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Differential Equations


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Table of Contents

Chapter 1 Introduction 1

1.1 Mathematical Models and Solutions 2

1.2 Qualitative Methods: Phase Lines and Direction Fields 12

1.3 Definitions, Classification, and Terminology 28

Chapter 2 First Order Differential Equations 37

2.1 Separable Equations 38

2.2 Linear Equations: Method of Integrating Factors 45

2.3 Modeling with First Order Equations 55

2.4 Differences Between Linear and Nonlinear Equations 70

2.5 Autonomous Equations and Population Dynamics 80

2.6 Exact Equations and Integrating Factors 93

2.7 Substitution Methods 101


2.P.1 Harvesting a Renewable Resource 110

2.P.2 A Mathematical Model of a Groundwater Contaminant Source 111

2.P.3 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin 113

Chapter 3 Systems of Two First Order Equations 116

3.1 Systems of Two Linear Algebraic Equations 117

3.2 Systems of Two First Order Linear Differential Equations 129

3.3 Homogeneous Linear Systems with Constant Coefficients 145

3.4 Complex Eigenvalues 167

3.5 Repeated Eigenvalues 178

3.6 A Brief Introduction to Nonlinear Systems 189


3.P.1 Estimating Rate Constants for an Open Two-Compartment Model 199

3.P.2 A Blood-Brain Pharmacokinetic Model 201

Chapter 4 Second Order Linear Equations 203

4.1 Definitions and Examples 203

4.2 Theory of Second Order Linear Homogeneous Equations 216

4.3 Linear Homogeneous Equations with Constant Coefficients 228

4.4 Mechanical and Electrical Vibrations 241

4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients 252

4.6 Forced Vibrations, Frequency Response, and Resonance 261

4.7 Variation of Parameters 274


4.P.1 A Vibration Insulation Problem 285

4.P.2 Linearization of a Nonlinear Mechanical System 286

4.P.3 A Spring-Mass Event Problem 288

4.P.4 Euler-Lagrange Equations 289

Chapter 5 The Laplace Transform 294

5.1 Definition of the Laplace Transform 295

5.2 Properties of the Laplace Transform 304

5.3 The Inverse Laplace Transform 311

5.4 Solving Differential Equations with Laplace Transforms 320

5.5 Discontinuous Functions and Periodic Functions 328

5.6 Differential Equations with Discontinuous Forcing Functions 337

5.7 Impulse Functions 344

5.8 Convolution Integrals and Their Applications 351

5.9 Linear Systems and Feedback Control 361


5.P.1 An Electric Circuit Problem 371

5.P.2 The Watt Governor, Feedback Control, and Stability 372

Chapter 6 Systems of First Order Linear Equations 377

6.1 Definitions and Examples 378

6.2 Basic Theory of First Order Linear Systems 389

6.3 Homogeneous Linear Systems with Constant Coefficients 399

6.4 Nondefective Matrices with Complex Eigenvalues 410

6.5 Fundamental Matrices and the Exponential of a Matrix 420

6.6 Nonhomogeneous Linear Systems 431

6.7 Defective Matrices 438


6.P.1 Earthquakes and Tall Buildings 446

6.P.2 Controlling a Spring-Mass System to Equilibrium 449

Chapter 7 Nonlinear Differential Equations and Stability 456

7.1 Autonomous Systems and Stability 456

7.2 Almost Linear Systems 466

7.3 Competing Species 476

7.4 Predator-Prey Equations 488

7.5 Periodic Solutions and Limit Cycles 496

7.6 Chaos and Strange Attractors: The Lorenz Equations 506


7.P.1 Modeling of Epidemics 514

7.P.2 Harvesting in a Competitive Environment 516

7.P.3 The Roessler System 518

Chapter 8 Numerical Methods 519

8.1 Numerical Approximations: Euler's Method 519

8.2 Accuracy of Numerical Methods 530

8.3 Improved Euler and Runge-Kutta Methods 537

8.4 Numerical Methods for Systems of First Order Equations 546


8.P.1 Designing a Drip Dispenser for a Hydrology Experiment 550

8.P.2 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin 551

Chapter 9 Series Solutions of Second Order Equations (online only)

9.1 Review of Power Series

9.2 Series Solutions Near an Ordinary Point, Part I

9.3 Series Solutions Near an Ordinary Point, Part II

9.4 Regular Singular Points

9.5 Series Solutions Near a Regular Singular Point, Part I

9.6 Series Solutions Near a Regular Singular Point, Part II

9.7 Bessel's Equation


9.P.1 Diffraction Through a Circular Aperature

9.P.2 Hermite Polynomials and the Quantum Mechanical Harmonic Oscillator

9.P.3 Perturbation Methods

Chapter 10 Orthogonal Functions, Fourier Series, and Boundary Value Problems (online only)

10.1 Orthogonal Families in the Space PC[a, b]

10.2 Fourier Series

10.3 Elementary Two-Point Boundary Value Problems

10.4 General Sturm-Liouville Boundary Value Problems

10.5 Generalized Fourier Series and Eigenfunction Expansions

10.6 Singular Boundary Value Problems

10.7 Convergence Issues

Chapter 11 Elementary Partial Differential Equations (online only)

11.1 Heat Conduction in a Rod-Homogeneous Case

11.2 Heat Conduction in a Rod-Nonhomogeneous Case

11.3 Wave Equation-Vibrations of an Elastic String

11.4 Wave Equation-Vibrations of a Circular Membrane

11.5 Laplace's Equation


11.P.1 Estimating the Diffusion Coefficient in the Heat Equation

11.P.2 The Transmission Line Problem

11.P.3 Solving Poisson's Equation by Finite Differences

11.P.4 Dynamic Behavior of a Hanging Cable

11.P.5 Advection Dispersion: A Model for Solute Transport in Saturated Porous Media

11.P.6 Fisher's Equation for Population Growth and Dispersion

Appendices (available on companion web site)

11.A Derivation of the Heat Equation

11.B Derivation of the Wave Equation

Appendix A Matrices and Linear Algebra 555

A.1 Matrices 555

A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank 564

A.3 Determinants and Inverses 581

A.4 The Eigenvalue Problem 590

Appendix B Complex Variables (online only)

Review of Integration (online only)

Answers 601

References 664

Index 666

About the Author

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

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