Introduction to Ordinary Differential Equations with Mathematica
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1 ALFRED GRAY MICHAEL MEZZINO MARKA. PINSKY Introduction to Ordinary Differential Equations with Mathematica An Integrated Multimedia Approach %JmT} Web-Enhanced Includes CD-ROM
2 TABLE OF CONTENTS Preface Acknowledgments vii xiii 1. Basic Concepts The Notion ofa Differential Equation Sources of Differential Equations Solving Differential Equations 7 2. Using Mathematica Getting Started with Mathematica Mathematica Notation versus Ordinary Mathematical Notation Plotting in Mathematica First-Order Differential Equations Introduction to First-Order Equations 35 xv
3 XVi Table of Contents 3.2 First-Order Linear Equations Separable Equations Exact Equations and Integrating Factors Homogeneous First-Order Equations Bernoulli Equations The Package ODE.m Getting Started with ODE Features o/ode Plotting with ODE First-Order Linear Equations via ODE Separable Equations via ODE First-Order Equations with Integrating Factors via ODE First-Order Homogeneous Equations via ODE Bernoulli Equations via ODE Clairaut and Lagrange Equations via ODE Nonelementary Integrals Using ODE to Define New Functions Riccati Equations Existence and Uniqueness of Solutions of First-Order Differential Equations The Existence and Uniqueness Theorem Explosions and a Criterionfor Global Existence Picard Iteration Proofs of Existence Theorems Direction Fields and Differential Equations Stability Analysis of Nonlinear First-Order Equations 145
4 Table of Contents XVM 6. Applications of First-Order Equations Population Models with Constant Growth Rate Population Models with Variable Growth Rate Logistic Model of Population Growth Population Growth with Harvesting Population Models for the United States Temperature Equalization Models Applications of First-Order Equations II Application of First-Order Equations to Elementary Mechanics Rocket Propulsion Electrical Circuits Mixing Problems Pursuit Curves Second-Order Linear Differential Equations General Forms and Examples Existence and Uniqueness Theory Fundamental Sets of Solutions to the Homogeneous Equation The Wronskian Linear Independence and the Wronskian Reduction of Order Equations with Given Solutions 260
5 XVIII Table of Contents 9. Second-Order Linear Differential Equations with Constant Coefficients Constant-Coefficient Second-Order Homogeneous Equations Complex Constant-Coefficient Second-Order Homogeneous Equations The Method of Undetermined Coefficients The Method of Variation of Parameters Using ODE to Solve Second-Order Linear Differential Equations Using ODE to Solve Second-Order Constant-Coefficient Equations Details of ODE for Second-Order Constant-Coefficient Equations Reduction of Order and Trial Solutions via ODE Equations with Given Solutions via ODE Applications of Linear Second-Order Equations Mass-Spring Systems Forced Vibrations of Mass-Spring Systems Electrical Circuits Sound Higher-Order Linear Differential Equations General Forms Constant-Coefficient Higher-Order Homogeneous Equations Variation of Parameters for Higher-Order Equations 393
6 Table of Contents xix 12.4 Higher-Order Differential Equations via ODE Seminumerical Solutions of Higher-Order Constant-Coefficient Equations Numerical Solutions of Differential Equations The Euler Method The Heun Method The Runge-Kutta Method Solving Differential Equations Numerically with ODE ODE's Implementation of Numerical Methods Using NDSolve Adaptive Step Size and Error Control The Numerov Method The Laplace Transform Definition and Properties ofthe Laplace Transform Piecewise Continuous Functions Using the Laplace Transform to Solve Initial Value Problems The Gamma Function Computation of Laplace Transforms Step Functions Second-Order Equations with Piecewise Continuous Forcing Functions Impulse Functions Convolution Laplace Transforms via ODE 489
7 XX Table of Contents 15. Systems of Linear Differential Equations Notation and Definitions for Systems Existence and Uniqueness Theorems for Systems Solution of Upper Triangulär Systems by Elimination Homogeneous Linear Systems Constant-Coefficient Homogeneous Systems The Method of Undetermined Coefficientsfor Systems The Method of Variation of Parameters for Systems Solving Systems Using the Laplace Transform Phase Portraits of Linear Systems Phase Portraits of Two-Dimensional Linear Systems Using ODE to Solve Linear Systems Phase Portraits of Two-Dimensional Linear Systems via ODE Stability of Nonlinear Systems Curves Autonomous Systems Critical Points of Systems of Differential Equations Stability and Asymptotic Stability of Nonlinear Systems Stability by Linearized Approximation Lyapunov Stability Theory Applications of Linear Systems Coupled Systems of Oscillators Electrical Circuits Markov Chains 634
8 Table of Contents xxi 19. Applications of Nonlinear Systems Numerical Solutions of Systems of Differential Equations Predator-Prey Modeling The Van Der Pol Equation The Simple Pendulum The Fundamental Theorem of Plane Curves Power Series Solutions of Second-Order Equations Review of Power Series Power Series via Mathematica Power Series Solutions about an Ordinary Point The Airy Equation The Legendre Equation Convergence of Series Solutions Series Solutions of Differential Equations Using ODE Frobenius Solutions of Second-Order Equations Solutions about a Regulär Singular Point The Cauchy-Euler Equation Method of Frobenius: The First Solution Bessel Functions I Method of Frobenius: The Second Solution Bessel Functions II Bessel Functions via Mathematica 744
9 Table of Contents 21.8 An Aging Spring 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
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