Differential Equations with Mathematica THIRD EDITION Martha L. Abell James P. Braselton ELSEVIER ACADEMIC PRESS Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo
Contents Preface xiii I Introduction to Differential Equations 1.1 Definitions and Concepts 1.2 Solutions of Differential Equations 1.3 Initial and Boundary-Value Problems 1.4 Direction Fields 2 6 18 26 2 First-Order Ordinary Differential Equations 2.1 Theory of First-Order Equations: A Brief Discussion 2.2 Separation of Variables Application: Kidney Dialysis 2.3 Homogeneous Equations Application: Models of Pursuit 2.4 Exact Equations 2.5 Linear Equations 2.5.1 Integrating Factor Approach 2.5.2 Variation of Parameters and the Method of Undetermined Coefficients Application: Antibiotic Production 2.6 Numerical Approximations of Solutions to First -Order Equations 2.6.1 Built-In Methods 41 41 46 55 59 64 69 74 75 86 89 92 92
vi Contents Application: Modeling the Spread of a Disease 97 2.6.2 Other Numerical Methods 103 3 Applications of First-Order Ordinary Differential Equations 119 3.1 Orthogonal Trajectories 119 Application: Oblique Trajectories 129 3.2 Population Growth and Decay 132 3.2.1 The Malthus Model 132 3.2.2 The Logistic Equation 138 Application: Harvesting 148 Application: The Logistic Differente Equation 152 3.3 Newton's Law of Cooling 157 3.4 Free-Falling Bodies 163 4 Higher-Order Differential Equations 175 4.1 Preliminary Definitions and Notation 175 4.1.1 Introduction 175 4.1.2 The nth-order Ordinary Linear Differential Equation 180 4.1.3 Fundamental Set of Solutions 185 4.1.4 Existente of a Fundamental Set of Solutions 191 4.1.5 Reduction of Order 193 4.2 Solving Homogeneous Equations with Constant Coefficients 196 4.2.1 Second-Order Equations 196 4.2.2 Higher-Order Equations 200 Application: Testing for Diabetes 211 4.3 Introduction to Solving Nonhomogeneous Equations with Constant Coefficients 216 4.4 Nonhomogeneous Equations with Constant Coefficients: The Method of Undetermined Coefficients 222 4.4.1 Second-Order Equations 223 4.4.2 Higher-Order Equations 239 4.5 Nonhomogeneous Equations with Constant Coefficients: Variation of Parameters 248 4.5.1 Second-Order Equations 248 4.5.2 Higher-Order Nonhomogeneous Equations 252
Contents +Aro. Vii 91114110,.);1n n.. 4.6 Cauchy Euler Equations 255 4.6.1 Second-Order Cauchy Euler Equations 255 4.6.2 Higher-Order Cauchy Euler Equations 261 4.6.3 Variation of Parameters 265 4.7 Series Solutions 268 4.7.1 Power Series Solutions about Ordinary Points 268 4.7.2 Series Solutions about Regular Singular Points 281 4.7.3 Method of Frobenius 283 Application: Zeros of the Bessel Functions of the First Kind 295 Application: The Wave Equation an a Circular Plate 298 4.8 Nonlinear Equations 304 5 Applications of Higher-Order Differential Equations 321 5.1 Harmonic Motion 321 5.1.1 Simple Harmonic Motion 321 5.1.2 Damped Motion 332 5.1.3 Forced Motion 346 5.1.4 Soft Springs 365 5.1.5 Hard Springs 368 5.1.6 Aging Springs 370 Application: Hearing Beats and Resonance 372 5.2 The Pendulum Problem 373 5.3 Other Applications 387 5.3.1 L R C Circuits 387 5.3.2 Deflection of a Beam 390 5.3.3 Bode Plots 393 5.3.4 The Catenary 398 6 Systems of Ordinary Differential Equations 41 1 6.1 Review of Matrix Algebra and Calculus 411 6.1.1 Defining Nested Lists, Matrices, and Vectors 411 6.1.2 Extracting Elements of Matrices 416 6.1.3 Basic Computations with Matrices 419 6.1.4 Eigenvalues and Eigenvectors 422 6.1.5 Matrix Calculus 426 6.2 Systems of Equations: Preliminary Definitions and Theory 427 6.2.1 Preliminary Theory 429
viii Contents 6.2.2 Linear Systems 446 6.3 Homogeneous Linear Systems with Constant Coefficients 454 6.3.1 Distinct Real Eigenvalues 454 6.3.2 Complex Conjugate Eigenvalues 461 6.3.3 Alternate Method for Solving Initial-Value Problems 474 6.3.4 Repeated Eigenvalues 477 6.4 Nonhomogeneous First-Order Systems: Undetermined Coefficients, Variation of Parameters, and the Matrix Exponential 485 6.4.1 Undetermined Coefficients 485 6.4.2 Variation of Parameters 490 6.4.3 The Matrix Exponential 498 6.5 Numerical Methods 506 6.5.1 Built-In Methods 506 Application: Controlling the Spread of a Disease 513 6.5.2 Euler's Method 525 6.5.3 Runge Kutta Method 531 6.6 Nonlinear Systems, Linearization, and Classification of Equilibrium. Points 535 6.6.1 Real Distinct Eigenvalues 535 6.6.2 Repeated Eigenvalues 543 6.6.3 Complex Conjugate Eigenvalues 548 6.6.4 Nonlinear Systems 552 7 Applications of Systems of Ordinary Differential Equations 567 7.1 Mechanical and Electrical Problems with First-Order Linear Systems 567 7.1.1 L R C Circuits with Loops 567 7.1.2 L R C Circuit with One Loop 568 7.1.3 L R C Circuit with Two Loops 571 7.1.4 Spring Mass Systems 574 7.2 Diffusion and Population Problems with First-Order Linear Systems 576 7.2.1 Diffusion through a Membrane 576 7.2.2 Diffusion through a Double-Walled Membrane 578 7,2.3 Population Problems 583 7.3 Applications that Lead to Nonlinear Systems 587 7.3.1 Biological Systems: Predator-Prey Inferactions, The Lotka-Volterra System, and Food Chains in the Chemostat 587
Contents ix 7.3.2 Physical Systems: Variable Damping 604 7.3.3 Differential Geometry: Curvature 611 8 Laplace Transform Methods 617 8.1 The Laplace Transform 618 8.1.1 Definition of the Laplace Transform 618 8.1.2 Exponential Order 621 8.1.3 Properties of the Laplace Transform 623 8.2 The Inverse Laplace Transform 629 8.2.1 Definition of the Inverse Laplace Transform 629 8.2.2 Laplace Transform of an Integral 635 8.3 Solving Initial-Value Problems with the Laplace Transform 637 8.4 Laplace Transforms of Step and Periodic Functions 645 8.4.1 Piecewise-Defined Functions: The Unit Step Function 645 8.4.2 Solving Initial-Value Problems 649 8.4.3 Periodic Functions 652 8.4.4 Impulse Functions: The Delta Function 661 8.5 The Convolution Theorem 667 8.5.1 The Convolution Theorem 667 8.5.2 Integral and Integrodifferential Equations 669 8.6 Applications of Laplace Transforms, Part I 672 8.6.1 Spring Mass Systems Revisited 672 8.6.2 L R C Circuits Revisited 679 8.6.3 Population Problems Revisited 687 Application: The Tautochrone 689 8.7 Laplace Transform Methods for Systems 691 8.8 Applications of Laplace Transforms, Part II 708 8.8.1 Coupled Spring Mass Systems 708 8.8.2 The Double Pendulum 714 Application: Free Vibration of a Three-Story Building 720 9 Eigenvalue Problems and Fourier Series 727 9.1 Boundary-Value Problems, Eigenvalue Problems, Sturm Liouville Problems 727 9.1.1 Boundary-Value Problems 727
x Contents 9.1.2 Eigenvalue Problems 730 9.1.3 Sturm Liouville Problems 735 9.2 Fourier Sine Series and Cosine Series 737 9.2.1 Fourier Sine Series 737 9.2.2 Fourier Cosine Series 746 9.3 Fourier Series 749 9.3.1 Fourier Series 749 9.3.2 Even, Odd, and Periodic Extensions 758 9.3.3 Differentiation and Integration of Fourier Series 764 9.3.4 Parseval's Equality 768 9.4 Generalized Fourier Series 770 10 Partial Differential Equations 783 10.1 Introduction to Partial Differential Equations and Separation of Variables 783 10.1.1 Introduction 783 10.1.2 Separation of Variables 785 10.2 The One-Dimensional Heat Equation 787 10.2.1 The Heat Equation with Hornogeneous Boundary Conditions 787 10.2.2 Nonhomogeneous Boundary Conditions 791 10.2.3 Insulated Boundary 795 10.3 The One-Dimensional Wave Equation 799 10.3.1 The Wave Equation 799 10.3.2 D'Alembert's Solution 806 10.4 Problems in Two Dimensions: Laplace's Equation 810 10.4.1 Laplace's Equation 810 10.5 Two-Dimensional Problems in a Circular Region 817 10.5.1 Laplace's Equation in a Circular Region 817 10.5.2 The Wave Equation in a Circular Region 821 10.5.3 Other Partial Differential Equations 836 Appendix: Getting Started 841 Introduction to Mathematica 841 A Note Regarding Different Versions of Mathematica 843 Getting Started with Mathematica 843 Five Basic Rules of Mathematica Syntax 849
Contents xi Loading Packages 850 A Word of Caution 853 Getting Help from Mathematica 854 Mathematica Help 858 The Mathematica Menu 863 Bibliography 865 Index 867