The Sixth Edition of this acclaimed differential equations book remains the same classic volume it's always been, but has been polished and sharpened to serve readers even more effectively. Offers  precise and clear-cut statements of fundamental existence and uniqueness theorems to allow understanding of their role in this subject. Features a strong numerical approach that emphasizes that the effective and reliable use of numerical methods often requires preliminary analysis using standard elementary techniques. Inserts new graphics and text where needed for improved accessibility. A useful reference for readers who need to brush up on differential equations.

Preface vii

CHAPTER

1 First-Order Differential Equations 1

1.1 Differential Equations and Mathematical Models 1

1.2 Integrals as General and Particular Solutions 10

1.3 Slope Fields and Solution Curves 19

1.4 Separable Equations and Applications 32

1.5 Linear First-Order Equations 46

1.6 Substitution Methods and Exact Equations 59

1.7 Population Models 74

1.8 Acceleration-Velocity Models 85

CHAPTER

2 Linear Equations of Higher Order 100

2.1 Introduction: Second-Order Linear Equations 100

2.2 General Solutions of Linear Equations 113

2.3 Homogeneous Equations with Constant Coefficients 124

2.4 Mechanical Vibrations 135

2.5 Nonhomogeneous Equations and Undetermined Coefficients 148

2.6 Forced Oscillations and Resonance 162

2.7 Electrical Circuits 173

2.8 Endpoint Problems and Eigenvalues 180

CHAPTER

3 Power Series Methods 194

3.1 Introduction and Review of Power Series 194

3.2 Series Solutions Near Ordinary Points 207

3.3 Regular Singular Points 218

3.4 Method of Frobenius: The Exceptional Cases 233

3.5 Bessel's Equation 248

3.6 Applications of Bessel Functions 257

v

vi Contents

CHAPTER

4 Laplace Transform Methods 266

4.1 Laplace Transforms and Inverse Transforms 266

4.2 Transformation of Initial Value Problems 277

4.3 Translation and Partial Fractions 289

4.4 Derivatives, Integrals, and Products of Transforms 297

4.5 Periodic and Piecewise Continuous Input Functions 304

4.6 Impulses and Delta Functions 316

CHAPTER

5 Linear Systems of Differential Equations 326

5.1 First-Order Systems and Applications 326

5.2 The Method of Elimination 338

5.3 Matrices and Linear Systems 347

5.4 The Eigenvalue Method for Homogeneous Systems 366

5.5 Second-Order Systems and Mechanical Applications 381

5.6 Multiple Eigenvalue Solutions 393

5.7 Matrix Exponentials and Linear Systems 407

5.8 Nonhomogeneous Linear Systems 420

CHAPTER

6 Numerical Methods 430

6.1 Numerical Approximation: Euler's Method 430

6.2 A Closer Look at the Euler Method 442

6.3 The Runge-Kutta Method 453

6.4 Numerical Methods for Systems 464

CHAPTER

7 Nonlinear Systems and Phenomena 480

7.1 Equilibrium Solutions and Stability 480

7.2 Stability and the Phase Plane 488

7.3 Linear and Almost Linear Systems 500

7.4 Ecological Models: Predators and Competitors 513

7.5 Nonlinear Mechanical Systems 526

7.6 Chaos in Dynamical Systems 542

References for Further Study 555

Appendix: Existence and Uniqueness of Solutions 559

Answers to Selected Problems 573

Index I-1