Introductory Differential Equations (eBook)

Front Cover 1
Introductory Differential Equations 4
Copyright 5
Contents 6
Preface 8
Technology 9
Applications 9
Style 9
Features 10
Pedagogical Features 10
Content 12
Chapter 1: Introduction to Differential Equations 14
1.1 Introduction to Differential Equations: Vocabulary 16
Exercises 1.1 23
1.2 A Graphical Approach to Solutions: Slope Fields and Direction Fields 28
Exercises 1.2 32
Chapter 1 Summary: Essential Concepts and Formulas 35
Chapter 1 Review Exercises 36
Chapter 2: First-Order Equations 40
2.1 Introduction to First-Order Equations 40
Exercises 2.1 45
2.2 Separable Equations 46
Exercises 2.2 52
2.3 First-Order Linear Equations 57
Exercises 2.3 62
2.4 Exact Differential Equations 66
Exercises 2.4 70
2.5 Substitution Methodsand Special Equations 73
Exercises 2.5 77
2.6 Numerical Methods for First-Order Equations 82
Exercises 2.6 91
Chapter 2 Summary: Essential Concepts and Formulas 93
Chapter 2 Review Exercises 93
Differential Equations at Work 96
A. Modeling the Spread of a Disease 96
B. Linear Population Model with Harvesting 97
C. Logistic Model with Harvesting 98
D. Logistic Model with Predation 99
References 100
Chapter 3: Applications of First-Order Differential Equations 102
3.1 Population Growth and Decay 102
Exercises 3.1 110
3.2 Newton's Law of Cooling and Related Problems 115
Exercises 3.2 120
3.3 Free-Falling Bodies 122
Exercises 3.3 128
Chapter 3 Summary: Essential Concepts and Formulas 131
Chapter 3 Review Exercises 132
Differential Equations at Work 137
A. Mathematics of Finance 137
B. Algae Growth 139
C. Dialysis 140
D. Antibiotic Production 142
Chapter 4: Higher Order Equations 144
4.1 Second-Order Equations: An Introduction 145
Exercises 4.1 154
4.2 Solutions of Second-Order Linear Homogeneous Equations with Constant Coefficients 156
Exercises 4.2 160
4.3 Solving Second-Order Linear Equations: Undetermined Coefficients 162
Exercises 4.3 168
4.4 Solving Second-Order Linear Equations: Variation of Parameters 171
Exercises 4.4 176
4.5 Solving Higher Order Linear Homogeneous Equations 179
Exercises 4.5 188
4.6 Solving Higher Order Linear Equations: Undetermined Coefficients and Variation of Parameters 192
Exercises 4.6 200
4.7 Cauchy-Euler Equations 202
Exercises 4.7 208
4.8 Power Series Solutions of Ordinary Differential Equations 210
Exercises 4.8 216
4.9 Series Solutions of Ordinary Differential Equations 219
Exercises 4.9 228
Chapter 4 Summary: Essential Concepts and Formulas 231
Chapter 4 Review Exercises 232
Differential Equations at Work 234
A. Testing for Diabetes 234
B. Modeling the Motion of a Skier 235
C. The Schrödinger Equation 237
Chapter 5: Applications of Higher Order Differential Equations 240
5.1 Simple Harmonic Motion 240
Exercises 5.1 245
5.2 Damped Motion 247
Exercises 5.2 254
5.3 Forced Motion 256
Exercises 5.3 262
5.4 Other Applications 265
Exercises 5.4 270
5.5 The Pendulum Problem 272
Exercises 5.5 275
Chapter 5 Summary: Essential Concepts and Formulas 278
Chapter 5 Review Exercises 278
Differential Equations at Work 282
A. Rack-and-Gear Systems 282
B. Soft, Hard, and Aging Springs 283
C. Bodé Plots 284
D. The Catenary 285
E. The Wave Equation on a Circular Plate 285
F. Duffing’s Equation 286
G. Suspending an Object from a Cable 286
H. Can Resonance Impact Machinery? 287
I. Inventory Management 287
J. Heat Transfer 287
Chapter 6: Systems of Differential Equations 290
6.1 Introduction 290
Exercises 6.1 295
6.2 Review of Matrix Algebra and Calculus 298
Exercises 6.2 306
6.3 An Introduction to Linear Systems 308
Exercises 6.3 314
6.4 First-Order Linear Homogeneous Systems with Constant Coefficients 317
Exercises 6.4 329
6.5 First-Order Linear Nonhomogeneous Systems: Undetermined Coefficients and Variation of Parameters 332
Exercises 6.5 338
6.6 Phase Portraits 342
Exercises 6.6 352
6.7 Nonlinear Systems 354
Exercises 6.7 358
6.8 Numerical Methods 363
Exercises 6.8 368
Chapter 6 Summary: Essential Concepts and Formulas 370
Chapter 6 Review Exercises 370
Differential Equations at Work 372
A. Modeling a Fox Population in Which Rabies Is Present 372
B. Controlling the Spread of a Disease 373
C. FitzHugh-Nagumo Model 375
D. An Agricultural Model 375
E. Modeling the Spread of Dengue in Indonesia 376
Chapter 7: Applications of Systems of Ordinary Differential Equations 378
7.1 Mechanical and Electrical Problems with First-Order Linear Systems 378
Exercises 7.1 383
7.2 Diffusion and Population Problems with First-Order Linear Systems 385
Exercises 7.2 391
7.3 Nonlinear Systems of Equations 393
Exercises 7.3 397
Chapter 7 Summary: Essential Concepts and Formulas 402
Chapter 7 Review Exercises 402
Differential Equations at Work 405
A. Competing Species 405
B. Food Chains 405
C. Chemical Reactor 407
D. Food Chains in a Chemostat 408
E. The Rössler System and Attractor 409
F. Cell Dynamics in Colon Cancer 410
Chapter 8: Introduction to the Laplace Transform 412
8.1 The Laplace Transform: Preliminary Definitions and Notation 413
Exercises 8.1 418
8.2 The Inverse Laplace Transform 420
Exercises 8.2 423
8.3 Solving Initial-Value Problems with the Laplace Transform 424
Exercises 8.3 427
8.4 Laplace Transforms of Several Important Functions 428
Exercises 8.4 437
8.5 The Convolution Theorem 440
Exercises 8.5 443
8.6 Laplace Transform Methods for Solving Systems 444
Exercises 8.6 447
8.7 Some Applications Using Laplace Transforms 448
Exercises 8.7 455
Chapter 8 Summary: Essential Concepts and Formulas 464
Chapter 8 Review Exercises 465
Differential Equations at Work 468
A. The Tautochrone 468
B. Vibration Absorbers 469
C. Airplane Wing 470
D. Free Vibration of a Three-Story Building 471
E. Control Systems 473
Answers to Selected Exercises 474
Bibliography 520
Appendices 522
Index 526