Elementary Differential Equations and Boundary Value Problems

Cover......Page 1
Introduction......Page 2
Copyright......Page 6
Dedication......Page 7
About the Authors......Page 8
Preface......Page 9
Table of Contents......Page 16
1.1 Some Basic Mathematical Models; Direction Fields......Page 19
1.1 Problems......Page 26
1.2 Solutions of Some Differential Equations......Page 27
1.2 Problems......Page 32
1.3 Classification of Differential Equations......Page 35
1.3 Problems......Page 40
1.4 Historical Remarks......Page 41
2.1 Linear Equations with Variable Coefficients......Page 47
2.1 Problems......Page 56
2.2 Separable Equations......Page 58
2.2 Problems......Page 63
2.3 Modeling with First Order Equations......Page 65
2.3 Problems......Page 75
2.4 Differences Between Linear and Nonlinear Equations......Page 82
2.4 Problems......Page 90
2.5 Autonomous Equations and Population Dynamics......Page 92
2.5 Problems......Page 102
2.6 Exact Equations and Integrating Factors......Page 107
2.6 Problems......Page 113
2.7 Numerical Approximations: Euler’s Method......Page 114
2.7 Problems......Page 121
2.8 The Existence and Uniqueness Theorem......Page 123
2.8 Problems......Page 131
2.9 First Order Difference Equations......Page 133
2.9 Problems......Page 142
3.1 Homogeneous Equations with Constant Coefficients......Page 147
3.1 Problems......Page 154
3.2 Fundamental Solutions of Linear Homogeneous Equations......Page 155
3.2 Problems......Page 163
3.3 Linear Independence and the Wronskian......Page 165
3.3 Problems......Page 170
3.4 Complex Roots of the Characteristic Equation......Page 171
3.4 Problems......Page 176
3.5 Repeated Roots; Reduction of Order......Page 178
3.5 Problems......Page 184
3.6 Nonhomogeneous Equations; Method of Undetermined Coefficients......Page 187
3.6 Problems......Page 196
3.7 Variation of Parameters......Page 197
3.7 Problems......Page 201
3.8 Mechanical and Electrical Vibrations......Page 204
3.8 Problems......Page 215
3.9 Forced Vibrations......Page 218
3.9 Problems......Page 223
4.1 General Theory of nth Order Linear Equations......Page 227
4.1 Problems......Page 230
4.2 Homogeneous Equations with Constant Coeffients......Page 232
4.2 Problems......Page 237
4.3 The Method of Undetermined Coefficients......Page 240
4.3 Problems......Page 242
4.4 The Method of Variation of Parameters......Page 244
4.4 Problems......Page 247
5.1 Review of Power Series......Page 249
5.1 Problems......Page 255
5.2 Series Solutions near an Ordinary Point, Part I......Page 256
5.2 Problems......Page 265
5.3 Series Solutions near an Ordinary Point, Part II......Page 267
5.3 Problems......Page 271
5.4 Regular Singular Points......Page 273
5.4 Problems......Page 277
5.5 Euler Equations......Page 278
5.5 Problems......Page 283
5.6 Series Solutions near a Regular Singular Point, Part I......Page 285
5.6 Problems......Page 289
5.7 Series Solutions near a Regular Singular Point, Part II......Page 290
5.7 Problems......Page 296
5.8 Bessel’s Equation......Page 298
5.8 Problems......Page 307
6.1 Definition of the Laplace Transform......Page 311
6.1 Problems......Page 316
6.2 Solution of Initial Value Problems......Page 317
6.2 Problems......Page 325
6.3 Step Functions......Page 328
6.3 Problems......Page 332
6.4 Differential Equations with Discontinuous Forcing Functions......Page 335
6.4 Problems......Page 339
6.5 Impulse Functions......Page 342
6.5 Problems......Page 346
6.6 The Convolution Integral......Page 348
6.6 Problems......Page 353
7.1 Introduction......Page 357
7.1 Problems......Page 362
7.2 Review of Matrices......Page 366
7.2 Problems......Page 373
7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors......Page 375
7.3 Problems......Page 384
7.4 Basic Theory of Systems of First Order Linear Equations......Page 386
7.4 Problems......Page 389
7.5 Homogeneous Linear Systems with Constant Coefficients......Page 391
7.5 Problems......Page 399
7.6 Complex Eigenvalues......Page 402
7.6 Problems......Page 408
7.7 Fundamental Matrices......Page 411
7.7 Problems......Page 418
7.8 Repeated Eigenvalues......Page 419
7.8 Problems......Page 425
7.9 Nonhomogeneous Linear Systems......Page 429
7.9 Problems......Page 435
8.1 The Euler or Tangent Line Method......Page 437
8.1 Problems......Page 445
8.2 Improvements on the Euler Method......Page 448
8.2 Problems......Page 452
8.3 The Runge–Kutta Method......Page 453
8.3 Problems......Page 456
8.4 Multistep Methods......Page 457
8.4 Problems......Page 462
8.5 More on Errors; Stability......Page 463
8.5 Problems......Page 472
8.6 Systems of First Order Equations......Page 473
8.6 Problems......Page 475
9.1 The Phase Plane; Linear Systems......Page 477
9.1 Problems......Page 486
9.2 Autonomous Systems and Stability......Page 489
9.2 Problems......Page 495
9.3 Almost Linear Systems......Page 497
9.3 Problems......Page 505
9.4 Competing Species......Page 509
9.4 Problems......Page 519
9.5 Predator–Prey Equations......Page 521
9.5 Problems......Page 527
9.6 Liapunov’s Second Method......Page 529
9.6 Problems......Page 537
9.7 Periodic Solutions and Limit Cycles......Page 539
9.7 Problems......Page 548
9.8 Chaos and Strange Attractors; the Lorenz Equations......Page 550
9.8 Problems......Page 556
10.1 Two-Point Boundary Valve Problems......Page 559
10.2 Fourier Series......Page 565
10.2 Problems......Page 573
10.3 The Fourier Convergence Theorem......Page 576
10.3 Problems......Page 580
10.4 Even and Odd Functions......Page 582
10.4 Problems......Page 588
10.5 Separation of Variables; Heat Conduction in a Rod......Page 591
10.5 Problems......Page 597
10.6 Other Heat Conduction Problems......Page 599
10.6 Problems......Page 606
10.7 The Wave Equation; Vibrations of an Elastic String......Page 609
10.7 Problems......Page 618
10.8 Laplace’s Equation......Page 622
10.8 Problems......Page 629
Appendix A. Derivation of the Heat Conduction Equation......Page 632
Appendix B. Derivation of the Wave Equation......Page 635
11.1 The Occurrence of Two Point Boundary Value Problems......Page 639
11.1 Problems......Page 644
11.2 Sturm–Liouville Boundary Value Problems......Page 647
11.2 Problems......Page 657
11.3 Nonhomogeneous Boundary Value Problems......Page 659
11.3 Problems......Page 669
11.4 Singular Sturm–Liouville Problems......Page 674
11.4 Problems......Page 679
11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion......Page 681
11.5 Problems......Page 684
11.6 Series of Orthogonal Functions: Mean Convergence......Page 687
11.6 Problems......Page 693
Ch 1......Page 697
Ch 2......Page 698
Ch 3......Page 705
Ch 4......Page 711
Ch 5......Page 713
Ch 6......Page 721
Ch 7......Page 724
Ch 8......Page 733
Ch 9......Page 737
Ch 10......Page 742
Ch 11......Page 749
Index......Page 755
SSM - Introduction......Page 764
SSM - Table of Contents......Page 768
Student Solutions Manual......Page 769
Ch 2......Page 774
Ch 3......Page 803
Ch 4......Page 832
Ch 5......Page 841
Ch 6......Page 871
Ch 7......Page 889
Ch 8......Page 928
Ch 9......Page 942
Ch 10......Page 971
Ch 11......Page 1003
ODE Workbook Introduction......Page 1025
copyright......Page 1028
Workbook Preface......Page 1029
Acknowledgments......Page 1032
Modules/Chapters Overview......Page 1033
Contents......Page 1038
1. Modeling with the ODE Architect......Page 1042
Building a Model of the Pacific Sardine Population......Page 1043
The Logistic Equation......Page 1051
Introducing Harvesting via Landing Data......Page 1053
How to Model in Eight Steps......Page 1056
Explorations......Page 1058
2. Introduction to ODEs......Page 1066
Solutions to Differential Equations......Page 1067
Slope Fields......Page 1068
Finding a Solution Formula......Page 1069
The Juggler......Page 1071
The Sky Diver......Page 1072
Explorations......Page 1076
3. Some Cool ODEs......Page 1084
Finding a General Solution......Page 1085
Time-Dependent Outside Temperature......Page 1087
Air Conditioning a Room......Page 1088
The Case of the Melting Snowman......Page 1090
Explorations......Page 1092
4. Second-Order Linear Equations......Page 1098
Undamped Oscillations......Page 1099
The Effect of Damping......Page 1101
Forced Oscillations......Page 1102
Beats......Page 1104
Seismographs......Page 1105
Explorations......Page 1110
5. Models of Motion......Page 1118
Vectors......Page 1119
Forces and Newton\'s Laws......Page 1120
Dunk Tank......Page 1121
Longer to Rise or to Fall?......Page 1122
Indiana Newton......Page 1123
Ski Jumping......Page 1125
Explorations......Page 1126
6. First-Order Linear Systems......Page 1134
Examples of Systems: Pizza and Video, Coupled Springs......Page 1135
Linear Systems with Constant Coefficients......Page 1136
Solution Formulas: Eigenvalues and Eigenvectors......Page 1138
Calculating Eigenvalues and Eigenvectors......Page 1139
Phase Portraits......Page 1140
Using ODE Architect to Find Eigenvalues and Eigenvectors......Page 1143
Parameter Movies......Page 1144
Explorations......Page 1146
7. Nonlinear Systems......Page 1156
The Geometry of Nonlinear Systems......Page 1157
Linearization......Page 1158
Separatrices and Saddle Points......Page 1161
Behavior of Solutions Away from Equilibrium Points......Page 1162
Bifurcation to a Limit Cycle......Page 1163
Spinning Bodies: Stability of Steady Rotations......Page 1164
The Planar Double Pendulum......Page 1167
Explorations......Page 1170
8. Compartment Models......Page 1176
Lake Polution......Page 1177
Allergy Relief......Page 1178
Lead in the Body......Page 1180
Equilibrium......Page 1182
The Autocatalator and a Hopf Bifurcation......Page 1183
Explorations......Page 1188
9. Population Models......Page 1196
The Logistic Model......Page 1197
Two-Species Population Models......Page 1199
Predator and Prey......Page 1200
Species Competition......Page 1201
Mathematical Epidemiology: The SIR Model......Page 1202
Explorations......Page 1204
10. The Pendulum and Its Friends......Page 1214
Modeling Pendulum Motion......Page 1215
Conservative Systems: Integrals of Motion......Page 1217
The Effect of Damping......Page 1218
Separatrices......Page 1221
Writing the Equations of Motion for Pumping a Swing......Page 1223
Geodesics......Page 1226
Geodesics on a Surface of Revolution......Page 1227
Geodesics on a Torus......Page 1229
Explorations......Page 1234
11. Applications of Series Solutions......Page 1244
Recurrence Formulas......Page 1245
Ordinary Points......Page 1247
Regular Singular Points......Page 1248
Bessel Functions......Page 1250
Transforming Bessel\'s Equation to the Aging Spring Equation......Page 1251
Explorations......Page 1254
12. Chaos and Control......Page 1262
Solutions as Functions of Time......Page 1263
Poincare Sections......Page 1264
The Unforced Pendulum......Page 1265
Tangled Basins, the Wada Property......Page 1267
Gaining Control......Page 1269
Explorations......Page 1272
13. Discrete Dynamical Systems......Page 1274
Equilibrium States......Page 1276
Linear vs. Nonlinear Dynamics......Page 1277
Stability of a Discrete Dynamical System......Page 1278
Bifurcations......Page 1279
Periodic and Chaotic Dynamics......Page 1281
What is Chaos?......Page 1282
Complex Numbers and Functions......Page 1283
Iterating a Complex Function......Page 1284
Julia Sets, the Mandelbrot Set, and Cantor Dust......Page 1285
Explorations......Page 1290
Glossary......Page 1298