This book presents a systematic and comprehensive introduction to ordinary differential equations for engineering students and practitioners. Mathematical concepts and various techniques are presented in a clear, logical, and concise manner. Various visual features are used to highlight focus areas. Complete illustrative diagrams are used to facilitate mathematical modeling of application problems. Readers are motivated by a focus on the relevance of differential equations through their applications in various engineering disciplines. Studies of various types of differential equations are determined by engineering applications. Theory and techniques for solving differential equations are then applied to solve practical engineering problems. Detailed step-by-step analysis is presented to model the engineering problems using differential equations from physical principles and to solve the differential equations using the easiest possible method. Such a detailed, step-by-step approach, especially when applied to practical engineering problems, helps the readers to develop problem-solving skills. This book is suitable for use not only as a textbook on ordinary differential equations for undergraduate students in an engineering program but also as a guide to self-study. It can also be used as a reference after students have completed learning the subject.

Preface xiii

1 Introduction 1

1.1 Motivating Examples 1

1.2 General Concepts and Definitions 6

2 First-Order and Simple Higer-Orde Differential Equations 16

2.1 The Method of Separation of Variables 16

2.2 Method of Transformation of Variables 20

      2.2.1 Homogeneous Equations 20

      2.2.2 Special Transformations 25

2.3 Exact Differential Equations and Integrating Factors 31

      2.3.1 Exact Differential Equations 32

      2.3.2 Integrating Factors 39

      2.3.3 Method of Inspection 45

      2.3-4 Integrating Factors by Groups 48

2.4 Linear First-Order Equations 55

      2.4.1 Linear First-Order Equations 55

      2.4.2 Bernoulli Differential Equations 58

2.5 Equations Solvable for the Independent or Dependent Variable 61

2.6 Simple Higher-Order Differential Equations 68

      2.6.1 Equations Immediately Integrable 68

      2.6.2 The Dependent Variable Absent

      2.6.3 The Independent Variable Absent

2.7 Summary Problems

3 Applications of First-Order Simple Higher-Order Equations 87

3.1 Heating and Cooling 87

3.2 Motion of a Particle in a Resisting Medium 91

3.3 Hanging Cables 97

      3.3.1 The Suspension Bridge 97

      3.3.2 Cable under Self Weight 102

3.4 Electric Circuits 108

3.5 Natural Purification in a Stream 114

3.6 Various Application Problems Problems 120

4 Linear Differential Equations 130

4.1 General Linear Ordinary Differential Equations 140

4.2 Complementary Solutions 143

      4.2.1 Characteristic Equation Having Real Distinct Roots 143

      4.2.2 Characteristic Equation Having Complex Roots 147

      4.2.3 Characteristic Equation Having Repeated Roots 151

4.3 Particular Solutions 153

      4.3.1 Method of Undetermined Coefficients 153

      4.3.2 Method of Operators 162

      4.3.3 Method of Variation of Parameters 173

4.4 Euler Differential Equations 178

4.5 Summary Problems 180

5 Applications of Linear Differential Equations 183

5.1 Vibration of a Single Degree of Freedom System 188

      5.1.1 Formulation-Equation of Motion 188

      5.1.2 Response of a Single Degree-oιFreedom System 193

      5.1.2.1 Free Vibration-Complementary Solution 193

      5.1.2.2 Forced Vibration-Particular Solution 200

5.2 Electric Circuits 209

5.3 Vibration of a Vehicle Passing a Speed Bump 213

5.4 Beam-Columns 218

5.5 Various Application Problems 223

Prolems 232

6 The Laplace Transform and Its Applications 244

6.1 The Laplace Transform 244

6.2 The Heaviside Step Function 249

6.3 Impulse Functions and the Dirac Delta Function 254

6.4 The Inverse Laplace Transform 257

6.5 Solving Differential Equations Using the Laplace Transform 263

6.6 Applications of the Laplace Transform 268

      6.6.1 Response of a Single Degree-oιFreedom System 275

      6.6.2 Other Applications 283

      6.6.3 Beams on Elastic Foundation 289

6.7 Summary Problems 291

7 Systems of Linear Differential Equations 300

7.1 Introduction 300

7.2 The Method of Operator 304

      7.2.1 Complementary Solutions 307

      7.2.2 Particular Solutions 318

7.3 The Method of Laplace Transform 325

7.4 The Matrix Method 326

      7.4.1 Complementary Solutions 334

      7.4.2 Particular Solutions 344

      7.4.3 Response of Multiple Degrees-oιFreedom Systems 347

7.5 Sumrηary 347

      7.5.1 The Method of Operator 348

      7.5.2 The Method of Laplace Transform 349

      7.5.3 The Matrix Method Problems 351

8 Applications of Systems of Linear Differential Equations 357

8.1 Mathematical Modeling of Mechanical Vibrations 357

8.2 Vibration Absorbers or Tuned Mass Dampers 366

8.3 An Electric Circuit 372

8.4 Vibration of a Two-Story Shear Building 377

      8.4.1 Free Vibration -Complementary Solutions 378

      8.4.2 Forced Vibration-General Solutions 380

      Problems 384

9 Series Solutions of Differential Equations 390

9.1 Review of Power Series 391

9.2 Series Solution about an Ordinary Point 394

9.3 Series Solution about a Regular Singular Point 403

      9.3.1 Bessel’SEquation and Its Applications 408

      9.3.1.1 Solutions of Bessel’s Equation 418

      9.3.2 Applications of Bessel’sEquation 424

9.4 Summary 424

Problems 426

10 Numerical Solutions of Differential Equations 431

10.1 Numerical Solutions of First-Order Initial Value Problems 431

      10.1.1 The Euler Method or Constant Slope Method 432

      10.1.2 Error Analysis 434

      10.1.3 The Backward Euler Method 436

      10.1.4 Improved Euler Method-Average Slope Method 437

      10.1.5 The Runge-Kutta Methods 440

10.2 Numerical Solutions of Systems of Differential Equations 445

10.3 Stiff Differential Equations 449

10.4 Summary 452

Problems 454

11 Patial Differential Equations 457

11.1 Simple Partial Differential Equations 457

11.2 Method of Separation of Variables 458

11.3 Application- Flexural Motion of Beams 465

      11.3.1 Formulation -Equation of Motion 465

      11.3.2 Free Vibration 466

      11.3.3 Forced Vibration 471

11.4 Application- Heat Conduction 473

      11.4.1 Formulation -Heat Equation 473

      11.4.2 Two-Dimensional Steady-State Heat Conduction 476

      11.4.3 One-Dimensional Transient Heat Conduction 480

      11.4.4 One-Dimensional Transient Heat Conduction on a Semi-Infinite Interval 483

      11.4.5 Three-Dimensional Steady-State Heat Conduction 488

11.5 Summary 491

Problems 493

12 Solving Ordinary Differential Euations Using Maple 498

12.1 Closed-Form Solutions of Differential Equations 499

      12.1.1 Simple Ordinary Differential Equations 499

      12.1.2 Linear Ordinary Differential Equations 506

      12.1.3 The Laplace Transform 507

      12.1.4 Systems of Ordinary Differential Equations 509

12.2 Series Solutions of Differential Equations 512

12.3 Numerical Solutions of Differential Equations 517

Problems 526

Appendix A Tables of Mathematical Formulas 531

A.I Table of Trigonometric Identities 531

A.2 Table of Derivatives 533

A.3 Table of Integrals 534

A.4 Table of Laplace Transforms 537

A.5 Table of Inverse Laplace Transforms 539

Index 542

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