrations of an Elastic String...450

11.4 The Wave Equation: Vibrations of a Circular Membrane...459

11.5 Laplace's Equation...459

About the Author

James R. Brannan is the author of Differential Equations: An Introduction to Modern Methods and Applications 3E Student Solutions Manual, published by Wiley. William E. Boyce is the author of Differential Equations: An Introduction to Modern Methods and Applications 3E Student Solutions Manual, published by Wiley.