Differential Equations with Mathematica THIRD EDITION This Page Intentionally Left Blank Differential Equations with Mathematica THIRD EDITION Martha L Abell James P Braselton Amsterdam Boston Heidelberg London New York Oxford San Diego San Francisco Singapore Sydney Tokyo Paris Senior Acquisition Editor: Associate Project Manager: Associate Editor: Marketing Manager: Cover Design: Composition: Printer: Barbara Holland Brandy Palacios Tom Singer Linda Beattie Eric Decicco Integra Maple Vail Press Elsevier Academic Press 200 Wheeler Road, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper Copyright c 2004, Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.com.uk You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Application submitted British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 0-12-041562-3 For all information on all Academic Press publications visit our web site at www.books.elsevier.com Printed in the United States of America 03 04 05 06 07 08 Contents Preface xiii Introduction to Differential Equations 1.1 Definitions and Concepts 1.2 Solutions of Differential Equations 1.3 Initial and Boundary-Value Problems 18 1.4 Direction Fields 26 First-Order Ordinary Differential Equations 41 2.1 Theory of First-Order Equations: A Brief Discussion 41 2.2 Separation of Variables Application: Kidney Dialysis 46 55 2.3 Homogeneous Equations Application: Models of Pursuit 59 64 2.4 Exact Equations 69 Linear Equations 2.5.1 Integrating Factor Approach 2.5.2 Variation of Parameters and the Method of Undetermined Coefficients Application: Antibiotic Production 74 75 86 89 Numerical Approximations of Solutions to First-Order Equations 2.6.1 Built-In Methods 92 92 2.5 2.6 v vi Contents Application: Modeling the Spread of a Disease 2.6.2 Other Numerical Methods 97 103 Applications of First-Order Ordinary Differential Equations 119 3.1 3.2 Orthogonal Trajectories 119 Application: Oblique Trajectories 129 Population Growth and Decay 132 132 138 148 152 3.3 Newton’s Law of Cooling 157 3.4 Free-Falling Bodies 163 3.2.1 The Malthus Model 3.2.2 The Logistic Equation Application: Harvesting Application: The Logistic Difference Equation Higher-Order Differential Equations 175 4.1 Preliminary Definitions and Notation Introduction The nth-Order Ordinary Linear Differential Equation Fundamental Set of Solutions Existence of a Fundamental Set of Solutions Reduction of Order 175 175 180 185 191 193 4.2 Solving Homogeneous Equations with Constant Coefficients 4.2.1 Second-Order Equations 4.2.2 Higher-Order Equations Application: Testing for Diabetes 196 196 200 211 4.3 Introduction to Solving Nonhomogeneous Equations with Constant Coefficients 216 Nonhomogeneous Equations with Constant Coefficients: The Method of Undetermined Coefficients 4.4.1 Second-Order Equations 4.4.2 Higher-Order Equations 222 223 239 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.4 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 4.6 4.7 vii Cauchy–Euler Equations 4.6.1 Second-Order Cauchy–Euler Equations 4.6.2 Higher-Order Cauchy–Euler Equations 4.6.3 Variation of Parameters 255 255 261 265 Series Solutions 268 268 281 283 295 298 Nonlinear Equations 304 4.7.1 Power Series Solutions about Ordinary Points 4.7.2 Series Solutions about Regular Singular Points 4.7.3 Method of Frobenius Application: Zeros of the Bessel Functions of the First Kind Application: The Wave Equation on a Circular Plate 4.8 Applications of Higher-Order Differential Equations 321 5.1 Harmonic Motion 5.1.1 Simple Harmonic Motion 5.1.2 Damped Motion 5.1.3 Forced Motion 5.1.4 Soft Springs 5.1.5 Hard Springs 5.1.6 Aging Springs Application: Hearing Beats and Resonance 321 321 332 346 365 368 370 372 5.2 The Pendulum Problem 373 5.3 Other Applications 5.3.1 L–R–C Circuits 5.3.2 Deflection of a Beam 5.3.3 Bod´e Plots 5.3.4 The Catenary 387 387 390 393 398 Systems of Ordinary Differential Equations 411 6.1 Review of Matrix Algebra and Calculus Defining Nested Lists, Matrices, and Vectors Extracting Elements of Matrices Basic Computations with Matrices Eigenvalues and Eigenvectors Matrix Calculus 411 411 416 419 422 426 Systems of Equations: Preliminary Definitions and Theory 6.2.1 Preliminary Theory 427 429 6.1.1 6.1.2 6.1.3 6.1.4 6.1.5 6.2 viii Contents 6.2.2 Linear Systems 6.3 Homogeneous Linear Systems with Constant Coefficients 6.3.1 Distinct Real Eigenvalues 6.3.2 Complex Conjugate Eigenvalues 6.3.3 Alternate Method for Solving Initial-Value Problems 6.3.4 Repeated Eigenvalues 454 454 461 474 477 6.4 Nonhomogeneous First-Order Systems: Undetermined Coefficients, Variation of Parameters, and the Matrix Exponential 6.4.1 Undetermined Coefficients 6.4.2 Variation of Parameters 6.4.3 The Matrix Exponential 485 485 490 498 Numerical Methods 6.5 6.6 Nonlinear Systems, Linearization, and Classification of Equilibrium Points 6.6.1 Real Distinct Eigenvalues 6.6.2 Repeated Eigenvalues 6.6.3 Complex Conjugate Eigenvalues 6.6.4 Nonlinear Systems 535 535 543 548 552 6.5.1 Built-In Methods Application: Controlling the Spread of a Disease 6.5.2 Euler’s Method 6.5.3 Runge–Kutta Method 446 506 506 513 525 531 Applications of Systems of Ordinary Differential Equations 567 7.1 7.2 7.3 Mechanical and Electrical Problems with First-Order Linear Systems 7.1.1 L–R–C Circuits with Loops 7.1.2 L–R–C Circuit with One Loop 7.1.3 L–R–C Circuit with Two Loops 7.1.4 Spring–Mass Systems Diffusion and Population Problems with First-Order Linear Systems 7.2.1 Diffusion through a Membrane 7.2.2 Diffusion through a Double-Walled Membrane 7.2.3 Population Problems 576 576 578 583 Applications that Lead to Nonlinear Systems 7.3.1 567 567 568 571 574 587 Biological Systems: Predator–Prey Interactions, The Lotka–Volterra System, and Food Chains in the Chemostat 587 Contents 7.3.2 7.3.3 ix Physical Systems: Variable Damping Differential Geometry: Curvature 604 611 Laplace Transform Methods 617 8.1 8.2 The Laplace Transform 8.1.1 Definition of the Laplace Transform 8.1.2 Exponential Order 8.1.3 Properties of the Laplace Transform 618 618 621 623 The Inverse Laplace Transform 629 Definition of the Inverse Laplace Transform 629 Laplace Transform of an Integral 635 8.2.1 8.2.2 8.3 8.4 Solving Initial-Value Problems with the Laplace Transform 637 Laplace Transforms of Step and Periodic Functions Piecewise-Defined Functions: The Unit Step Function Solving Initial-Value Problems Periodic Functions Impulse Functions: The Delta Function 645 645 649 652 661 8.4.1 8.4.2 8.4.3 8.4.4 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 8.6.1 Spring–Mass Systems Revisited 8.6.2 L–R–C Circuits Revisited 8.6.3 Population Problems Revisited Application: The Tautochrone 672 672 679 687 689 8.7 Laplace Transform Methods for Systems 691 Applications of Laplace Transforms, Part II 708 708 714 720 8.8 8.8.1 Coupled Spring–Mass Systems 8.8.2 The Double Pendulum Application: Free Vibration of a Three-Story Building Eigenvalue Problems and Fourier Series 727 9.1 Boundary-Value Problems, Eigenvalue Problems, Sturm–Liouville Problems 9.1.1 Boundary-Value Problems 727 727 This Page Intentionally Left Blank The Mathematica Menu File Edit Cell Format Input Kernel Find Window Help 863 This Page Intentionally Left Blank Bibliography [1] Abell, Martha and Braselton, James, Mathematica By Example, Third Edition, Academic Press, 2004 [2] Abell, Martha and Braselton, James, Modern Differential Equations, Second Edition, Harcourt, 2001 [3] Apostol, Tom, Mathematical Analysis, Second Edition, Addison-Wesley, 1974 [4] Barnsley, Michael, Fractals Everywhere, Second Edition, Morgan Kaufmann, 2000 [5] Boyce, William E and DiPrima, Richard C., Elementary Differential Equations and Boundary-Value Problems, Seventh Edition, John Wiley & Sons, 2000 [6] Coddington, Earl and Levinson, Norman, Theory of Ordinary Differential Equations, Robert E Krieger Publishing Company/McGraw Hill, 1955/1984 [7] Corduneanu, C., Principles of Differential and Integral Equations, Chelsea Publishing, 1977 [8] Devaney, Robert L and Keen, Linda (eds.), Chaos and Fractals: The Mathematics Behind the Computer Graphics, Proceedings of Symposia in Applied Mathematics, Volume 39, American Mathematical Society, 1989 [9] Edwards, C Henry and Penney, David E Calculus with Analytic Geometry, Fifth Edition, Prentice Hall, 1998 [10] Edwards, C Henry and Penney, David E Differential Equations and Boundary Value Problems: Computing and Modeling, Third Edition, Pearson/Prentice Hall, 2004 [11] Gaylord, Richard J., Kamin, Samuel N., and Wellin, Paul R., Introduction to Programming with Mathematica, Second Edition, TELOS/Springer-Verlag, 1996 865 866 Bibliography [12] Giordano, Frank R., Weir, Maurice D., and Fox, William P., A First Course in Mathematical Modeling, Third Edition, Thomson/Brooks/Cole, 2003 [13] Graff, Karl F., Wave Motion in Elastic Solids, Oxford University Press/Dover, 1975/1991 [14] Gray, Alfred, Modern Differential Geometry of Curves and Surfaces, Second Edition, CRC Press, 1997 [15] Gray, John W., Mastering Mathematica: Programming Methods and Applications, Second Edition, Academic Press, 1997 [16] Herriott, Scott R., College Algebra through Functions and Models, Preliminary Edition, Brooks/Cole, 2002 [17] Jordan, D.W and Smith, Peter, Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, Third Edition, Oxford Applied and Engineering Mathematics, Oxford University Press, 1999 [18] Kyreszig, Erwin, Advanced Engineering Mathematics, Eighth Edition, John Wiley & Sons, 1999 [19] Larson, Roland E., Hostetler, Robert P., and Edwards, Bruce H., Calculus with Analytic Geometry, Sixth Edition, Houghton Mifflin, 1998 [20] Maeder, Roman E., The Mathematica Programmer II, Academic Press, 1996 [21] Maeder, Roman E., Programming in Mathematica, Third Edition, Addison-Wesley, 1996 [22] Rabenstein, Albert L., Introduction to Ordinary Differential Equations, Academic Press, 1966 [23] Robinson, Clark, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Second Edition, CRC Press, 1999 [24] Smith, Hal L and Waltman, P., The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995 [25] Stewart, James, Calculus: Concepts and Contexts, Second Edition, Brooks/Cole, 2001 [26] Weisstein, Eric W., CRC Concise Encyclopedia of Mathematics, CRC Press, 1999 [27] Wolfram, Stephen, A New Kind of Science, Wolfram Media, 2002 [28] Wolfram, Stephen, The Mathematica Book, Fourth Edition, Wolfram Media, 2004 [29] Zwillinger, Daniel, Handbook of Differential Equations, Second Edition, Academic Press, 1992 Index Symbols $Version 843 $VersionNumber 843 ’7 ’’ ( ) 850 420 / 22, 29, 53 // 213 /; 622 /@ 331 == ? 854, 857 ?? 854 [ ] 850 [[ ]] 9, 416, 850 { } 850 Π 845 A Abs 744 Add-Ons 850 Algebraic Manipulation 466 TrigToExp 466 Animate Selected Graphics 326 Antibiotic production 89 Apart 53, 634, 704 Application antibiotic production 89 controlling spread of a disease 513 drug concentration 80 exponential growth 87 free vibration of a three-story building 720 harvesting 148 hearing beats and resonance 372 kidney dialysis 55 logistic difference equation 152 modeling spread of a disease 97 models of pursuit 64 oblique trajectories 129 radioactive decay 133 tautochrone 689 testing for diabetes 211 wave equation on circular plate 298 zeros of Bessel functions of first kind 295 Arc length function 612 Array 411, 414 Attractor ă Rossler 441 Autonomous system 433, 535, 553 Auxiliary condition 18 Auxiliary equation 256, 261 Average period of infectivity 97 B BasicInput 845, 846 BasicTypesetting 412, 415, 848 Beam deflection 390 Beats 356 hearing 372 Becquerel, Henri 133 Bendixson’s theorem 558 Bessel, Friedrich Wilhelm 289 Bessel function of first kind 290, 291 modified 299 zeros of 295 of second kind 291 Bessel’s equation 2, 282, 289, 770 Bessel-Fourier series 771 convergence 775 BesselI 299 BesselJ 282, 290 BesselJZeros 296, 771, 824, 829 BesselY 282, 291, 292 BesselZeros 295, 774, 823, 829 BesselJZeros 296, 771, 824, 829 Bod´e plots 393 Boundary conditions homogeneous 728 Boundary-value problem 23, 727 867 868 INDEX Break-even concentration 145 Building vibration 720 C C 18 Cases (/;) 622 Catenary 398 Cauchy–Euler equation definition 255 higher-order 261 characteristic/auxiliary equation 261 second-order 255 characteristic/auxiliary equation 256 variation of parameters 265 Cell Animate Selected Graphics 326 ConvertTo 847 Center 548 Characteristic (auxiliary) equation 197, 256, 261 definition 200 Characteristic polynomial 422 Characteristic system 836 CharacteristicPolynomial 422, 423 Characteristics method of 836 Charge steady-state 389 Chemostat growth 144 scaled equations for 144 long food chain 602 models simple food chain 594 Chop 208, 830 Circuit L–R–C 387, 679 with loops 567 one loop 568 two loops 571 Circular plate wave equation 298 Circular region Laplace’s equation 817 wave equation 821 Clear ClipFill 838 Coefficients constant 180 Collect 214 Command key and cursor 395 Command-Y 326 Competing species 35 Compile 834 CompiledFunction 834 ComplexExpand 213, 240, 474 Compositions of function 154 Concentration break-even 145 Constant coefficients 180 Contact number 97 Continuous piecewise definition 622 ContourPlot 9, 33, 42, 50, 61, 73, 124 Axes 10, 73 AxesOrigin 10, 73 AxesStyle 10, 73 Contours 10, 61, 124 ContourShading 10, 73 ContourStyle 122, 125 DisplayFunction 61 Evaluate 10 Frame 10, 73 PlotPoints 10, 73, 122 Controlling spread of disease 513 Convergence of a power series solution 276 ConvertTo 847 Convolution integral 667 Convolution theorem 667 Cooling Newton’s law of 87, 157 Corresponding eigenfunction 730 homogeneous equation 75, 180 Cosine series Fourier 746 Coupled spring-mass system 708 Cramer’s rule 249, 252 Create Table/ Matrix/Palette 412, 415, 421 Matrix 413 Curie, Marie 133 Curie, Pierre 133 Curvature 612 Curve(s) orthogonal definition 119 parametrization with respect to arc length 612 smooth length of 399 Cycle limit 40, 429, 608 Cycloheximide 89 D D 80, 182, 187, 275, 426 Daily contact rate 97 Daily death removal rate 97 Daily recovery removal rate 97 D’Alembert’s solution 806 Damped motion 332, 574 critically damped 333 overdamped 332 underdamped 333 Damping nonlinear 175 variable 604 Decay radioactive 133 Decibel 394 Deflection of a beam 390 Degenerate stable node 543, 544 unstable node 543, 544 Delta function 662 DensityPlot 790 Dependence linear 181, 449 Derivative of Laplace transform 626 of matrix 426 Det 182, 419 Determinant 419 Diabetes mellitus 55, 211 testing for 211 Dialysate 55 Dialyser 55 Dialysis 55 Difference equation logistic 152 Differential equation definition exact definition 69 homogeneous definition 59 INDEX linear see Linear differential equation nonlinear 304 definition order of definition reduction of 193 ordinary definition first-order 41 system definition 429 partial see Partial differential equation (PDE) separable definition 46 solution see Solution system linear see Linear system nonlinear see Nonlinear system ordinary definition 429 Differential geometry curvature 612 Differentiation implicit Diffusion through double-walled membrane 578 through membrane 576 Dirac delta function 662 Laplace transform of 663 DiracDelta 662 Direction field 26 Dirichlet problem 811, 818 Discontinuity jump definition 621 Disease controlling spread of 513 endemic 514 epidemic 513 modeling spread of 97 Do 326, 713, 803, 827 Double pendulum 714 Drug concentration 80 Drumhead 821 DSolve 13, 23, 29, 32, 47, 60, 70, 79, 92, 122, 175, 431, 456, 715, 836, 860 Dt Duffing’s equation 310, 557 869 E E 392, 680 Eigenfunctions corresponding 730 orthogonality of 736 Eigensystem 422, 424, 455, 536 Eigenvalue 422, 606, 730 Eigenvalue problem 728, 730 Eigenvalues 422, 423, 598 Eigenvector 422 Eigenvectors 422, 424 Endemic disease 514 Enter 845 Envelope function 354 Epidemic disease 513 Equation Bessel’s 2, 282, 289, 770 Cauchy–Euler definition 255 higher-order 261 characteristic/ auxiliary equation 261 second-order 255 characteristic/ auxiliary equation 256 variation of parameters 265 characteristic 197, 256, 261 definition 200 Duffing’s 310, 557 FitzHugh–Nagumo 522 heat 4, 787 homogeneous boundary conditions 787 insulated boundary 795 nonhomogeneous boundary conditions 791 indicial 284, 287 integral 669 integrodifferential 670 Laplace’s 3, 4, 784, 810 in circular region 817 Legendre’s 276 Lorenz 560 potential 810 ¨ Rossler 441 Sturm–Liouville 736 Van-der-Pol 175, 427, 606 wave 3, 11, 799 on circular plate 298 in circular region 821 D’Alembert’s solution 806 normal modes 12 Equilibrium (rest) point 514 classification 553 definition 535 locally stable 554 unstable 554 Errors when loading packages 853 Euler equation see Cauchy–Euler equation Euler’s formula 461 Euler’s method 103, 525, 533 improved 108 Evaluate 10, 17, 493 Even extension 759 Exact differential equation definition 69 Existence and Uniqueness theorem 41, 180, 438 Expand 211, 570 Exponential matrix definition 498 Exponential growth 87 Exponential order definition 621 Exponents 288 ExpToTrig 474, 503 Extension even 759 odd 759 periodic 759 F Factor 197, 201 GaussianIntegers 198, 203 Fasting blood sugar test 211 Feedback inhibition 89 File Palettes Algebraic Manipulation 466 BasicTypesetting 412, 415, 848 Quit 849 Save 849 Save As Special 849 Find in Help 860 FindRoot 136, 171, 209, 300, 735 First-order equations exact 69 870 INDEX homogeneous 59 linear 74 standard form 75 separable 46 theory 41 FitzHugh–Nagumo equation 522 Flatten 153, 456 Flight paths 64 Force resistive 163 Forced motion 346, 672 damped 356 undamped 347 Forcing function 5, 180 Fourier 313 Fourier cosine series 746 series 749 convergence 752 definition 751 differentiation 764 generalized 770 integration 765 sine series 738 transform 312 Free-falling bodies 163 Frequency natural 349 FresnelS 21 Frobenius method of 283 FullSimplify 503 Function arc length 612 Bessel see Bessel function composition 154 Dirac delta 662 envelope 354 even extension 759 forcing 5, 180 generalized 662 impulse 661 idealized 662 Jacobi elliptic 435 linear combination 187 linearly dependent 181 linearly independent 181, 736 natural log 47 odd extension 759 periodic definition 652 Laplace transform of 652 periodic extension 759 piecewise continuous definition 622 piecewise smooth definition 764 unit impulse 662 unit step 79 definition 645 solving initial-value problems with 649 weighting 736 Fundamental matrix definition 452 Fundamental set of solutions 452 definition 185, 449 existence 191, 452 Fundamental theorm of calculus 75 G Gain 394 Generalized Fourier series 770 Generalized function 662 Gibbs phenomenon 744 Glucose tolerance test (GTT) 211 GramSchmidt 853 Graphics animation 326 Graphics Graphics LogListPlot 315 LogLogPlot 396 ImplicitPlot ImplicitPlot 10, 71 PlotField 516 PlotVectorField 27, 33, 54, 97, 433, 455, 457, 459, 507 Shapes 851, 852 GraphicsArray 13, 852 Graylevel 19, 188 Growth exponential 87 Growth constant 87 H Half-life 133 Half-wave rectification 655 Harmonic motion 321 aging springs 370 damped motion 332, 574 forced motion 346, 672 damped 356 undamped 347 hard springs 368 simple 321, 329 soft springs 365 Harvesting 148 Heat equation 4, 787 homogeneous boundary conditions 787 insulated boundary 795 nonhomogeneous boundary conditions 791 Help 13, 858 ? 854, 857 ?? 854 Add-Ons 850 Find in Help 860 Information 855 Options 855 Help Browser 850, 858, 860, 861 BesselZeros 296, 824 DSolve 15, 860 Go 860 JacobiSN 437 MatrixManipulation 418 NDSolve 21 PlotField 27 Herd immunity 515 Homogeneous boundary conditions 728 Homogeneous differential equation corresponding 75, 180 definition 59 linear 5, 75, 180 with constant coefficients higher order 200 second-order 196 linear partial 783 Homogeneous linear system 447 with constant coefficients 454, 535 complex conjugate eigenvalues 461, 548 distinct real eigenvalues 454, 535 repeated eigenvalues 477, 543 corresponding 447 Hooke’s Law 321 HypergeometricU 287 Hypertension 55 INDEX I I 392, 683 Idealized unit impulse function 662 Immunity herd 515 Implicit differentiation Implicit solution ImplicitPlot ImplicitPlot 10, 71 ImplicitPlot 10, 71 AxesOption 71 PlotPoints 71 Improved Euler’s method 108 Impulse 661 idealized 662 Independence linear 181, 449, 736 Indicial equation 284, 287 Indicial roots 288 Information 855 Initial condition 19 Initial-boundary value problem 788 Initial-value problem 19, 23 nth-order 180 Input Create Table/ Matrix/Palette 412, 415, 421 Matrix 413 InputForm 333, 847, 848 Integral convolution 667 of matrix 426 Integral equation 669 Integrate 50, 426, 618, 619, 656 Integrating factor 75 Integration by parts 773 Integrodifferential equation 670 InterpolatingFunction 85, 94 Inverse 419, 476 Inverse Laplace transform definition 629 determination of existence 636 linearity property 630 use of partial fractions in finding irreducible quadratic factors 635 871 nonrepeated linear factors 632 repeated linear factors 634 Inverse of matrix 419 InverseLaplaceTransform 630, 717 Irregular singular point definition 281 Isotherms 123 J Jacobi elliptic functions 435 Jacobian matrix 553, 554, 588, 606 JacobiSN 435 Jump discontinuity definition 621 K Kidney dialysis 55 Kirchhoff’s law 387, 388 current 567 voltage 567 L L–R–C circuit 387, 679 with loops 567 one loop 568 two loops 571 LaguerreL 287 Laplace transform applications double pendulum 714 L–R–C circuits 679 population problems 687 spring-mass systems 672 coupled 708 tautochrone 689 vibration of building 720 definition 618 derivatives of 626 of Dirac delta function 663 of first derivative 624 of higher derivatives 625 of integral 635 inverse definition 629 determination of existence 636 linearity property 630 use of partial fractions in finding irreducible quadratic factors 635 nonrepeated linear factors 632 repeated linear factors 634 linearity property 621 methods for systems 691 of periodic function 652 shifting property 624 solving initial-value problems with 637 sufficient condition for existence 622 Laplace’s equation 3, 4, 784, 810 in circular region 817 LaplaceTransform 618, 619, 620, 628, 716 Laplacian in polar coordinates 298 Legendre, Adrien Marie 276 Legendre polynomial of degree n 279 orthogonality condition 280 LegendreP 279 LegendreQ 279 Legendre’s equation 276 Length 18 Length of smooth curve 399 Life expentancy 97 Limit cycle 40, 429, 608 Linear Algebra MatrixManipulation 418 TakeColumns 419 TakeRows 419 Orthogonalization 853 GramSchmidt 853 Normalize 853 Linear combination of functions 187 Linear differential equation definition 5, 180 first-order definition 74 standard form 75 homogeneous 5, 75, 180 with constant coefficients higher order 200 second-order 196 corresponding 75, 180 nonhomogeneous 180 nonhomogeneous with constant coefficients 216 872 INDEX general solution 217 method of undetermined coefficients 222, 352 higher-order equations 239 second-order equations 223 variation of parameters 248 higher-order equations 252 second-order equations 248 nonhomogeneous partial 783 nth-order 180 particular solution 86, 248 definition 216 see also Differential equation; Linear system Linear system 6, 430, 446 homogeneous 447 with constant coefficients 454, 535 complex conjugate eigenvalues 461, 548 distinct real eigenvalues 454, 535 repeated eigenvalues 477, 543 corresponding 447 Laplace transform methods 691 nonhomogeneous 447 first-order 485 matrix exponential 498 method of undetermined coefficients 485 variation of parameters 490 Linearity property of inverse Laplace transform 630 of Laplace transform 621 Linearization 552 Linearly dependent functions 181 vectors 449 Linearly independent functions 181, 736 vectors 449 LinearSolve 480 List nested 414 ListPlot 106, 153 DisplayFunction 114 PlotJoined 155 PlotStyle 106 Log 47 LogicalExpand 274 Logistic difference equation 152 Logistic equation 54, 138 with predation 141 LogListPlot 315 LogLogPlot 396 Long food chain in chemostat 602 Lorenz equations 560 Lotka-Volterra system 587 Method of characteristics 836 Method of Frobenius 283 ă Mobius strip 852 Modeling spread of disease 97 Models of pursuit 64 Momentum 163 Motion damped 332, 574 forced 346, 672 damped 356 undamped 347 harmonic 321 simple 321, 329 Newton’s second law of 163 Multiplicity definition 200 M Maclaurin series 269, 375 Malthus, Thomas R 132 Malthus model 132 Map 82, 95, 187, 331, 434, 608, 628, 718, 856 Master Index 861 MathSource 850 Matrix 413 Matrix 411 basic computations 419 derivative 426 determinant 419 extracting elements 416 fundamental definition 452 integral 426 inverse 419 Jacobian 553, 554, 588, 606 product 420 transpose 417, 419 variational 553, 588 Matrix exponential definition 498 MatrixExp 498, 503 MatrixForm 186, 412, 416 MatrixManipulation 418 TakeColumns 419 TakeRows 419 Membrane diffusion through 576 double-walled diffusion through 578 permeability 576 Menu 863 N N 84, 496, 845 Names 856, 857 SpellingCorrection 857 Natural frequency 349 Natural logarithm function 47 NDSolve 20, 22, 25, 38, 84, 92, 175, 431, 506, 522, 606 MaxSteps 560 Nest 154 Nested list 414 Neumann problem 811 Newton’s law of cooling 87, 157 Newton’s second law of motion 163 NIntegrate 825, 829 Node stable 535 degenerate 543, 544 unstable 536 degenerate 543, 544 Nonhomogeneous linear differential equation 180 with constant coefficients 216 general solution definition 217 method of undetermined coefficients 222, 352 INDEX higher-order equations 239 second-order equations 223 variation of parameters 248 higher-order equations 252 second-order equations 248 partial 783 Nonhomogeneous linear system 447 first-order 485 matrix exponential 498 method of undetermined coefficients 485 variation of parameters 490 Nonlinear damping 175 Nonlinear differential equation 304 definition Nonlinear system 6, 535, 552 autonomous 535, 553 classification of equilibrium points 553 linearized system corresponding 552 NonlinearFit 215 NonlinearFit NonlinearFit 215 Nontrivial solution 185 Normal 276 Normal modes 12, 298 Normalize 853 NRoots 723 Nuclides 133 Numerical methods Euler’s method 103, 525, 533 improved Euler’s method 108 NDSolve 92, 506 Runge-Kutta method 111, 531 of order four 115 of order two 111 NumericalMath BesselZeros 295, 774, 823, 829 BesselJZeros 296, 771, 824, 829 873 O Oblique trajectories 129 Odd extension 759 Options 28, 380, 855 Order of differential equation reduction of 193 exponential definition 621 Ordinary differential equation definition first-order 41 system definition 429 Ordinary point definition 268 Orthogonal curves definition 119 Orthogonal trajectories 119 family of 121 Orthogonality condition 280, 736 Orthogonalization 853 GramSchmidt 853 Normalize 853 OutputForm 848 P p-series 770 Packages 850 errors messages 853 Palettes Algebraic Manipulation 466 BasicTypesetting 412, 415, 848 Parameters variation of 86, 248 Cauchy–Euler equations 265 higher-order equations 252 second-order equations 248 systems 490 ParametricPlot 39, 429, 456, 690 ParametricPlot3D 302, 471, 834 PlotPoints 523 Parametrization of curve 612 Parseval’s equality 769 Part ([[ ]]) 9, 416, 850 Partial differential equation (PDE) heat 4, 787 homogeneous boundary conditions 787 insulated boundary 795 nonhomogeneous boundary conditions 791 Laplace’s equation 3, 4, 784, 810 in circular region 817 linear second-order 783 homogeneous 783 nonhomogeneous 783 potential equation 810 quasi-linear first-order 836 almost linear 836 homogeneous 836 linear 836 solution definition 784 wave equation 3, 11, 799 on circular plate 298 in circular region 821 D’Alembert’s solution 806 normal modes 12 Particular solution 86, 248 definition 216 to system 485 Partition 178 Pendulum 373 damped 378, 557 double 714 equation 5, 557 Period 652 Periodic extension 759 Periodic function definition 652 Laplace transform of 652 Permeability 576 Perpendicular curves 119 Phase shift 394 Pi 845 Piecewise continuous definition 622 Piecewise smooth definition 764 “Pitchfork diagram” 155 Play 372 Plot Bod´e 393 Poincar´e 317 Plot 7, 17, 53, 135, 260, 854 874 INDEX AspectRatio 30, 121 AxesLabel 30 AxesStyle 30 DisplayFunction 17, 122 Evaluate 17, 493 PlotRange 29 PlotStyle 19, 30, 188, 219 Plot3D 13 ClipFill 838 PlotField 516 PlotVectorField 27, 33, 54, 97, 433, 455, 457, 459, 507 PlotVectorField 27, 33, 54, 97, 433, 455, 457, 459, 507 Axes 28 AxesOrigin 28 DisplayFunction 507 HeadLength 28 ScaleFunction 28 Poincar´e plot (return) 317 Point equilibrium 514 classification 553 definition 535 locally stable 554 unstable 554 ordinary definition 268 saddle 536 singular definition 268 irregular definition 281 regular definition 281 Polar coordinates Laplacian 298 Polynomial characteristic 422 PolynomialLCM 856 Pond motion in 170 Population problems 687 growth and decay 132 United States 136, 139 three neighbouring territories 585 two neighbouring territories 583 Potential equation 810 Power spectrum 313 PowerExpand 213 Predator–prey 440, 587 equations 3, 587 see also Chemostat Principle of superposition 187, 190, 785 Problem boundary-value 23, 727 Dirichlet 811, 818 eigenvalue 728, 730 initial-boundary value 788 initial-value 19, 23 nth-order 180 Neumann 811 Sturm-Liouville 735 self-adjoint form 736 Product Log function 50 Product method 785 ProductLog 50 Pursuit models 64 Q Quasiperiod 337 Quit 849 R Radially symmetric solution 821 Radioactive decay 133 Radioactivity 133 Rayleigh’s equation 25, 37 Reduction of order 193 Regular singular point definition 281 Remove 853 ReplaceAll (/.) 22, 29, 53 Replacement rule 16 ReplaceRepeated (//.) 213 Resistive force 163 Resonance 350 hearing 372 Rest point see Equilibrium (rest) point Return 845 Root 723 ă Rossler attractor 441 RowReduce 424 Runge-Kutta method 111, 531 of order four 115 of order two 111 S Saddle point 536 Sarcophagus 134 Save 849 Save As Special 849 Second law of motion Newton’s 163 Self-adjoint form Sturm–Liouville problem 736 Separable differential equation definition 46 Separation of variables 46, 785 Series 273 Series Bessel–Fourier 771 convergence 775 Fourier 749 convergence 752 cosine 746 definition 751 differentiation 764 generalized 770 integration 765 sine 738 p- 770 Series solutions 268 about ordinary points 268 convergence 276 about regular singular points 281 method of Frobenius 283 Shapes 851, 852 Shift phase 394 Shift-Return 845 Short 18, 495 Show 31, 122, 380 DisplayFunction 101, 380, 384 GraphicsArray 13, 852 PointSize 380 Simple food chain in chemostat 594 Simple harmonic motion 321, 329 Simplify 7, 47, 474 Sine series Fourier 738 Singular point definition 268 of equation with polynomial coefficients definition 281 irregular definition 281 INDEX regular definition 281 SIR model with vital dynamics 514 without vital dynamics 513 SIS model 98 Smooth piecewise definition 764 Solution D’Alembert’s 806 definition envelope 354 fundamental set 452 definition 185, 449 existence 191, 452 general 18, 217, 452 definition 187 global behavior 556 implicit interesting 440 local behavior 556 meaningful 440 method of Frobenius 283 nontrivial 185 numerical methods Euler’s method 103, 525, 533 improved Euler’s method 108 NDSolve 92, 506 Runge–Kutta method 111, 531 of order four 115 of order two 111 partial differential equation definition 784 particular 86, 248 definition 216 to system 485 radially symmetric 821 series 268 about ordinary points 268 convergence 276 about regular singular points 281 method of Frobenius 283 steady-state 357 of system 429, 449 transient 357 vector definition 449 875 Solve 9, 79, 202, 480, 514, 856 SolveAlways 224, 240, 633 Sound 372 Spectrum power 313 Sphere 852 Spiral stable 548 unstable 548 Spring aging 370 elastic 321 hard 368 soft 365 Spring–mass system 574, 672 coupled 708 Stable node 535 degenerate 543, 544 spiral 548 star 543 Standard form definition 268 StandardForm 847, 848 Star stable 543 unstable 543 Statistics NonlinearFit Nonlinearfit 215 Steady-state charge 389 Steady-state solution 357 Steady-state temperature 792 Step size 103 Streptomyces griseus 89 Sturm–Liouville problem 735 self-adjoint form 736 Sum 654, 657 Superposition principle of 187, 190, 785 Susceptible-infected– susceptible model 98 Syntax basic rules 849 System autonomous 433, 535, 553 characteristic 836 of differential equations 3, 429 solution 429 linear see Linear system nonlinear see Nonlinear system T Table 13, 18, 29, 53, 188, 243, 414, 607 GrayLevel 19, 188 Length 18 Short 18 TableForm 90, 628 TakeColumns 419 TakeRows 419 Tangent vector 611 Tautochrone 689 Taylor series 112 Temperature 123 steady-state 792 transient 792 see also Newton’s law of cooling Ten-Minute Tutorial 859 Testing for diabetes 211 Together 654 Torus 852 TraditionalForm 847, 848 Trajectories oblique 129 orthogonal 119 family of 121 Transform Fourier 312 Laplace see Laplace transform Transient solution 357 Transient temperature 792 Transpose 417, 419 Transpose of matrix 417, 419 Traveling wave solution 810 TrigToExp 466 Two-point boundary value problem 728 U Undetermined coefficients method of 86, 222, 352 higher-order equations 239 second-order equations 223 systems 485 Union 516, 607 Uniqueness and Existence theorem 41, 180, 438 Unit impulse function definition 662 Unit step function 79 definition 645 876 INDEX solving initial-value problems with 649 Unit tangent vector 611 United States population of 136, 139 UnitStep 79, 622, 645 Unstable node 536 degenerate 543, 544 spiral 548 star 543 V Van-der-Pol equation 175, 427, 606 Variation of parameters 86, 248 Cauchy–Euler equations 265 higher-order equations 252 second-order equations 248 systems 490 Variational matrix 553, 588 Vector 415 column 415 constant 485 row 415 solution definition 449 unit tangent 611 Verhulst, Pierre 138 Verhulst equation (logistic equation) 54, 138 with predation 141 Vibration of building 720 Violin string 25 W Wave equation 3, 11, 799 on circular plate 298 in circular region 821 D’Alembert’s solution 806 normal modes 12 Weighting function 736 Welcome Screen 859 Wronskian 450 definition 181 ... covered in Differential Equations with Mathematica, Third Edition, includes typical examples solved by traditional methods and examples solved using Mathematica Differential Equations with Mathematica. . .Differential Equations with Mathematica THIRD EDITION This Page Intentionally Left Blank Differential Equations with Mathematica THIRD EDITION Martha L Abell James P Braselton Amsterdam... Compatibility All examples illustrated in Differential Equations with Mathematica, Third Edition, were completed using Version of Mathematica Although most computations can continue to be carried out with