ALFRED GRAY MICHAEL MEZZINO MARKA. PINSKY Introduction to Ordinary Differential Equations with Mathematica An Integrated Multimedia Approach %JmT} Web-Enhanced Includes CD-ROM
TABLE OF CONTENTS Preface Acknowledgments vii xiii 1. Basic Concepts 1 1.1 The Notion ofa Differential Equation 1 1.2 Sources of Differential Equations 4 1.3 Solving Differential Equations 7 2. Using Mathematica 17 2.1 Getting Started with Mathematica 17 2.2 Mathematica Notation versus Ordinary Mathematical Notation 21 2.3 Plotting in Mathematica 29 3. First-Order Differential Equations 35 3.1 Introduction to First-Order Equations 35 xv
XVi Table of Contents 3.2 First-Order Linear Equations 39 3.3 Separable Equations 47 3.4 Exact Equations and Integrating Factors 53 3.5 Homogeneous First-Order Equations 63 3.6 Bernoulli Equations 67 4. The Package ODE.m 73 4.1 Getting Started with ODE 74 4.2 Features o/ode 77 4.3 Plotting with ODE 81 4.4 First-Order Linear Equations via ODE 85 4.5 Separable Equations via ODE 91 4.6 First-Order Equations with Integrating Factors via ODE 94 4.7 First-Order Homogeneous Equations via ODE 97 4.8 Bernoulli Equations via ODE 100 4.9 Clairaut and Lagrange Equations via ODE 102 4.10 Nonelementary Integrals 108 4.11 Using ODE to Define New Functions 111 4.12 Riccati Equations 113 5. Existence and Uniqueness of Solutions of First-Order Differential Equations 119 5.1 The Existence and Uniqueness Theorem 120 5.2 Explosions and a Criterionfor Global Existence 727 5.3 Picard Iteration 131 5.4 Proofs of Existence Theorems 137 5.5 Direction Fields and Differential Equations 141 5.6 Stability Analysis of Nonlinear First-Order Equations 145
Table of Contents XVM 6. Applications of First-Order Equations 1 155 6.1 Population Models with Constant Growth Rate 156 6.2 Population Models with Variable Growth Rate 166 6.3 Logistic Model of Population Growth 170 6.4 Population Growth with Harvesting 178 6.5 Population Models for the United States 184 6.6 Temperature Equalization Models 193 7. Applications of First-Order Equations II 203 7.1 Application of First-Order Equations to Elementary Mechanics 203 7.2 Rocket Propulsion 214 7.3 Electrical Circuits 218 7.4 Mixing Problems 225 7.5 Pursuit Curves 231 8. Second-Order Linear Differential Equations 235 8.1 General Forms and Examples 236 8.2 Existence and Uniqueness Theory 241 8.3 Fundamental Sets of Solutions to the Homogeneous Equation 245 8.4 The Wronskian 247 8.5 Linear Independence and the Wronskian 254 8.6 Reduction of Order 258 8.7 Equations with Given Solutions 260
XVIII Table of Contents 9. Second-Order Linear Differential Equations with Constant Coefficients 265 9.1 Constant-Coefficient Second-Order Homogeneous Equations 266 9.2 Complex Constant-Coefficient Second-Order Homogeneous Equations 276 9.3 The Method of Undetermined Coefficients 283 9.4 The Method of Variation of Parameters 294 10. Using ODE to Solve Second-Order Linear Differential Equations 303 10.1 Using ODE to Solve Second-Order Constant-Coefficient Equations 304 10.2 Details of ODE for Second-Order Constant-Coefficient Equations 316 10.3 Reduction of Order and Trial Solutions via ODE 320 10.4 Equations with Given Solutions via ODE 323 11. Applications of Linear Second-Order Equations.. 325 11.1 Mass-Spring Systems 325 11.2 Forced Vibrations of Mass-Spring Systems 350 11.3 Electrical Circuits 368 11.4 Sound 377 12. Higher-Order Linear Differential Equations 381 12.1 General Forms 382 12.2 Constant-Coefficient Higher-Order Homogeneous Equations 386 12.3 Variation of Parameters for Higher-Order Equations 393
Table of Contents xix 12.4 Higher-Order Differential Equations via ODE 398 12.5 Seminumerical Solutions of Higher-Order Constant-Coefficient Equations 400 13. Numerical Solutions of Differential Equations... 407 13.1 The Euler Method 408 13.2 The Heun Method 415 13.3 The Runge-Kutta Method 420 13.4 Solving Differential Equations Numerically with ODE 424 13.5 ODE's Implementation of Numerical Methods 428 13.6 Using NDSolve 434 13.7 Adaptive Step Size and Error Control 439 13.8 The Numerov Method 443 14. The Laplace Transform 449 14.1 Definition and Properties ofthe Laplace Transform 450 14.2 Piecewise Continuous Functions 456 14.3 Using the Laplace Transform to Solve Initial Value Problems 459 14.4 The Gamma Function 466 14.5 Computation of Laplace Transforms 470 14.6 Step Functions 474 14.7 Second-Order Equations with Piecewise Continuous Forcing Functions 479 14.8 Impulse Functions 482 14.9 Convolution 487 14.10 Laplace Transforms via ODE 489
XX Table of Contents 15. Systems of Linear Differential Equations 495 15.1 Notation and Definitions for Systems 496 15.2 Existence and Uniqueness Theorems for Systems 500 15.3 Solution of Upper Triangulär Systems by Elimination 505 15.4 Homogeneous Linear Systems 507 15.5 Constant-Coefficient Homogeneous Systems 515 15.6 The Method of Undetermined Coefficientsfor Systems 531 15.7 The Method of Variation of Parameters for Systems 543 15.8 Solving Systems Using the Laplace Transform 546 16. Phase Portraits of Linear Systems 557 16.1 Phase Portraits of Two-Dimensional Linear Systems 551 16.2 Using ODE to Solve Linear Systems 569 16.3 Phase Portraits of Two-Dimensional Linear Systems via ODE 577 17. Stability of Nonlinear Systems 581 17.1 Curves 582 17.2 Autonomous Systems 583 17.3 Critical Points of Systems of Differential Equations 586 17.4 Stability and Asymptotic Stability of Nonlinear Systems 592 17.5 Stability by Linearized Approximation 595 17.6 Lyapunov Stability Theory 604 18. Applications of Linear Systems 617 18.1 Coupled Systems of Oscillators 617 18.2 Electrical Circuits 628 18.3 Markov Chains 634
Table of Contents xxi 19. Applications of Nonlinear Systems 647 19.1 Numerical Solutions of Systems of Differential Equations 648 19.2 Predator-Prey Modeling 654 19.3 The Van Der Pol Equation 660 19.4 The Simple Pendulum 663 19.5 The Fundamental Theorem of Plane Curves 674 20. Power Series Solutions of Second-Order Equations 679 20.1 Review of Power Series 681 20.2 Power Series via Mathematica 688 20.3 Power Series Solutions about an Ordinary Point 697 20.4 The Airy Equation 698 20.5 The Legendre Equation 702 20.6 Convergence of Series Solutions 709 20.7 Series Solutions of Differential Equations Using ODE 777 21. Frobenius Solutions of Second-Order Equations 715 21.1 Solutions about a Regulär Singular Point 776 27.2 The Cauchy-Euler Equation 77 7 27.3 Method of Frobenius: The First Solution 726 21.4 Bessel Functions I 730 21.5 Method of Frobenius: The Second Solution 735 21.6 Bessel Functions II 741 21.7 Bessel Functions via Mathematica 744
Table of Contents 21.8 An Aging Spring 748 21.9 The Hypergeometric Equation 752 A. Appendix: Review of Linear Algebra and Matrix Theory 759 A.l Vector and Matrix Notation 759 A.2 Determinants and Inverses 763 A3 Systems of Linear Equations and Determinants 768 A.4 Eigenvalues and Eigenvectors 773 A.5 The Exponential of a Matrix 784 A.6 Abstract Vector Spaces 787 A.7 Vectors and Matrices with Mathematica 791 A.8 Solving Equations with Mathematica 797 A.9 Eigenvalues and Eigenvectors with Mathematica 802 B. Appendix: Systems of Units 807 Answers 811 Bibliography 869 General Index 875 Name Index 885 Miniprogram and Mathematica Index 887