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Preface to the Second Edition
From the earliest times, humankind has constructed models to explain the inner workings of nature, to predict what the future portends, or to exert some measure of control
over the environment. For example, the geometry of Euclid helped ancient Egyptians
and Greeks to predict astronomical events with commercial significance. How effective were these models? Given the inherent complexity of the natural world, some
models worked surprisingly well and others less so.
To build models one needs tools. The earliest tools were physical and metaphysical in nature but over the millennia humans created abstract tools that, when coupled
with observation and experimentation, sometimes produced spectacular results. For
example, Isaac Newton invented calculus in the seventeenth century as a tool to model
the motion of material bodies under the action of gravity and other forces. Over time,
and with the help of many scientists and mathematicians, calculus gave rise to the field
of differential equations, a highly successful tool for constructing models of dynamically changing systems such as our solar system. Differential equations provided a
tool which made it possible to land men on the moon in the twentieth century.
A great deal of modern mathematics since the seventeenth century has roots in
the analysis of models of dynamical systems. These abstract mathematical structures
eventually took on a life of their own, divorced from their origins. However, with the
powerful and inexpensive computing tools currently available, we think that now is
the right time to reconnect differential equations to its roots via an introductory course
from a modeling perspective. That is the goal of this text.
Features Retained from the First Edition
•
Novel features from the first edition were retained in this new edition. First and
foremost, modeling of dynamically changing systems, and visualization of solutions
continue to be central themes. We also continue to use qualitative, analytical, and
graphical approaches in our study of solutions of differential equations and differential systems. Differential systems still come up early in a completely natural way when
modeling dynamical systems with more than one state variable. We continue to give
advice from time to time on how to use computers effectively to produce useful graphical displays. Finally, this text continues to offer a rich source of modeling problems
from physics, chemistry, engineering, and biology.
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Preface to the Second Edition
Features New to this Edition
•
The main goal of this revision was to make the text easier for both students and faculty
to use. Great care in the revision process and some new features ensure this goal:
Gentle Introduction to Modeling. We put the basic features of modeling into a new
Chapter 1 and illustrated them with simple and familiar models: vertical motion, population dynamics, radioactive decay and linear cascades.
Delayed Use of Solvers. Solvers do not come up until Chapter 2. Thus, we introduce
modeling, analytic solution techniques, and qualitative analysis before solvers are even
mentioned.
Focus of Sections, Spotlight Sections. Each section is now focused so that the main
points can be covered in one lecture. In some cases we extracted peripheral material
and inserted it into special Spotlight sections at the ends of chapters. Spotlight sections
are optional. Additional Spotlight sections appear on the Wiley Web site and also on
a CD-ROM included with this text. The titles and locations of all Spotlight sections
appear in the table of contents.
Problem Sets. The problems at the end of each section are organized under headings
that identify the problem type. Each problem set begins with several skill-building
exercises. Hints are often provided, and additional information often appears in a
banner under a heading. Problems are now individually numbered.
Level of Presentation. The level of presentation increases gradually within a section,
from section to section, and from chapter to chapter. Problem sets have been carefully
graded in order of difficulty. Also, a better use of color and formatting make the text
easier to read.
Models. More models appear at every level. Many are in Spotlight sections (or the
included CD-ROM) and can be covered as desired for maximum flexibility.
Examples. We include new examples to further illustrate principles and solution techniques. We kept many examples from the first edition, edited to make them more
focused and easier to understand.
Procedures. Solution techniques are often summarized in the form of a step-by-step
procedure for easy reference.
Prerequisites
•
To meet the needs of diverse student populations we try to strike a balance in our
approach: each chapter starts with a carefully developed model, and in following sections we give the theoretical underpinnings. We assume a calculus background and
some familiarity with vectors and matrices (often, these days, from high school). Two
Spotlight sections review the basic concepts from linear algebra used in the text. A
few sections (mostly in the later chapters) require some basic knowledge of partial
differentiation. We assume no knowledge of computer programming.
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Preface to the Second Edition
Use of This Text
•
This edition is designed to allow an instructor great flexibility to cover the material that
suits individual student needs. Spotlight sections can be inserted where appropriate to
customize models to student backgrounds and needs. Spotlights are also effective as
take-home group lab projects.
There are four Spotlights in particular that should be considered when designing
a syllabus. For more emphasis on numerical methods and the use of numerical solvers
you may want to cover the S POTLIGHT ON A PPROXIMATE N UMERICAL S OLUTIONS and
the S POTLIGHT ON C OMPUTER I MPLEMENTATION (both at the end of Chapter 2). If your
students need some basics from linear algebra then cover the S POTLIGHT ON V ECTORS ,
M ATRICES , I NDEPENDENCE and the S POTLIGHT ON L INEAR A LGEBRAIC E QUATIONS (both
at the end of Chapter 6, where it is needed).
Recommended one-semester course syllabi for four different student populations
appear on page ix. Each syllabus entails 35 lectures and leaves some time to cover
additional sections and Spotlights.
Throughout this text we adopt the modern view of differential systems as evolving
dynamical systems. We introduce modern topics such as long-term behavior, sensitivity, bifurcation and chaos, but we also present solution formula methods and theory
expected in a first course for a general audience that includes scientists and engineers.
The study of solutions of differential equations takes many forms. Sometimes a
formula for a solution is adequate. Often there is no way to find a solution formula,
but we still would like to know the qualitative behavior of solutions. Software tools
not only provide quantitative values of solutions but also allow us to visualize graphs
of solutions. We use all these approaches here because they work well together.
The basic theory of initial value problems for differential equations and differential
systems is gradually developed so as not to overwhelm students with the most general
version right from the start. Wherever possible we give constructive proofs, but we
do not always prove everything in generality. Whether or not proofs are covered in a
course naturally depends on the student backgrounds.
Supplements
•
Technology Resource Manuals. Professor Jenny Switkes (California State Polytechnic University Pomona) has written three brief Technology Resource Manuals, one
each for Maple (ISBN 0-471-447846), Matlab (ISBN 0-471-483877), and Mathematica (ISBN 0-471-483869). These manuals provide sample code for virtually every type
of problem in this text requiring a computer.
Student Resource Manual. This manual (ISBN 0-471-433330) gives complete solutions (along with graphs) to nearly every odd-numbered problem.
Instructor’s Resource Manual. This manual (ISBN 0-471-447854) is available from
the publisher. It has complete solutions (with graphs) to nearly all problems.
ODE Architect. A new feature in this edition is a CD-ROM with the prize-winning
point-and-click numerical solver ODE Architect. A user’s guide is included on the
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Preface to the Second Edition
ODEA
ODEA
CD-ROM. Examples in the text whose graphs appear in the ODE Architect Library
are marked with the icon
. A few Examples and Spotlights use ODE Architect
and these are marked with the icon
.
Acknowledgments
•
We would like to express our gratitude to the many people who have helped to bring
this second edition to fruition. We are especially indebted to Jenny Switkes whose
help has been invaluable. Her mathematical skills and talented programming ability
contributed significantly at every level of our project. Kudos also to our LATEX gurus,
David Richards (Synopsys) and Lisa Wice (Harvey Mudd College) whose phenomenal
facility with this typesetting package created the clean layout and formatting of this
text. Thanks also to the following students for typesetting help and in suggesting
improvements: Steven Avery, Ben Brooks, Katherine Bryant, Melissa Federowicz,
Avani Gadani, Lena Kaloostian, Meredith McDonald, Stuart Mershon, Imad Muhi ElDdin, and Azusa Yabe. Our thanks also to Paul Damikolas and Bijan Dehbozorgi for
testing ODE Architect on examples and Spotlights. Our special thanks to the Wiley
editoral staff, especially to Laurie Rosatone, Anne Scanlan-Rohrer, and Kelly Boyle,
for help in guiding this project to completion, no mean feat.
We are indebted to the following reviewers for their useful comments and suggestions on various versions of this text: James Bradley (Calvin College), Mark Brittenham (University of Nebraska), John Cantwell (St. Louis University), Bohumil
Cenkl (Northeastern University), Alex Feldman (Boise State University), David Gines
(Agilent Technologies), David Gurarie (Case Western Reserve University), Grant B.
Gustafson (University of Utah), Hoon Hong (North Carolina State University), Michael
Huff (Austin Community College), Thomas W. Judson (Harvard University), Matthias
Kawski (Arizona State Univeristy), Mohamed A. Khamsi (University of Texas–El
Paso), Michael Kirby (Colorado State University), Julie Levandosky (Stanford University), Joseph Mahaffy, (San Diego State University), John McCuan (Georgia Tech),
Mark McKibben (Goucher College), Arthur Ogus (University of California–Berkeley),
V.S. Prasad (University of Massachussetts–Lowell), W. Proskurowski (University of
Southern California), Joe Thrash (University of Southern Mississippi), Lawrence Turyn (Wright State University), David A. Voss (Western Illiinois University), Robert E.
White (North Carolina State University).
We are grateful to Marie Vanisko (California State University - Stanislaus) and
Timothy Comar (Benedictine University) for their painstaking efforts in checking the
accuracy of solutions to the problems, and to James Bradley for his advice on writing style. We deeply appreciate the advice and encouragment of our colleagues at
Harvey Mudd College. Last, but not least, we thank our wives for their patience and
understanding during this seemingly unending project.
R.L. Borrelli
Borrelli@hmc.edu
C.S. Coleman Coleman@hmc.edu
Claremont, October 2003
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Preface to the Second Edition
Chapter Dependency Chart and Semester Course Syllabi
1
2
3
6
7
10
11
If Fourier Series and PDEs are
part of the syllabus, then the first
four sections of Chapter 10 can
replace the indicated sections of
Chapters 9 or 11.
4
8
5
9
For Engineering Students
Chapter 1 (3 lectures)
Chapter 2 (2.1–2.5, 2.7, 2.8)
Chapter 3 (6 lectures)
Chapter 4 (3 lectures)
Chapter 5 (4 lectures)
Chapter 6 (6.2–6.7)
Chapter 7 (7.1, 7.2)
Chapter 11 (4 lectures)
Total: 35 lectures
A Course Emphasizing Systems
Chapter 1 (3 lectures)
Chapter 2 (2.1–2.9)
Chapter 3 (3.1–3.3, 3.7, 3.8)
Chapter 4 (4.1)
Chapter 6 (6.1–6.7)
Chapter 7 (7.1–7.3)
Chapter 8 (8.1–8.3)
Chapter 9 (9.1–9.4)
Total: 35 lectures
For Math/Physics Students
Chapter 1 (2 lectures)
Chapter 2 (9 lectures)
Chapter 3 (8 lectures)
Chapter 4 (4.1)
Chapter 6 (6.2–6.4, 6.6–6.8)
Chapter 7 (7.1, 7.2)
Chapter 8 (8.1–8.3)
Chapter 9 (9.1–9.4)
or Chapter 11 (11.1–11.4)
Total: 35 lectures
For Biology/Pre-Med Students
Chapter 1 (4 lectures,
+ C OLD M EDICATION I)
Chapter 2 (2.1–2.9)
+ C OLD M EDICATION II)
Chapter 3 (3.2–3.5)
Chapter 4 (4.1)
Chapter 6 (6.1–6.7)
Chapter 7 (7.1–7.3)
+ S ENSITIVITY AND B IFURCATION)
Chapter 8 (2 lectures)
Chapter 9 (9.1, 9.3)
Total: 35 lectures
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Contents
1 • Modeling and Differential Equations
1.1
1.2
1.3
1.4
The Modeling Approach 1
A Modeling Adventure 9
Models and Initial Value Problems 15
The Modeling Process: Differential Systems
1
22
S POTLIGHT ON M ODELING : R ADIOCARBON DATING
S POTLIGHT ON M ODELING : C OLD M EDICATION I
31
37
2 • First-Order Differential Equations
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Linear Differential Equations 42
Linear Differential Equations: Qualitative Analysis 50
Existence and Uniqueness of Solutions 58
Visualizing Solution Curves: Slope Fields 67
Separable Differential Equations: Planar Systems 76
A Predator-Prey Model: the Lotka–Volterra System 85
Extension of Solutions: Long-Term Behavior 94
Qualitative Analysis: State Lines, Sign Analysis 102
Bifurcations: A Harvested Logistic Model 112
Snapshot on Solution Formula Techniques 120
S POTLIGHT ON A PPROXIMATE N UMERICAL S OLUTIONS 122
S POTLIGHT ON C OMPUTER I MPLEMENTATION 130
S POTLIGHT ON S TEADY S TATES : L INEAR ODE S 137
S POTLIGHT ON M ODELING : C OLD M EDICATION II 142
S POTLIGHT ON C HANGE OF VARIABLES : P URSUIT M ODELS 145
S POTLIGHT ON C ONTINUITY IN THE DATA 154
42
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Contents
3 • Second-Order Differential Equations
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
160
Models of Springs 160
Undriven Constant-Coefficient Linear Differential Equations 169
Visualizing Graphs of Solutions: Direction Fields 176
Periodic Solutions: Simple Harmonic Motion 185
Driven Linear ODEs: Undetermined Coefficients I 193
Driven Linear ODEs: Undetermined Coefficients II 207
Theory of Second-Order Linear Differential Equations 213
Nonlinear Second-Order Differential Equations 223
A Snapshot Look at Constant-Coefficient Polynomial Operators 237
S POTLIGHT ON M ODELING : V ERTICAL M OTION 238
S POTLIGHT ON M ODELING : S HOCK A BSORBERS 247
S POTLIGHT ON E INSTEIN ’ S F IELD E QUATIONS 251
4 • Applications of Second-Order Differential Equations
4.1
4.2
4.3
4.4
252
Newton’s Laws: The Pendulum 252
Beats and Resonance 264
Frequency Response Modeling 272
Electrical Circuits 282
Snapshot on Mechanical and Electrical Models 291
S POTLIGHT ON M ODELING : T UNING A C IRCUIT 293
5 • The Laplace Transform
5.1
5.2
5.3
5.4
295
The Laplace Transform: Solving IVPs 295
Working with the Transform 304
Transforms of Periodic Functions 314
Convolution 322
S POTLIGHT ON THE D ELTA F UNCTION 328
S POTLIGHT ON M ODELING : T IME D ELAYS AND C OLLISIONS
336
6 • Linear Systems of Differential Equations
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
Compartment Models: Tracking Lead 342
Eigenvalues, Eigenvectors, and Eigenspaces of Matrices
Undriven Linear Differential Systems: Real Eigenvalues
Undriven Linear Systems: Complex Eigenvalues 374
Orbital Portraits for Planar Systems 384
Driven Systems: The Matrix Exponential 394
Steady States 405
The Theory of General Linear Systems 413
S POTLIGHT ON V ECTORS , M ATRICES , I NDEPENDENCE 421
S POTLIGHT ON L INEAR A LGEBRAIC E QUATIONS 430
S POTLIGHT ON B IFURCATIONS : S ENSITIVITY 440
342
352
363
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Contents
7 • Nonlinear Differential Systems
7.1
7.2
7.3
444
Chemical Kinetics: The Fundamental Theorem 444
Properties of Autonomous Systems, Direction Fields 452
Interacting Species: Cooperation, Competition 464
S POTLIGHT ON M ODELING : D ESTRUCTIVE C OMPETITION 474
S POTLIGHT ON M ODELING : B IFURCATION AND S ENSITIVITY 478
8 • Stability
8.1
8.2
8.3
479
Stability and Linear Autonomous Systems 479
Stability and Nonlinear Autonomous Systems 486
Stability of Planar Nonlinear Systems 499
Conservative Systems 500
S POTLIGHT ON LYAPUNOV F UNCTIONS 508
S POTLIGHT ON ROTATING B ODIES 516
9 • Nonlinear Systems: Cycles and Chaos
9.1
9.2
9.3
9.4
Cycles 521
Solution Behavior in Planar Autonomous Systems
Bifurcations 538
Chaos 546
S POTLIGHT ON C HAOTIC S YSTEMS 557
521
529
10 • Fourier Series and Partial Differential Equations
10.1
10.2
10.3
10.4
10.5
10.6
Vibrations of a Guitar String 560
Fourier Trigonometric Series 573
Half-Range and Exponential Fourier Series 582
Temperature in a Thin Rod 589
Sturm–Liouville Problems 600
The Method of Eigenfunction Expansions 606
S POTLIGHT ON D ECAY E STIMATES 612
S POTLIGHT ON THE O PTIMAL D EPTH FOR A W INE C ELLAR
S POTLIGHT ON A PPROXIMATION OF F UNCTIONS 616
11 • Series Solutions
11.1
11.2
11.3
11.4
560
The Method of Power Series 633
Series Solutions near an Ordinary Point 641
Regular Singular Points: The Euler ODE 647
Series Solutions near Regular Singular Points 654
S POTLIGHT ON L EGENDRE P OLYNOMIALS 661
S POTLIGHT ON B ESSEL F UNCTIONS 668
614
633
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A • Basic Theory of Initial Value Problems
A.1
A.2
A.3
Uniqueness 675
The Picard Process for Solving an Initial Value Problem
Extension of Solutions 685
675
677
B • Background Information
B.1
B.2
Power Series 687
Results from Calculus
• Answers
691
to Selected Problems
• Index
www • Spotlights
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
S POTLIGHT ON
687
696
711
S LOPE F IELDS I www-1
S LOPE F IELDS II www-2
S ENDING A M ESSAGE VIA A C OAXIAL C ABLE www-3
E XACT ODE S www-4
S ENSITIVITY OF S OLUTIONS TO THE DATA www-11
C OMPLEX -VALUED F UNCTIONS www-18
M INIMAL T IME OF D ESCENT www-21
T IME -S TATE C URVES AND P ROJECTIONS www-23
O PTICAL I LLUSIONS www-24
O RBITS OF S ATELLITES www-27
M ODELING : N OISE F ILTER www-37
L OCATING E IGENVALUES www-42
M ODELING : T HE S HAPE OF A PAPER C UTTER www-47
D EFICIENT E IGENSPACES www-51
M ODELING : C OUPLED S PRINGS www-59
C OUPLED S PRINGS : N ORMAL M ODES www-64
P ENDULUM M OTION : S ENSITIVITY www-68
M ODELING : T HE P OSSUM P LAGUE www-70
S CALING AND U NITS www-75
Q UADRATIC R ATES AND L IMIT C YCLES www-80
C HAOS IN A N ONLINEAR C IRCUIT www-81
C HAOS IN N UMERICS www-85
P ROPERTIES OF THE WAVE E QUATION www-96
I NITIAL VALUE P ROBLEMS www-102
L APLACE ’ S E QUATION www-106
THE E XTENDED M ETHOD OF F ROBENIUS
www-113
S TEADY T EMPERATURES www-122
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Contents