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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 18

1.4 Separable Equations and Applications 31

1.5 Linear First-Order Equations 46

1.6 Substitution Methods and Exact Equations 58

CHAPTER 2 Mathematical Models and Numerical Methods 77

2.1 Population Models 77

2.2 Equilibrium Solutions and Stability 90

2.3 Acceleration-Velocity Models 98

2.4 Numerical Approximation：Euler’s Method 110

2.5 A CLoser Look at the Euler Method 122

2.6 The Runge-Kutta Method 132

CHAPTER 3 Linear Equations of Higher Order 144

3.1 Introduction： Second-Order Linear Equations 144

3.2 General Solutions of Linear Equations 158

3.3 Homogeneous Equations with Constant Coefficients 170

3.4 Mechanical Vibrations 182

3.5 Nonhomogeneous Equations and Undetermined Coefficients 195

3.6 Forced Oscillations and Resonance 209

3.7 Electrical Circuits 222

3.8 Endpoint Problems and Eigenvalues 229

CHAPTER 4 Introduction to Systems of Differential Equations 242

4.1 First-Order Systems and Applications 242

4.2 The Method of Elimination 254

4.3 Numerical Methods for Systems 265

CHAPTER 5 Linear Systems of Differential Equations 281

5.1 Matrices and Linear Systems 281

5.2 The Eigenvalue Method for Homogeneous Systems 300

5.3 Second-Order Systems and Mechanical Applications 315

5.4 Multiple Eigenvalue Solutions 328

5.5 Matrix Exponentials and Linear Systems 344

5.6 Nonhomogeneous Linear Systems 358

CHAPTER 6 Nonlinear Systems and Phenomena 366

6.1 Stability and the Phase Plane 366

6.2 Linear and Almost Linear Systems 378

6.3 Ecological Models： Predators and Competitors 393

6.4 Nonlinear Mechanical Systems 406

6.5 Chaos in Dynamical Systems 423

CHAPTER 7 Laplace Transform Methods 435

7.1 Laplace Transforms and Inverse Transforms 435

7.2 Transformation of Initial Value Problems 446

7.3 Translation and Partial Fractions 457

7.4 Derivatives， Integrals，and Products of Transforms 467

7.5 Periodic and Piecewise Continuous Input Functions 475

7.6 Impulses and Delta Functions 486

CHAPTER 8 Power Series Methods 497

8.1 Introduction and Review of Power Series 497

8.2 Series Solutions Near Ordinary Points 510

8.3 Regular Singular Points 523

8.4 Method of Frobenius：The Exceptional Cases 539

8.5 Bessel’s Equation 554

8.6 Applications of Bessel Functions 563

CHAPTER 9 Fourier Series Methods 572

9.1 Periodic Functions and Trigonometric Series 572

9.2 General Fourier Series and Convergence 581

9.3 Fourier Sine and Cosine Series 589

9.4 Applications of Fourier Series 601

9.5 Heat Conduction and Separation of Variables 606

9.6 Vibrating Strings and the One-Dimensional Wave Equation 621

9.7 Steady-State Temperature and Laplace’s Equatiion 635

CHAPTER 10 Eigenvalues and Boundary Value Problems 645

10.1 Sturm-Liouville Problems and Eigenfunction Expansions 645

10.2 Applications of Eigenfunction Series 658

10.3 Steady Periodic Solutions and Natural Frequencies 668

10.4 Cylindrical Coordinate Problems 678

10.5 Higher-Dimensional Phenomena 693

References for Further Study 711

Appendix： Existence and Uniqueness of Solutions 714

Answers to Selected Problems 729