A First Course in Differential Equations with Modeling Applications

Cover......Page 1
Contents......Page 4
Preface......Page 8
Chapter 1: Introduction to Differential Equations......Page 13
1.1 Denitions and Terminology......Page 14
1.2 Initial-Value Problems......Page 26
1.3 Differential Equations as Mathematical Models......Page 33
Chapter 1 In Review......Page 45
Chapter 2: First-Order Differential Equations......Page 47
2.1.1 Direction Fields......Page 48
2.1.2 Autonomous First-Order DEs......Page 50
2.2 Separable Equations......Page 58
2.3 Linear Equations......Page 66
2.4 Exact Equations......Page 75
2.5 Solutions by Substitutions......Page 83
2.6 A Numerical Method......Page 87
Chapter 2 In Review......Page 92
Chapter 3: Modeling with First-Order Differential Equations......Page 95
3.1 Linear Models......Page 96
3.2 Nonlinear Models......Page 107
3.3 Modeling with Systems of First-Order DEs......Page 118
Chapter 3 In Review......Page 125
Chapter 4: Higher-Order Differential Equations......Page 129
4.1.1 Initial-Value and Boundary-Value Problems......Page 130
4.1.2 Homogeneous Equations......Page 132
4.1.3 Nonhomogeneous Equations......Page 138
4.2 Reduction of Order......Page 143
4.3 Homogeneous Linear Equations with Constant
 Coefcients......Page 146
4.4 Undetermined Coefcients—Superposition Approach......Page 153
4.5 Undetermined Coefcients—Annihilator Approach......Page 163
4.6 Variation of Parameters......Page 170
4.7 Cauchy-Euler Equations......Page 177
4.8.1 Initial-Value Problems......Page 184
4.8.2 Boundary-Value Problems......Page 190
4.9 Solving Systems of Linear DEs byElimination......Page 194
4.10 Nonlinear Differential Equations......Page 199
Chapter 4 In Review......Page 204
Chapter 5: Modeling with Higher-Order Differential Equations......Page 207
5.1.1 Spring/Mass Systems: Free Undamped Motion......Page 208
5.1.2 Spring/Mass Systems: Free Damped Motion......Page 213
5.1.3 Spring/Mass Systems: Driven Motion......Page 215
5.1.4 Series Circuit Analogue......Page 218
5.2 Linear Models: Boundary-Value Problems......Page 224
5.3 Nonlinear Models......Page 233
Chapter 5 In Review......Page 243
Chapter 6: Series Solutions of Linear Equations......Page 247
6.1 Review of Power Series......Page 248
6.2 Solutions About Ordinary Points......Page 254
6.3 Solutions About Singular Points......Page 263
6.4 Special Functions......Page 273
Chapter 6 In Review......Page 287
Chapter 7: The Laplace Transform......Page 289
7.1 Denition of the Laplace Transform......Page 290
7.2.1 Inverse Transforms......Page 297
7.2.2 Transforms of Derivatives......Page 300
7.3 Operational Properties I......Page 305
7.3.1 Translation on the S-Axis......Page 306
7.3.2 Translation on the t-Axis......Page 309
7.4.1 Derivatives of a Transform......Page 317
7.4.2 Transforms of Integrals......Page 318
7.4.3 Transform of a Periodic Function......Page 324
7.5 The Dirac Delta Function......Page 329
7.6 Systems of Linear Differential Equations......Page 333
Chapter 7 In Review......Page 338
Chapter 8: Systems of Linear First-Order Differential Equations......Page 343
8.1 Preliminary Theory—Linear Systems......Page 344
8.2 Homogeneous Linear Systems......Page 351
8.2.1 Distinct Real Eigenvalues......Page 352
8.2.2 Repeated Eigenvalues......Page 355
8.2.3 Complex Eigenvalues......Page 359
8.3.1 Undetermined Coefcients......Page 366
8.3.2 Variation of Parameters......Page 368
8.4 Matrix Exponential......Page 373
Chapter 8 In Review......Page 377
Chapter 9: Numerical Solutions of Ordinary Differential Equations......Page 379
9.1 Euler Methods and Error Analysis......Page 380
9.2 Runge-Kutta Methods......Page 385
9.3 Multistep Methods......Page 389
9.4 Higher-Order Equations and Systems......Page 392
9.5 Second-Order Boundary-Value Problems......Page 396
Chapter 9 In Review......Page 400
Appendices......Page 401
Appendix A: Integral-Defined Functions......Page 402
Appendix B: Matrices......Page 410
Appendix C: Laplace Transforms......Page 428
Answers for Selected Odd-Numbered Problems......Page 431
Index......Page 449