Differential Equations for Engineers

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Background......Page 15
Scope......Page 16
Acknowledgments......Page 18
1.1 Motivating Examples......Page 19
1.2 General Concepts and Definitions......Page 24
General and Particular Solutions......Page 26
Boundary Value Problem......Page 28
Existence and Uniqueness of Solutions......Page 29
2.1 The Method of Separation of Variables......Page 34
2.2.1 Homogeneous Equations......Page 38
2.2.2 Special Transformations......Page 43
2.3 Exact Differential Equations and Integrating Factors......Page 49
2.3.1 Exact Differential Equations......Page 50
Method 2: The Method of Grouping Terms......Page 52
2.3.2 Integrating Factors......Page 57
2.3.3 Method of Inspection......Page 63
2.3.4 Integrating Factors by Groups......Page 66
2.4.1 Linear First-Order Equations......Page 73
2.4.2 Bernoulli Differential Equations......Page 76
Case 1. Equation Solvable for Variable y......Page 79
Case 2. Equation Solvable for Variable x......Page 80
The Clairaut Equation......Page 85
2.6.1 Equations Immediately Integrable......Page 86
2.6.2 The Dependent Variable Absent......Page 88
2.6.3 The Independent Variable Absent......Page 90
2.7 Summary......Page 92
First-Order Ordinary Differential Equation......Page 93
Simple Higher-Order Differential Equation......Page 95
Homogeneous and Special Transformations......Page 96
Integrating Factors......Page 97
Method of Inspection......Page 98
Linear First-Order Equations......Page 99
Bernoulli Differential Equations......Page 100
Simple Higher-Order Differential Equations......Page 101
Review Problems......Page 102
3.1 Heating and Cooling Simple Higher-Order Equations......Page 105
Case I: Upward Motion......Page 109
Case II: Downward Motion......Page 111
3.3.1 The Suspension Bridge......Page 115
3.3.2 Cable under Self-Weight......Page 120
3.4 Electric Circuits......Page 126
Series RC Circuit......Page 128
3.5 Natural Purification in a Stream......Page 132
3.6 Various Application Problems......Page 138
Problems......Page 148
The D-Operator......Page 158
Properties of the D-Operator......Page 159
4.2.1 Characteristic Equation Having Real Distinct Roots......Page 161
4.2.2 Characteristic Equation Having Complex Roots......Page 165
4.2.3 Characteristic Equation Having Repeated Roots......Page 169
4.3.1 Method of Undetermined Coefficients......Page 171
4.3.2 Method of Operators......Page 180
4.3.3 Method of Variation of Parameters......Page 191
4.4 Euler Differential Equations......Page 196
Complementary Solution......Page 198
Particular Solution......Page 199
Complementary Solutions......Page 201
Particular Solutions — D-Operator Method......Page 202
General Solutions......Page 203
Method of Variation of Parameters......Page 204
Euler Differential Equations......Page 205
5.1.1 Formulation—Equation of Motion......Page 206
1. Vibration of a Shear Building under Externally Applied Force F(t)......Page 207
2. Vibration of a Shear Building under Base Excitation x0(t)......Page 208
5.1.2.1 Free Vibration—Complementary Solution......Page 211
Special Case: Undamped System ζ = 0, ωd = ω0......Page 212
Underdamped Free Vibration......Page 214
Case 3: Overdamped System ζ >1......Page 217
5.1.2.2 Forced Vibration—Particular Solution......Page 218
Resonance......Page 221
Undamped System under Sinusoidal Excitation......Page 222
Parallel RLC Circuit......Page 227
5.3 Vibration of a Vehicle Passing a Speed Bump......Page 231
Phase 1: On the Speed Bump 0 ≤ t ≤ T, T = b/U......Page 232
Phase 2: Passed the Speed Bump t ≥ T, T = b/U......Page 234
Numerical Results......Page 235
5.4 Beam-Columns......Page 236
5.5 Various Application Problems......Page 241
Problems......Page 250
6.1 The Laplace Transform......Page 262
6.2 The Heaviside Step Function......Page 267
6.3 Impulse Functions and the Dirac Delta Function......Page 272
6.4 The Inverse Laplace Transform......Page 275
6.5 Solving Differential Equations Using the Laplace Transform......Page 281
6.6.1 Response of a Single Degree-of-Freedom System......Page 286
Free Vibration......Page 287
Forced Vibration......Page 288
6.6.2 Other Applications......Page 293
Case 1. Underdamped System......Page 296
Case 2. Critically Damped System......Page 297
Case 3. Overdamped System......Page 298
6.6.3 Beams on Elastic Foundation......Page 301
6.7 Summary......Page 307
Problems......Page 309
7.1 Introduction......Page 318
7.2.1 Complementary Solutions......Page 322
7.2.2 Particular Solutions......Page 325
Method of Variation of Parameters......Page 332
7.3 The Method of Laplace Transform......Page 336
7.4 The Matrix Method......Page 343
Distinct Eigenvalues......Page 344
Complex Eigenvalues......Page 346
Multiple Eigenvalues......Page 348
7.4.2 Particular Solutions......Page 352
Undamped Free Vibration......Page 362
Orthogonality of Mode Shapes......Page 363
Damped Forced Vibration......Page 364
7.5.1 The Method of Operator......Page 365
7.5.2 The Method of Laplace Transform......Page 366
Case 3. Multiple Eigenvalues......Page 367
General Solutions......Page 368
The Method of Operator......Page 369
The Method of Laplace Transform......Page 370
The Matrix Method......Page 372
8.1 Mathematical Modeling of Mechanical Vibrations......Page 375
8.2 Vibration Absorbers or Tuned Mass Dampers......Page 384
8.3 An Electric Circuit......Page 390
Method of Laplace Transform......Page 391
Matrix Method......Page 393
8.4 Vibration of a Two-Story Shear Building......Page 395
8.4.1 Free Vibration—Complementary Solutions......Page 396
8.4.2 Vibration GeneralForced— Solutions......Page 398
Problems......Page 402
9 Series Solutions of Differential Equations......Page 408
9.1 Review of Power Series......Page 409
9.2 Series Solution about an Ordinary Point......Page 412
9.3 Series Solution about a Regular Singular Point......Page 421
9.3.1.1 Solutions of Bessel’s Equation......Page 426
Case 2. ν = 0, then α1 = α2 = 0......Page 429
Case 3. ν is a positive integer......Page 431
9.3.2 Applications of Bessel’s Equation......Page 436
9.4 Summary......Page 442
Problems......Page 444
10.1 Numerical Solutions of First-Order Initial Value......Page 449
10.1.1 The Euler Method or Constant Slope Method......Page 450
10.1.2 Error Analysis......Page 452
Truncation Error of the Euler Method......Page 453
10.1.3 The Backward Euler Method......Page 454
10.1.4 Improved Euler Method—Average Slope Method......Page 455
10.1.5 The Runge-Kutta Methods......Page 458
10.2 Numerical Solutions of Systems of Differential Equations......Page 463
10.3 Stiff Differential Equations......Page 467
Explicit Methods......Page 470
Implicit Methods......Page 471
Problems......Page 472
11.1 Simple Partial Differential Equations......Page 475
11.2 Method of Separation of Variables......Page 476
11.3.1 Formulation—Equation of Motion......Page 483
11.3.2 Free Vibration......Page 484
11.3.3 Forced Vibration......Page 489
Equation of Heat Conduction......Page 491
11.4.2 Two-Dimensional Steady-State Heat Conduction......Page 494
11.4.3 One-Dimensional Transient Heat Conduction......Page 498
11.4.4 One-Dimensional Transient Heat Conduction on a Semi-Infinite Interval......Page 501
11.4.5 Three-Dimensional Steady-State Heat Conduction......Page 506
11.5 Summary......Page 510
Problems......Page 511
12 Solving Ordinary Differential Equations Using Maple......Page 516
12.1.1 Simple Ordinary Differential Equations......Page 517
12.1.2 Linear Ordinary Differential Equations......Page 524
12.1.3 The Laplace Transform......Page 525
12.1.4 Systems of Ordinary Differential Equations......Page 527
Eigenvalues and Eigenvectors of a Matrix......Page 528
Series Solutions of Differential Equations......Page 532
12.2 Series Solutions of Differential Equations......Page 530
12.3 Numerical Solutions of Differential Equations......Page 535
Problems......Page 544
A.1 Table of Trigonometric Identities......Page 549
A.2 Table of Derivatives......Page 551
A.3 Table of Integrals......Page 552
A.4 Table of Laplace Transforms......Page 555
A.5 Table of Inverse Laplace Transforms......Page 557
Index......Page 560