Differential Equations: An Introduction with Mathematica®

444 Pages·2004·12.85 MB·English

by  Clay C. Ross (auth.)

Undergraduate Texts in Mathematics
Editors
S. Axler
F.W. Gehring
K.A. Ribet
Undergraduate Texts in Mathematics
Abbott: Understanding Analysis. Childs: A Concrete Introduction to
Anglin: Mathematics: A Concise History Higher Algebra. Second edition.
and Philosophy. Chung/AitSahlia: Elementary Probability
Readings in Mathematics. Theory: With Stochastic Processes and
Anglin/Lambek: The Heritage of an Introduction to Mathematical
Thales. Finance. Fourth edition.
Readings in Mathematics. Cox/Little/O'Shea: Ideals, Varieties,
Apostol: Introduction to Analytic and Algorithms. Second edition.
Number Theory. Second edition. Croom: Basic Concepts of Algebraic
Armstrong: Basic Topology. Topology.
Armstrong: Groups and Symmetry. Curtis: Linear Algebra: An Introductory
Axler: Linear Algebra Done Right. Approach. Fourth edition.
Second edition. Daepp/Gorkin: Reading, Writing, and
Reardon: Limits: A New Approach to Proving: A Closer Look at
Real Analysis. Mathematics.
Bak!Newman: Complex Analysis. Devlin: The Joy of Sets: Fundamentals
Second edition. of Contemporary Set Theory.
Banchoff/Wermer: Linear Algebra Second edition.
Through Geometry. Second edition. Dixmier: General Topology.
Berberian: A First Course in Real Driver: Why Math?
Analysis. Ebbinghaus/Fium/Thomas:
Bix: Conics and Cubics: A Mathematical Logic. Second edition.
Concrete Introduction to Algebraic Edgar: Measure, Topology, and Fractal
Curves. Geometry.
Bremaud: An Introduction to Elaydi: An Introduction to Difference
Probabilistic Modeling. Equations. Second edition.
Bressoud: Factorization and Primality Erdi:ls/Suninyi: Topics in the Theory of
Testing. Numbers.
Bressoud: Second Year Calculus. Estep: Practical Analysis in One Variable.
Readings in Mathematics. Exner: An Accompaniment to Higher
Brickman: Mathematical Introduction Mathematics.
to Linear Programming and Game Exner: Inside Calculus.
Theory. Fine/Rosenberger: The Fundamental
Browder: Mathematical Analysis: Theory of Algebra.
An Introduction. Fischer: Intermediate Real Analysis.
Buchmann: Introduction to Flanigan/Kazdan: Calculus Two: Linear
Cryptography. and Nonlinear Functions. Second
Buskes/van Rooij: Topological Spaces: edition.
From Distance to Neighborhood. Fleming: Functions of Several Variables.
Callahan: The Geometry of Spacetime: Second edition.
An Introduction to Special and General Foulds: Combinatorial Optimization for
Relavitity. Undergraduates.
Carter/van Brunt: The Lebesgue Foulds: Optimization Techniques: An
Stieltjes Integral: A Practical Introduction.
Introduction.
(continued after index)
Cederberg: A Course in Modern
Geometries. Second edition.
Clay C. Ross
Differential Equations
An Introduction with Mathematica®
Second Edition
With 88 Figures
~Springer
Clay C. Ross
Department of Mathematics
The University of the South
735 University Avenue
Sewanee, TN 37383
USA
[email protected]
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
Mathematics Department Mathematics Department Mathematics Department
San Francisco State East Hali University of California,
University University of Michigan Berkeley
San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 947.20-3840
USA USA USA
Mathematics Subject Classification (2000): 34-01, 65Lxx
Library of Congress Cataloging-in-Publication Data
Ross, Clay C.
Differential equations : an introduction with Mathematica / Clay C. Ross. - 2nd ed.
p. cm. - (Undergraduate texts in mathematics)
Includes bibliographical references and index.
ISBN 978-1-4419-1941-0 ISBN 978-1-4757-3949-7 (eBook)
DOI 10.1007/978-1-4757-3949-7
1. Differential equations. 2. Mathematica (Computer file) 1. Title. TI. Series.
QA371.R595 2004
515'.35-dc22 2004048203
ISBN 978-1-4419-1941-0 Printed on acid-free paper.
© 2004, 1995 Spinger Science+Business Media New York
Originally published by Springer Science+ Business Media , loc. in 2004
Softcover reprint of the hardcover lod edition 2004
AII rights reserved. This work may not be translated or copied in whole or in part with
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except for brief excerpts in connection with
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This book is dedicated to my parents,
Vera K. and Clay C. Ross,
and to my wife,
Andrea,
with special gratitude to each.
Preface
Goals and Emphasis of the Book
Mathematicians have begun to find productive ways to incorporate computing power
into the mathematics curriculum. There is no attempt here to use computing to avoid
doing differential equations and linear algebra. The goal is to make some first ex
plorations in the subject accessible to students who have had one year of calculus.
Some of the sciences are now using the symbol-manipulative power of Mathemat
ica to make more of their subject accessible. This book is one way of doing so for
differential equations and linear algebra.
I believe that if a student's first exposure to a subject is pleasant and exciting,
then that student will seek out ways to continue the study of the subject. The theory
of differential equations and of linear algebra permeates the discussion. Every topic
is supported by a statement of the theory. But the primary thrust here is obtaining
solutions and information about solutions, rather than proving theorems. There are
other courses where proving theorems is central. The goals of this text are to establish
a solid understanding of the notion of solution, and an appreciation for the confidence
that the theory gives during a search for solutions. Later the student can have the same
confidence while personally developing the theory.
When a study of the book has been completed, many important elementary con
cepts of differential equations and linear algebra will have been encountered. In
addition, the use of Mathematica makes it possible to analyze problems that are
formidable without computational assistance. Mathematica is an integral part of the
presentation, because in introductory differential equations or linear algebra courses
it is too often true that simple tasks like finding an antiderivative, or finding the roots
of a polynomial of relatively high degree-even when the roots are all rational
completely obscure the mathematics that is being studied. The complications en
countered in the manual solution of a realistic problem of four first-order linear equa
tions with constant coefficients can totally obscure the beauty and centrality of the
theory. But having Mathematica available to carry out the complicated steps frees
the student to think about what is happening, how the ideas work together, and what
everything means.
VIII Preface
The text contains many examples. Most are followed immediately by the same
example done in Mathematica. The form of a Mathematica notebook is reproduced
almost exactly so that the student knows what to expect when trying problems by
him/herself. Having solutions by Mathematica included in the text also provides a
sort of encyclopedia of working approaches to doing things in Mathernatica. In ad
dition, each of these examples exists as a real Mathematica notebook that can be
executed, studied, printed out, or modified to do some other problem. Other Math
ematica notebooks may be provided by the instructor. Occasionally a problem will
request that new methods be tried, but by the time these occur, students should be
able to write effective Mathematica code of their own.
Mathematica can carry the bulk of the computational burden, but this does not
relieve the student of knowing whether or not what is being done is correct. For that
reason, periodic checking of results is stressed. Often an independent manual calcu
lation will keep a Mathematica calculation safely on course. Mathematica, itself, can
and should do much of the checking, because as the problems get more complex, the
calculations get more and more complicated. A calculation that is internally consis
tent stands a good chance of being correct when the concepts that are guiding the
process are correct.
Since all of the problems except those that are of a theoretical nature can be
solved and checked in Mathematica, very few of the exercises have answers supplied.
As the student solves the problems in each section, they should save the notebooks
to disk-where they can serve as an answer book and study guide if the solutions
have been properly checked. A Mathematica package is a collection of functions
that are designed to perform certain operations. Several notebooks depend heavily
on a package that has been provided. Most of the packages supplied undertake very
complicated tasks, where the functions are genuinely intimidating, so the code does
not appear in the text of study notebooks.
What Is New in This Edition
The changes are two-fold:
1. Rearrange and restate some topics (Linear algebra has now been gathered into
a separate chapter, and series methods for systems have been eliminated.) Many
typographical errors have been corrected.
2. Completely rewrite, and occasionally expand, the Mathematica code using ver
sion 5 of Mathematica.
In addition, since Mathematica now includes a complete and fully on-line Help sub
system, several appendices have been eliminated.
Topics Receiving Lesser Emphasis
The solutions of most differential equations are not simple, so the solutions of such
equations are often examined numerically. We indicate some ways to have Math
ematica solve differential equations numerically. Also, properties of a solution are
Preface IX
often deduced from careful examination of the differential equation itself, but an
extended study of qualitative differential equations must wait for a more advanced
course. The best advice is to use the NDSol ve function when a numerical solution
is required.
Some differential equations have solutions that are very hard to describe either
analytically or numerically because the equations are sensitive to small changes in
the initial values. Chaotic behavior is a topic of great current interest; we present
some examples of such equations, but do not fully develop the concepts.
Acknowledgments
I would like to thank those students and others who read the manuscript for the first
edition, and several science department colleagues for enduring questions and for
responding so kindly.
Reviews of the first edition were received from Professors Matthew Richey,
Margie Hale, Stephen L. Clark, Stan Wagon, and William Sit. Dave Withoff con
tributed to that edition his expert help on technical aspects of Mathematica program
ming. Any errors that remain in this edition are solely the responsibility of the author.
Sewanee, Tennessee, USA CLAY C. ROSS
November 2003
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
1 About Differential Equations ................................... .
1.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Uniqueness Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Differential Inclusions (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1 Familiar Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Abstract Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Differential Equations from Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Characteristic Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1 First-Order Linear Differential Equations. . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Linear Equations by M athematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Exact Equations............................................ 59
3.4 Variables Separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Homogeneous Nonlinear Differential Equations . . . . . . . . . . . . . . . . . 75
3.6 Bernoulli and Riccati Differential Equations (Optional)........... 79
3.7 Clairaut Differential Equations (Optional) . . . . . . . . . . . . . . . . . . . . . . 86
4 Applications of First-Order Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.1 Orthogonal Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2 Linear Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Nonlinear Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
XII Contents
5 Higher-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . 129
5.0 Introduction ............................................... 129
5.1 The Fundamental Theorem .................................. 130
5.2 Homogeneous Second-Order Linear Constant Coefficients ........ 139
5.3 Higher-Order Constant Coefficients (Homogeneous) ............. 152
5.4 The Method of Undetermined Coefficients ..................... 160
5.5 Variation of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6 Applications of Second-Order Equations. . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.1 Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.2 Damped Harmonic Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.3 Forced Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.4 Simple Electronic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6.5 Two Nonlinear Examples (Optional). . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.0 Introduction ............................................... 210
7.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.2 Properties of the Laplace Transform ........................... 214
7.3 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.4 Discontinous Functions and Their Transforms. . . . . . . . . . . . . . . . . . . 230
8 Higher-Order Differential Equations with Variable Coefficients . . . . . 240
8.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
8.1 Cauchy-Euler Differential Equations .......................... 241
8.2 Obtaining a Second Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.3 Sums, Products and Recursion Relations . . . . . . . . . . . . . . . . . . . . . . . 253
8.4 Series Solutions of Differential Equations . . . . . . . . . . . . . . . . . . . . . . 262
8.5 Series Solutions About Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . 269
8.6 Series Solution About Regular Singular Points . . . . . . . . . . . . . . . . . . 277
8.7 Important Classical Differential Equations and Functions ......... 293
9 Differential Systems: Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
9.0 Introduction ............................................... 297
9.1 Reduction to First-Order Systems ............................. 302
9.2 Theory of First-Order Systems ............................... 309
9.3 First- Order Constant Coefficients Systems . . . . . . . . . . . . . . . . . . . . . . 317
9.4 Repeated and Complex Roots ................................ 331
9.5 Nonhomogeneous Equations and Boundary-Value Problems ...... 344
9.6 Cauchy-Euler Systems ...................................... 358