1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1
1.1 Definitions and Terminology 2
1.2 Initial-Value Problems 15
1.3 Differential Equations as Mathematical Models 22
Chapter 1 in Review 37
2 FIRST-ORDER DIFFERENTIAL EQUATIONS 39
2.1 Solution Curves Without the Solution 40
2.2 Separable Variables 51
2.3 Linear Equations 60
2.4 Exact Equations 72
2.5 Solutions by Substitutions 80
2.6 A Numerical Solution 86
Chapter 2 in Review 92
3 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 95
3.1 Linear Equations 96
3.2 Nonlinear Equations 109
3.3 Systems of Linear and Nonlinear Differential Equations 121
Chapter 3 in Review 130
Project Module: Harvesting of Renewable Natural Resources, by Gilbert N. Lewis 133
4 HIGHER-ORDER DIFFERENTIAL EQUATIONS 138
4.1 Preliminary Theory: Linear Equations 139
4.1.1 Initial-Value and Boundary-Value Problems 139
4.1.2 Homogeneous Equations 142
4.1.3 Nonhomogeneous Equations 148
4.2 Reduction of Order 154
4.3 Homogeneous Linear Equations with Constant Coefficients 158
4.4 Undetermined Coefficients—Superposition Approach 167
4.5 Undetermined Coefficients—Annihilator Approach 178
4.6 Variation of Parameters 188
4.7 Cauchy-Euler Equation 193
4.8 Solving Systems of Linear Equations by Elimination 201
4.9 Nonlinear Equations 207
Chapter 4 in Review 212
5 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 215
5.1 Linear Equations: Initial-Value Problems 216
5.1.1 Spring/Mass Systems: Free Undamped Motion 216
5.1.2 Spring/Mass Systems: Free Damped Motion 220
5.1.3 Spring/Mass Systems: Driven Motion 224
5.1.4 Series Circuit Analogue 227
5.2 Linear Equations: Boundary-Value Problems 237
5.3 Nonlinear Equations 247
Chapter 5 in Review 259
Project Module: The Collapse of the Tacoma Narrows Suspension Bridge, by Gilbert N. Lewis 263
6 SERIES SOLUTIONS Of LINEAR EQUATIONS 267
6.1 Solutions About Ordinary Points 268
6.1.1 Review of Power Series 268
6.1.2 Power Series Solutions 271
6.2 Solutions About Singular Points 280
6.3 Two Special Equations 292
Chapter 6 in Review 304
7 THE LAPLACE TRANSFORM 306
7.1 Definition of the Laplace Transform 307
7.2 Inverse Transform and Transforms of Derivatives 314
7.3 Translation Theorems 324
7.3.1 Translation on the s-Axis 324
7.3.2 Translation on the t-Axis 328
7.4 Additional Operational Properties 338
7.5 Dirac Delta Function 351
7.6 Systems of Linear Equations 354
Chapter 7 in Review 361
8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 364
8.1 Preliminary Theory 365
8.2 Homogeneous Linear Systems with Constant Coefficients 375
8.2.1 Distinct Real Eigenvalues 376
8.2.2 Repeated Eigenvalues 380
8.2.3 Complex Eigenvalues 384
8.3 Variation of Parameters 393
8.4 Matrix Exponential 399
Chapter 8 in Review 404
Project Module: Earthquake Shaking of Multistory Buildings, by Gilbert N. Lewis 406
9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 410
9.1 Euler Methods and Error Analysis 411
9.2 Runge-Kutta Methods 417
9.3 Multistep Methods 424
9.4 Higher-Order Equations and Systems 427
9.5 Second-Order Boundary-Value Problems 433
Chapter 9 in Review 438
10 PLANE AUTONOMOUS SYSTEMS AND STABILITY 439
10.1 Autonomous Systems, Critical Points, and Periodic Solutions 440
10.2 Stability of Linear Systems 448
10.3 Linearization and Local Stability 458
10.4 Modeling Using Autonomous Systems 470
Chapter 10 in Review 480
11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 483
11.1 Orthogonal Functions 484
11.2 Fourier Series 489
11.3 Fourier Cosine and Sine Series 495
11.4 Sturm-Liouville Problem 504
11.5 Bessel and Legendre Series 511
11.5.1 Fourier-Bessel Series 512
11.5.2 Fourier-Legendre Series 515
Chapter 11 in Review 519
12 PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 521
12.1 Separable Partial Differential Equations 522
12.2 Classical Equations and Boundary-Value Problems 527
12.3 Heat Equation 533
12.4 Wave Equation 536
12.5 Laplace s Equation 542
12.6 Nonhomogeneous Equations and Boundary Conditions 547
12.7 Orthogonal Series Expansions 551
12.8 Boundary-Value Problems Involving Fourier Series in Two Variables 555
Chapter 12 in Review 559
13 BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS 561
13.1 Problems Involving Laplace s Equation in Polar Coordinates 562
13.2 Problems in Polar and Cylindrical Coordinates: Bessel Functions 567
13.3 Problems in Spherical Coordinates: Legendre Polynomials 575
Chapter 13 in Review 578
14 INTEGRAL TRANSFORM METHOD 581
14.1 Error Function 582
14.2 Applications of the Laplace Transform 584
14.3 Fourier Integral 595
14.4 Fourier Transforms 601
Chapter 14 in Review 607
15 NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 610
15.1 Elliptic Equations 611
15.2 Parabolic Equations 617
15.3 Hyperbolic Equations 625
Chapter 15 in Review 630