基于MATHEMATICA微分方程学习.pdf

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Differential Equations with Mathematica

Contents

Preface

Chapter 1. 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

Chapter 2. First-Order Ordinary Differential Equations

2.1 Theory of First-Order Equations: A Brief Discussion

2.2 Separation of Variables

2.3 Homogeneous Equations

2.4 Exact Equations

2.5 Linear Equations

2.6 Numerical Approximations of Solutions to First-Order Equations

Chapter 3. Applications of First-Order Ordinary Differential Equations

3.1 Orthogonal Trajectories

3.2 Population Growth and Decay

3.3 Newton’s Law of Cooling

3.4 Free-Falling Bodies

Chapter 4. Higher-Order Differential Equations

4.1 Preliminary Definitions and Notation

4.2 Solving Homogeneous Equations with Constant Coefficients

4.3 Introduction to Solving Nonhomogeneous Equations with Constant Coefficients

4.4 Nonhomogeneous Equations with Constant Coefficients: The Method of Undetermined Coefficients

4.5 Nonhomogeneous Equations with Constant Coefficients: Variation of Parameters

4.6 Cauchy–Euler Equations

4.7 Series Solutions

4.8 Nonlinear Equations

Chapter 5. Applications of Higher-Order Differential Equations

5.1 Harmonic Motion

5.2 The Pendulum Problem

5.3 Other Applications

Chapter 6. Systems of Ordinary Differential Equations

6.1 Review of Matrix Algebra and Calculus

6.2 Systems of Equations: Preliminary Definitions and Theory

6.3 Homogeneous Linear Systems with Constant Coefficients

6.4 Nonhomogeneous First-Order Systems: Undetermined Coefficients, Variation of Parameters, and the Matrix Exponential

6.5 Numerical Methods

6.6 Nonlinear Systems, Linearization, and Classification of Equilibrium Points

Chapter 7. Applications of Systems of Ordinary Differential Equations

7.1 Mechanical and Electrical Problems with First-Order Linear Systems

7.2 Diffusion and Population Problems with First-Order Linear Systems

7.3 Applications that Lead to Nonlinear Systems

Chapter 8. Laplace Transform Methods

8.1 The Laplace Transform

8.2 The Inverse Laplace Transform

8.3 Solving Initial-Value Problems with the Laplace Transform

8.4 Laplace Transforms of Step and Periodic Functions

8.5 The Convolution Theorem

8.6 Applications of Laplace Transforms, Part I

8.7 Laplace Transform Methods for Systems

8.8 Applications of Laplace Transforms, Part II

Chapter 9. Eigenvalue Problems and Fourier Series

9.1 Boundary-Value Problems, Eigenvalue Problems, Sturm–Liouville Problems

9.2 Fourier Sine Series and Cosine Series

9.3 Fourier Series

9.4 Generalized Fourier Series

Chapter 10. Partial Differential Equations

10.1 Introduction to Partial Differential Equations and Separation of Variables

10.2 The One-Dimensional Heat Equation

10.3 The One-Dimensional Wave Equation

10.4 Problems in Two Dimensions: Laplace’s Equation

10.5 Two-Dimensional Problems in a Circular Region

Appendix: Getting Started

Introduction to Mathematica

Loading Packages

Getting Help from Mathematica

The Mathematica Menu

Bibliography

Index

Differential Equations with Mathematica THIRD EDITION

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Differential Equations with Mathematica THIRD EDITION Martha L. Abell James P. Braselton Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo

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 9 8 7 6 5 4 3 2 1

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 Introduction to Differential Equations . . . . . . . . . . . . . . . . . . . . 1.1 Deﬁnitions and Concepts 1.2 Solutions of Differential Equations . . . . 1.3 1.4 Direction Fields . Initial and Boundary-Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 6 18 26 . . . . . . . . . . . . . . . . . . . . 2 First-Order Ordinary Differential Equations . . . . . . . . . . . . . . . 41 . . . . . . . . . . . . . . . . . . . Application: Kidney Dialysis . 2.3 Homogeneous Equations . Application: Models of Pursuit . . 2.1 Theory of First-Order Equations: A Brief Discussion . . . . . . . . . 2.2 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Numerical Approximations of Solutions to First-Order Equations . . . . . . . . Integrating Factor Approach . Variation of Parameters and the Method of Undetermined Coefﬁcients 2.5.1 2.5.2 Application: Antibiotic Production . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Exact Equations . 2.5 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . Built-In Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 46 55 59 64 69 74 75 86 89 92 92 v

vi Contents Application: Modeling the Spread of a Disease . . . . . 2.6.2 Other Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 103 3 Applications of First-Order Ordinary Differential Equations . . . . . . 119 . . . . Application: Oblique Trajectories 3.1 Orthogonal Trajectories . . . . . . . . . . . . . . . . 3.2 Population Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Malthus Model . . . . . The Logistic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 3.2.2 Application: Harvesting . . . . . Application: The Logistic Difference Equation . . . . . . . . . . 3.3 Newton’s Law of Cooling . . . . . 3.4 Free-Falling Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 129 132 132 138 148 152 157 163 4 Higher-Order Differential Equations . . . . . . . . . . . . . . . . . . . . 175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 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 . . . . . . 4.2 Solving Homogeneous Equations with Constant Coefﬁcients . . . . . . . . . . . . . . . . . . . . 4.2.1 Second-Order Equations 4.2.2 Higher-Order Equations Application: Testing for Diabetes . . . . Introduction to Solving Nonhomogeneous Equations with Constant Coefﬁcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 175 175 180 185 191 193 196 196 200 211 . . . 216 4.4 Nonhomogeneous Equations with Constant Coefﬁcients: The Method of Undetermined Coefﬁcients 4.4.1 Second-Order Equations 4.4.2 Higher-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Nonhomogeneous Equations with Constant Coefﬁcients: Variation of Parameters . . . . 4.5.1 4.5.2 Higher-Order Nonhomogeneous Equations Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 223 239 248 248 252

Contents vii . . . . . . . . . . 4.6.1 4.6.2 4.6.3 . . . . . . . . . . 4.7 Series Solutions . . . . . . . . . . . . . . . . . . . 4.6 Cauchy–Euler Equations . Second-Order Cauchy–Euler Equations . . . . Higher-Order Cauchy–Euler Equations . . . . Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . Power Series Solutions about Ordinary Points . . . . . . . . . . . . . . . . . . . . . Series Solutions about Regular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 4.7.2 4.7.3 Method of Frobenius Application: Zeros of the Bessel Functions of the First Kind . . . . Application: The Wave Equation on a Circular Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 255 261 265 268 268 281 283 295 298 304 5 Applications of Higher-Order Differential Equations . . . . . . . . . . 321 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forced Motion . . . Soft Springs Hard Springs . . Aging Springs . . Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . 5.1.1 . . 5.1.2 Damped Motion . . . . . 5.1.3 . . . . . . . . . . . . 5.1.4 . . . . . . . . . . . . 5.1.5 . . . . 5.1.6 Application: Hearing Beats and Resonance . . . . . . . . . . . . . . . . . . . 5.2 The Pendulum Problem . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L–R–C Circuits . . Deﬂection of a Beam . . Bod´e Plots . . The Catenary . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 5.3.2 5.3.3 5.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 321 332 346 365 368 370 372 373 387 387 390 393 398 6 Systems of Ordinary Differential Equations . . . . . . . . . . . . . . . . 411 . . . . . . . . . 6.1 Review of Matrix Algebra and Calculus . . . . 6.1.1 6.1.2 6.1.3 6.1.4 6.1.5 Matrix Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Systems of Equations: Preliminary Deﬁnitions and Theory . . . . . . . . . . Deﬁning Nested Lists, Matrices, and Vectors Extracting Elements of Matrices Basic Computations with Matrices Eigenvalues and Eigenvectors . . . . Preliminary Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 . . . . . . . . . . . . . 411 411 416 419 422 426 427 429

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