CONTENTS Preface xi 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1 1.1 Definitions and Terminology 2 1.2 Initial-Value Problems 13 1.3 Differential Equations as Mathematical Models 19 CHAPTER 1 IN REVIEW 32 2 FIRST-ORDER DIFFERENTIAL EQUATIONS 34 2.1 Solution Curves Without a Solution 35 2.1.1 Direction Fields 35 2.1.2 Autonomous First-Order DEs 37 2.2 Separable Variables 44 2.3 Linear Equations 53 2.4 Exact Equations 62 2.5 Solutions by Substitutions 70 2.6 A Numerical Method 75 CHAPTER 2 IN REVIEW 80 3 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 82 3.1 Linear Models 83 3.2 Nonlinear Models 94 3.3 Modeling with Systems of First-Order DEs 105 CHAPTER 3 IN REVIEW 113
3 v CONTENTS 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1 Preface xi 1.1 Definitions and Terminology 2 1.2 Initial-Value Problems 13 1.3 Differential Equations as Mathematical Models 19 CHAPTER 1 IN REVIEW 32 2 FIRST-ORDER DIFFERENTIAL EQUATIONS 34 2.1 Solution Curves Without a Solution 35 2.1.1 Direction Fields 35 2.1.2 Autonomous First-Order DEs 37 2.2 Separable Variables 44 2.3 Linear Equations 53 2.4 Exact Equations 62 2.5 Solutions by Substitutions 70 2.6 A Numerical Method 75 CHAPTER 2 IN REVIEW 80 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 82 3.1 Linear Models 83 3.2 Nonlinear Models 94 3.3 Modeling with Systems of First-Order DEs 105 CHAPTER 3 IN REVIEW 113
i ·CONTENTS 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS 117 4.1 Preliminary Theory-Linear Equations 118 4.1.1 Initial-Value and Boundary-Value Problems 118 4.1.2 Homogeneous Equations 120 4.1.3 Nonhomogeneous Equations 125 4.2 Reduction of Order 130 4.3 Homogeneous Linear Equations with Constant Coefficients 133 4.4 Undetermined Coefficients-Superposition Approach 140 4.5 Undetermined Coefficients-Annihilator Approach 150 4.6 Variation of Parameters 157 4.7 Cauchy-Euler Equation 162 4.8 Solving Systems of Linear DEs by Elimination 169 4.9 Nonlinear Differential Equations 174 CHAPTER 4 IN REVIEW 178 5 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 181 5.1 Linear Models:Initial-Value Problems 182 5.1.1 Spring/Mass Systems:Free Undamped Motion 182 5.1.2 Spring/Mass Systems:Free Damped Motion 186 5.1.3 Spring/Mass Systems:Driven Motion 189 5.1.4 Series Circuit Analogue 192 5.2 Linear Models:Boundary-Value Problems 199 5.3 Nonlinear Models 207 CHAPTER 5 IN REVIEW 216 6 SERIES SOLUTIONS OF LINEAR EQUATIONS 219 6.1 Solutions About Ordinary Points 220 6.1.1 Review of Power Series 220 6.1.2 Power Series Solutions 223 6.2 Solutions About Singular Points 231 6.3 Special Functions 241 6.3.1 Bessel's Equation 241 6.3.2 Legendre's Equation 248 CHAPTER 6 IN REVIEW 253
5 4 vi ● CONTENTS HIGHER-ORDER DIFFERENTIAL EQUATIONS 117 4.1 Preliminary Theory—Linear Equations 118 4.1.1 Initial-Value and Boundary-Value Problems 118 4.1.2 Homogeneous Equations 120 4.1.3 Nonhomogeneous Equations 125 4.2 Reduction of Order 130 4.3 Homogeneous Linear Equations with Constant Coefficients 133 4.4 Undetermined Coefficients—Superposition Approach 140 4.5 Undetermined Coefficients—Annihilator Approach 150 4.6 Variation of Parameters 157 4.7 Cauchy-Euler Equation 162 4.8 Solving Systems of Linear DEs by Elimination 169 4.9 Nonlinear Differential Equations 174 CHAPTER 4 IN REVIEW 178 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 181 5.1 Linear Models: Initial-Value Problems 182 5.1.1 Spring/Mass Systems: Free Undamped Motion 182 5.1.2 Spring/Mass Systems: Free Damped Motion 186 5.1.3 Spring/Mass Systems: Driven Motion 189 5.1.4 Series Circuit Analogue 192 5.2 Linear Models: Boundary-Value Problems 199 5.3 Nonlinear Models 207 CHAPTER 5 IN REVIEW 216 SERIES SOLUTIONS OF LINEAR EQUATIONS 219 6.1 Solutions About Ordinary Points 220 6.1.1 Review of Power Series 220 6.1.2 Power Series Solutions 223 6.2 Solutions About Singular Points 231 6.3 Special Functions 241 6.3.1 Bessel’s Equation 241 6.3.2 Legendre’s Equation 248 CHAPTER 6 IN REVIEW 253 6
CONTENTS vii 7 THE LAPLACE TRANSFORM 255 7.1 Definition of the Laplace Transform 256 7.2 Inverse Transforms and Transforms of Derivatives 262 7.2.1 Inverse Transforms 262 7.2.2 Transforms of Derivatives 265 7.3 Operational Properties I 270 7.3.1 Translation on the s-Axis 271 7.3.2 Translation on the t-Axis 274 7.4 Operational Properties II 282 7.4.1 Derivatives of a Transform 282 7.4.2 Transforms of Integrals 283 7.4.3 Transform of a Periodic Function 287 7.5 The Dirac Delta Function 292 7.6 Systems of Linear Differential Equations 295 CHAPTER 7 IN REVIEW 300 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 303 8.1 Preliminary Theory-Linear Systems 304 8.2 Homogeneous Linear Systems 311 8.2.1 Distinct Real Eigenvalues 312 8.2.2 Repeated Eigenvalues 315 8.2.3 Complex Eigenvalues 320 8.3 Nonhomogeneous Linear Systems 326 8.3.1 Undetermined Coefficients 326 8.3.2 Variation of Parameters 329 8.4 Matrix Exponential 334 CHAPTER 8 IN REVIEW 337 9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 339 9.1 Euler Methods and Error Analysis 340 9.2 Runge-Kutta Methods 345 9.3 Multistep Methods 350 9.4 Higher-Order Equations and Systems 353 9.5 Second-Order Boundary-Value Problems 358 CHAPTER 9 IN REVIEW 362
CONTENTS ● vii 7 THE LAPLACE TRANSFORM 255 7.1 Definition of the Laplace Transform 256 7.2 Inverse Transforms and Transforms of Derivatives 262 7.2.1 Inverse Transforms 262 7.2.2 Transforms of Derivatives 265 7.3 Operational Properties I 270 7.3.1 Translation on the s-Axis 271 7.3.2 Translation on the t-Axis 274 7.4 Operational Properties II 282 7.4.1 Derivatives of a Transform 282 7.4.2 Transforms of Integrals 283 7.4.3 Transform of a Periodic Function 287 7.5 The Dirac Delta Function 292 7.6 Systems of Linear Differential Equations 295 CHAPTER 7 IN REVIEW 300 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 303 8.1 Preliminary Theory—Linear Systems 304 8.2 Homogeneous Linear Systems 311 8.2.1 Distinct Real Eigenvalues 312 8.2.2 Repeated Eigenvalues 315 8.2.3 Complex Eigenvalues 320 8.3 Nonhomogeneous Linear Systems 326 8.3.1 Undetermined Coefficients 326 8.3.2 Variation of Parameters 329 8.4 Matrix Exponential 334 CHAPTER 8 IN REVIEW 337 9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 339 9.1 Euler Methods and Error Analysis 340 9.2 Runge-Kutta Methods 345 9.3 Multistep Methods 350 9.4 Higher-Order Equations and Systems 353 9.5 Second-Order Boundary-Value Problems 358 CHAPTER 9 IN REVIEW 362
viii CONTENTS 10 PLANE AUTONOMOUS SYSTEMS 363 10.1 Autonomous Systems 364 10.2 Stability of Linear Systems 370 10.3 Linearization and Local Stability 378 10.4 Autonomous Systems as Mathematical Models 388 CHAPTER 10 IN REVIEW 395 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 397 11.1 Orthogonal Functions 398 11.2 Fourier Series 403 11.3 Fourier Cosine and Sine Series 408 11.4 Sturm-Liouville Problem 416 11.5 Bessel and Legendre Series 423 11.5.1 Fourier-Bessel Series 424 11.5.2 Fourier-Legendre Series 427 CHAPTER 11 IN REVIEW 430 12 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 432 12.1 Separable Partial Differential Equations 433 12.2 Classical PDEs and Boundary-Value Problems 437 12.3 Heat Equation 443 12.4 Wave Equation 445 12.5 Laplace's Equation 450 12.6 Nonhomogeneous Boundary-Value Problems 455 12.7 Orthogonal Series Expansions 461 12.8 Higher-Dimensional Problems 466 CHAPTER 12 IN REVIEW 469
viii ● CONTENTS 10 PLANE AUTONOMOUS SYSTEMS 363 10.1 Autonomous Systems 364 10.2 Stability of Linear Systems 370 10.3 Linearization and Local Stability 378 10.4 Autonomous Systems as Mathematical Models 388 CHAPTER 10 IN REVIEW 395 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 397 11.1 Orthogonal Functions 398 11.2 Fourier Series 403 11.3 Fourier Cosine and Sine Series 408 11.4 Sturm-Liouville Problem 416 11.5 Bessel and Legendre Series 423 11.5.1 Fourier-Bessel Series 424 11.5.2 Fourier-Legendre Series 427 CHAPTER 11 IN REVIEW 430 12 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 432 12.1 Separable Partial Differential Equations 433 12.2 Classical PDEs and Boundary-Value Problems 437 12.3 Heat Equation 443 12.4 Wave Equation 445 12.5 Laplace’s Equation 450 12.6 Nonhomogeneous Boundary-Value Problems 455 12.7 Orthogonal Series Expansions 461 12.8 Higher-Dimensional Problems 466 CHAPTER 12 IN REVIEW 469
CONTENTS 。i 13 BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS 471 13.1 Polar Coordinates 472 13.2 Polar and Cylindrical Coordinates 477 13.3 Spherical Coordinates 483 CHAPTER 13 IN REVIEW 486 14 INTEGRAL TRANSFORMS 488 14.1 Error Function 489 14.2 Laplace Transform 490 14.3 Fourier Integral 498 14.4 Fourier Transforms 504 CHAPTER 14 IN REVIEW 510 15 NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 511 15.1 Laplace's Equation 512 15.2 Heat Equation 517 15.3 Wave Equation 522 CHAPTER 15 IN REVIEW 526 APPENDICES Gamma Function APP-1 I Matrices APP-3 Laplace Transforms APP-21 Answers for Selected Odd-Numbered Problems ANS-1 Index I-1
CONTENTS ● ix 13 BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS 471 13.1 Polar Coordinates 472 13.2 Polar and Cylindrical Coordinates 477 13.3 Spherical Coordinates 483 CHAPTER 13 IN REVIEW 486 14 INTEGRAL TRANSFORMS 488 14.1 Error Function 489 14.2 Laplace Transform 490 14.3 Fourier Integral 498 14.4 Fourier Transforms 504 CHAPTER 14 IN REVIEW 510 15 NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 511 15.1 Laplace’s Equation 512 15.2 Heat Equation 517 15.3 Wave Equation 522 CHAPTER 15 IN REVIEW 526 APPENDICES I Gamma Function APP-1 II Matrices APP-3 III Laplace Transforms APP-21 Answers for Selected Odd-Numbered Problems ANS-1 Index I-1