Preface vii
Chapter 1 Introduction 1
1.1 Some Basic Mathematical Models; Direction Fields 1
1.2 Solutions of Some Differential Equations 9
1.3 Classification of Differential Equations 17
1.4 Historical Remarks 23
Chapter 2 First Order Differential Equations 29
2.1 Linear Equations with Variable Coefficients 29
2.2 Separable Equations 40
2.3 Modeling with First Order Equations 47
2.4 Differences Between Linear and Nonlinear Equations 64
2.5 Autonomous Equations and Population Dynamics 74
2.6 Exact Equations and Integrating Factors 89
2.7 Numerical Approximations: Euler's Method 96
2.8 The Existence and Uniqueness Theorem 105
2.9 First Order Difference Equations 115
Chapter 3 Second Order Linear Equations 129
3.1 Homogeneous Equations with Constant Coefficients 129
3.2 Fundamental Solutions of Linear Homogeneous Equations 137
3.3 Linear Independence and the Wronskian 147
3.4 Complex Roots of the Characteristic Equation 153
3.5 Repeated Roots; Reduction of Order 160
3.6 Nonhomogeneous Equations; Method of Undetermined Coefficients 169
3.7 Variation of Parameters 179
3.8 Mechanical and Electrical Vibrations 186
3.9 Forced Vibrations 200
Chapter 4 Higher Order Linear Equations 209
4.1 General Theory of nth Order Linear Equations 209
4.2 Homogeneous Equations with Constant Coeffients 214
4.3 The Method of Undetermined Coefficients 222
4.4 The Method of Variation of Parameters 226
Chapter 5 Series Solutions of Second Order Linear Equations 231
5.1 Review of Power Series 231
5.2 Series Solutions near an Ordinary Point, Part I 238
5.3 Series Solutions near an Ordinary Point, Part II 249
5.4 Regular Singular Points 255
5.5 Euler Equations 260
5.6 Series Solutions near a Regular Singular Point, Part I 267
5.7 Series Solutions near a Regular Singular Point, Part II 272
5.8 Bessel's Equation 280
Chapter 6 The Laplace Transform 293
6.1 Definition of the Laplace Transform 293
6.2 Solution of Initial Value Problems 299
6.3 Step Functions 310
6.4 Differential Equations with Discontinuous Forcing Functions 317
6.5 Impulse Functions 324
6.6 The Convolution Integral 330
Chapter 7 Systems of First Order Linear Equations 339
7.1 Introduction 339
7.2 Review of Matrices 348
7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 357
7.4 Basic Theory of Systems of First Order Linear Equations 368
7.5 Homogeneous Linear Systems with Constant Coefficients 373
7.6 Complex Eigenvalues 384
7.7 Fundamental Matrices 393
7.8 Repeated Eigenvalues 401
7.9 Nonhomogeneous Linear Systems 411
Chapter 8 Numerical Methods 419
8.1 The Euler or Tangent Line Method 419
8.2 Improvements on the Euler Method 430
8.3 The Runge-Kutta Method 435
8.4 Multistep Methods 439
8.5 More on Errors; Stability 445
8.6 Systems of First Order Equations 455
Chapter 9 Nonlinear Differential Equations and Stability 459
9.1 The Phase Plane; Linear Systems 459
9.2 Autonomous Systems and Stability 471
9.3 Almost Linear Systems 479
9.4 Competing Species 491
9.5 Predator-Prey Equations 503
9.6 Liapunov's Second Method 511
9.7 Periodic Solutions and Limit Cycles 521
9.8 Chaos and Strange Attractors; the Lorenz Equations 532
Chapter 10 Partial Differential Equations and Fourier Series 541
10.1 Two-Point Boundary Valve Problems 541
10.2 Fourier Series 547
10.3 The Fourier Convergence Theorem 558
10.4 Even and Odd Functions 564
10.5 Separation of Variables; Heat Conduction in a Rod 573
10.6 Other Heat Conduction Problems 581
10.7 The Wave Equation; Vibrations of an Elastic String 591
10.8 Laplace's Equation 604
Appendix A. Derivation of the Heat Conduction Equation 614
Appendix B. Derivation of the Wave Equation 617
Chapter 11 Boundary Value Problems and Sturm-Liouville Theory 621
11.1 The Occurrence of Two Point Boundary Value Problems 621
11.2 Sturm-Liouville Boundary Value Problems 629
11.3 Nonhomogeneous Boundary Value Problems 641
11.4 Singular Sturm-Liouville Problems 656
11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 663
11.6 Series of Orthogonal Functions: Mean Convergence 669
Answers to Problems 679
Index 737
