Differential Equations with Boundary-Value Problems

Front Cover......Page 1
Title Page......Page 6
Copyright......Page 9
Contents......Page 10
Preface......Page 16
1 INTRODUCTION TO DIFFERENTIAL EQUATIONS......Page 22
1.1 Definitions and Terminology......Page 23
1.2 Initial-Value Problems......Page 34
1.3 Differential Equations as Mathematical Models......Page 40
CHAPTER 1 IN REVIEW......Page 53
2 FIRST-ORDER DIFFERENTIAL EQUATIONS......Page 55
2.1.1 Direction Fields......Page 56
2.1.2 Autonomous First-Order DEs......Page 58
2.2 Separable Variables......Page 65
2.3 Linear Equations......Page 74
2.4 Exact Equations......Page 83
2.5 Solutions by Substitutions......Page 91
2.6 A Numerical Method......Page 96
CHAPTER 2 IN REVIEW......Page 101
3 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS......Page 103
3.1 Linear Models......Page 104
3.2 Nonlinear Models......Page 115
3.3 Modeling with Systems of First-Order DEs......Page 126
CHAPTER 3 IN REVIEW......Page 134
4 HIGHER-ORDER DIFFERENTIAL EQUATIONS......Page 138
4.1.1 Initial-Value and Boundary-Value Problems......Page 139
4.1.2 Homogeneous Equations......Page 141
4.1.3 Nonhomogeneous Equations......Page 146
4.2 Reduction of Order......Page 151
4.3 Homogeneous Linear Equations with Constant Coefficients......Page 154
4.4 Undetermined Coefficients—Superposition Approach......Page 161
4.5 Undetermined Coefficients—Annihilator Approach......Page 171
4.6 Variation of Parameters......Page 178
4.7 Cauchy-Euler Equation......Page 183
4.8 Solving Systems of Linear DEs by Elimination......Page 190
4.9 Nonlinear Differential Equations......Page 195
CHAPTER 4 IN REVIEW......Page 199
5 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS......Page 202
5.1.1 Spring/Mass Systems: Free Undamped Motion......Page 203
5.1.2 Spring/Mass Systems: Free Damped Motion......Page 207
5.1.3 Spring/Mass Systems: Driven Motion......Page 210
5.1.4 Series Circuit Analogue......Page 213
5.2 Linear Models: Boundary-Value Problems......Page 220
5.3 Nonlinear Models......Page 228
CHAPTER 5 IN REVIEW......Page 237
6 SERIES SOLUTIONS OF LINEAR EQUATIONS......Page 240
6.1.1 Review of Power Series......Page 241
6.1.2 Power Series Solutions......Page 244
6.2 Solutions About Singular Points......Page 252
6.3.1 Bessel's Equation......Page 262
6.3.2 Legendre's Equation......Page 269
CHAPTER 6 IN REVIEW......Page 274
7 THE LAPLACE TRANSFORM......Page 276
7.1 Definition of the Laplace Transform......Page 277
7.2.1 Inverse Transforms......Page 283
7.2.2 Transforms of Derivatives......Page 286
7.3 Operational Properties I......Page 291
7.3.1 Translation on the s-Axis......Page 292
7.3.2 Translation on the t-Axis......Page 295
7.4.1 Derivatives of a Transform......Page 303
7.4.2 Transforms of Integrals......Page 304
7.4.3 Transform of a Periodic Function......Page 308
7.5 The Dirac Delta Function......Page 313
7.6 Systems of Linear Differential Equations......Page 316
CHAPTER 7 IN REVIEW......Page 321
8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS......Page 324
8.1 Preliminary Theory—Linear Systems......Page 325
8.2 Homogeneous Linear Systems......Page 332
8.2.1 Distinct Real Eigenvalues......Page 333
8.2.2 Repeated Eigenvalues......Page 336
8.2.3 Complex Eigenvalues......Page 341
8.3.1 Undetermined Coefficients......Page 347
8.3.2 Variation of Parameters......Page 350
8.4 Matrix Exponential......Page 355
CHAPTER 8 IN REVIEW......Page 358
9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS......Page 360
9.1 Euler Methods and Error Analysis......Page 361
9.2 Runge-Kutta Methods......Page 366
9.3 Multistep Methods......Page 371
9.4 Higher-Order Equations and Systems......Page 374
9.5 Second-Order Boundary-Value Problems......Page 379
CHAPTER 9 IN REVIEW......Page 383
10 PLANE AUTONOMOUS SYSTEMS......Page 384
10.1 Autonomous Systems......Page 385
10.2 Stability of Linear Systems......Page 391
10.3 Linearization and Local Stability......Page 399
10.4 Autonomous Systems as Mathematical Models......Page 409
CHAPTER 10 IN REVIEW......Page 416
11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES......Page 418
11.1 Orthogonal Functions......Page 419
11.2 Fourier Series......Page 424
11.3 Fourier Cosine and Sine Series......Page 429
11.4 Sturm-Liouville Problem......Page 437
11.5 Bessel and Legendre Series......Page 444
11.5.1 Fourier-Bessel Series......Page 445
11.5.2 Fourier-Legendre Series......Page 448
CHAPTER 11 IN REVIEW......Page 451
12 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES......Page 453
12.1 Separable Partial Differential Equations......Page 454
12.2 Classical PDEs and Boundary-Value Problems......Page 458
12.3 Heat Equation......Page 464
12.4 Wave Equation......Page 466
12.5 Laplace's Equation......Page 471
12.6 Nonhomogeneous Boundary-Value Problems......Page 476
12.7 Orthogonal Series Expansions......Page 482
12.8 Higher-Dimensional Problems......Page 487
CHAPTER 12 IN REVIEW......Page 490
13 BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS......Page 492
13.1 Polar Coordinates......Page 493
13.2 Polar and Cylindrical Coordinates......Page 498
13.3 Spherical Coordinates......Page 504
CHAPTER 13 IN REVIEW......Page 507
14 INTEGRAL TRANSFORMS......Page 509
14.1 Error Function......Page 510
14.2 Laplace Transform......Page 511
14.3 Fourier Integral......Page 519
14.4 Fourier Transforms......Page 525
CHAPTER 14 IN REVIEW......Page 531
15 NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS......Page 532
15.1 Laplace's Equation......Page 533
15.2 Heat Equation......Page 538
15.3 Wave Equation......Page 543
CHAPTER 15 IN REVIEW......Page 547
I: Gamma Function......Page 548
II: Matrices......Page 550
III: Laplace Transforms......Page 568
Answers for Selected Odd-Numbered Problems......Page 572
Index......Page 602