A First Course in Differential Equations

Cover......Page 1
EPP2......Page 2
EPP3......Page 3
Title Page
......Page 7
Copyright......Page 8
Statement......Page 9
Contents......Page 11
To the Instructor......Page 15
Student Resources......Page 16
Reviewers of Past Editions......Page 17
Reviewers of the Current Editions......Page 19
Is AIDS an Invariably Fatal Disease?......Page 23
Related Problems......Page 25
About the Author......Page 26
The Allee Effect......Page 27
About the Author......Page 29
Wolf Population Dynamics......Page 30
About the Author......Page 32
Bungee Jumping......Page 33
Related Problems......Page 34
About the Author......Page 35
The Collapse of the Tacoma Narrows Suspension Bridge......Page 36
Related Problems......Page 38
About the Author......Page 39
Murder at the Mayfair Diner......Page 40
Related Problems......Page 41
About the Author......Page 42
Earthquake Shaking of Multistory Buildings......Page 43
Related Problems......Page 46
Modeling Arms Races......Page 47
Related Problems......Page 48
About the Author......Page 50
Introduction......Page 51
1.1 Definitions and Terminology......Page 52
1.2 Initial-Value Problems......Page 63
1.3 Differential Equations as Mathematical Models......Page 70
Chapter 1 in Review......Page 83
Introduction......Page 85
2.1.1 Direction Fields......Page 86
2.1.2 Autonomous First-Order DEs......Page 88
2.2 Separable Equations......Page 96
2.3 Linear Equations......Page 104
2.4 Exact Equations......Page 113
2.5 Solutions by Substitutions......Page 121
2.6 A Numerical Method......Page 125
Chapter 2 in Review......Page 130
Introduction......Page 133
3.1 Linear Models......Page 134
3.2 Nonlinear Models......Page 145
3.3 Modeling with Systems of First-Order DEs......Page 156
Chapter 3 in Review......Page 163
Introduction......Page 166
4.1.1 Initial-Value and Boundary-Value Problems......Page 167
4.1.2 Homogeneous Equations......Page 169
4.1.3 Nonhomogeneous Equations......Page 174
4.2 Reduction of Order......Page 179
4.3 Homogeneous Linear Equations with Constant Coefficients......Page 182
4.4 Undetermined Coefficients—Superposition Approach......Page 189
4.5 Undetermined Coefficients—Annihilator Approach......Page 199
4.6 Variation of Parameters......Page 206
4.7 Cauchy-Euler Equation......Page 212
4.8.1 Initial-Value Problems......Page 219
4.8.2 Boundary-Value Problems......Page 226
4.9 Solving Systems of Linear DEs by Elimination......Page 230
4.10 Nonlinear Differential Equations......Page 235
Chapter 4 in Review......Page 240
Introduction......Page 242
5.1.1 Spring/Mass Systems: Free Undamped Motion......Page 243
5.1.2 Spring/Mass Systems: Free Damped Motion......Page 247
5.1.3 Spring/Mass Systems: Driven Motion......Page 250
5.1.4 Series Circuit Analogue......Page 253
5.2 Linear Models: Boundary-Value Problems......Page 260
5.3 Nonlinear Models......Page 268
Chapter 5 in Review......Page 278
Introduction......Page 281
6.1 Review of Power Series......Page 282
6.2 Solutions About Ordinary Points......Page 288
6.3 Solutions About Singular Points......Page 297
6.4 Special Functions......Page 307
Chapter 6 in Review......Page 321
Introduction......Page 323
7.1 Definition of the Laplace Transform......Page 324
7.2.1 Inverse Transforms......Page 331
7.2.2 Transforms of Derivatives......Page 334
7.3 Operational Properties I......Page 339
7.3.1 Translation on the s-Axis......Page 340
7.3.2 Translation on the t-Axis......Page 343
7.4.1 Derivatives of a Transform......Page 351
7.4.2 Transforms of Integrals......Page 352
7.4.3 Transform of a Periodic Function......Page 357
7.5 The Dirac Delta Function......Page 362
7.6 Systems of Linear Differential Equations......Page 365
Chapter 7 in Review......Page 370
Introduction......Page 375
8.1 Preliminary Theory—Linear Systems......Page 376
8.2 Homogeneous Linear Systems......Page 383
8.2.1 Distinct Real Eigenvalues......Page 384
8.2.2 Repeated Eigenvalues......Page 387
8.2.3 Complex Eigenvalues......Page 392
8.3.1 Undetermined Coefficients......Page 398
8.3.2 Variation of Parameters......Page 401
8.4 Matrix Exponential......Page 406
Chapter 8 in Review......Page 410
Introduction......Page 412
9.1 Euler Methods and Error Analysis......Page 413
9.2 Runge-Kutta Methods......Page 418
9.3 Multistep Methods......Page 423
9.4 Higher-Order Equations and Systems......Page 425
9.5 Second-Order Boundary-Value Problems......Page 430
Chapter 9 in Review......Page 434
Appendix I: Gamma Function......Page 435
Appendix II: Matrices......Page 437
Appendix III: Laplace Transforms......Page 455
Answers for Selected Odd-Numbered Problems......Page 459
C......Page 477
E......Page 478
H......Page 479
L......Page 480
M......Page 481
P......Page 482
S......Page 483
Z......Page 484
EPP6......Page 488
EPP7......Page 489