A first course in differential equations

294 Pages·2006·1.564 MB·English

by  J David Logan

J. David Logan
A First Course in
Differential Equations
With 55 Figures
J.DavidLogan
WillaCatherProfessorofMathematics
DepartmentofMathematics
UniversityofNebraskaatLincoln
Lincoln,NE68588-0130
USA
[email protected]
EditorialBoard
S.Axler K.A.Ribet
MathematicsDepartment DepartmentofMathematics
SanFranciscoStateUniversity UniversityofCaliforniaatBerkeley
SanFrancisco,CA94132 Berkeley,CA94720-3840
USA USA
[email protected] [email protected]
MathematicsSubjectClassification(2000):34-xx,15-xx
LibraryofCongressControlNumber:2005926697(hardcover);
LibraryofCongressControlNumber:2005926698(softcover)
ISBN-10:0-387-25963-5(hardcover)
ISBN-13:978-0387-25963-5
ISBN-10:0-387-25964-3(softcover)
ISBN-13:978-0387-25964-2
©2006SpringerScience+BusinessMedia,Inc.
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithout
thewrittenpermissionofthepublisher(SpringerScience+BusinessMedia,Inc.,233Spring
Street,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor
scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,
electronic adaptation, computer software, or by similar or dissimilar methodology now
knownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,
eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionas
towhetherornottheyaresubjecttoproprietaryrights.
PrintedintheUnitedStatesofAmerica. (SBA)
9 8 7 6 5 4 3 2 1
springeronline.com
Dedicated to—
Reece Charles Logan,
Jaren Logan Golightly
Contents
Preface ......................................................... xi
To the Student ................................................. xiii
1. Differential Equations and Models .......................... 1
1.1 Differential Equations..................................... 2
1.1.1 Equations and Solutions............................. 2
1.1.2 Geometrical Interpretation .......................... 9
1.2 Pure Time Equations ..................................... 13
1.3 Mathematical Models ..................................... 19
1.3.1 Particle Dynamics .................................. 21
1.3.2 Autonomous Differential Equations ................... 28
1.3.3 Stability and Bifurcation ............................ 41
1.3.4 Heat Transfer...................................... 45
1.3.5 Chemical Reactors.................................. 48
1.3.6 Electric Circuits.................................... 51
2. Analytic Solutions and Approximations..................... 55
2.1 Separation of Variables.................................... 55
2.2 First-Order Linear Equations .............................. 61
2.3 Approximation........................................... 70
2.3.1 Picard Iteration .................................... 71
2.3.2 Numerical Methods................................. 74
2.3.3 Error Analysis ..................................... 78
viii Contents
3. Second-Order Differential Equations ........................ 83
3.1 Particle Mechanics........................................ 84
3.2 Linear Equations with Constant Coefficients ................. 87
3.3 The Nonhomogeneous Equation ............................ 95
3.3.1 Undetermined Coefficients........................... 96
3.3.2 Resonance.........................................102
3.4 Variable Coefficients ......................................105
3.4.1 Cauchy–Euler Equation .............................106
3.4.2 Power Series Solutions ..............................109
3.4.3 Reduction of Order .................................111
3.4.4 Variation of Parameters .............................112
3.5 Boundary Value Problems and Heat Flow ...................117
3.6 Higher-Order Equations ...................................124
3.7 Summary and Review.....................................127
4. Laplace Transforms.........................................133
4.1 Definition and Basic Properties ............................133
4.2 Initial Value Problems ....................................140
4.3 The Convolution Property.................................145
4.4 Discontinuous Sources.....................................149
4.5 Point Sources ............................................152
4.6 Table of Laplace Transforms ...............................157
5. Linear Systems .............................................159
5.1 Introduction .............................................159
5.2 Matrices ................................................165
5.3 Two-Dimensional Systems .................................179
5.3.1 Solutions and Linear Orbits..........................179
5.3.2 The Eigenvalue Problem ............................185
5.3.3 Real Unequal Eigenvalues ...........................187
5.3.4 Complex Eigenvalues ...............................189
5.3.5 Real, Repeated Eigenvalues..........................191
5.3.6 Stability ..........................................194
5.4 Nonhomogeneous Systems .................................198
5.5 Three-Dimensional Systems................................204
6. Nonlinear Systems..........................................209
6.1 Nonlinear Models.........................................209
6.1.1 Phase Plane Phenomena ............................209
6.1.2 The Lotka–Volterra Model...........................217
6.1.3 Holling Functional Responses ........................221
6.1.4 An Epidemic Model ................................223
Contents ix
6.2 Numerical Methods.......................................229
6.3 Linearization and Stability.................................233
6.4 Periodic Solutions ........................................246
6.4.1 The Poincar´e–Bendixson Theorem....................249
Appendix A. References ........................................255
Appendix B. Computer Algebra Systems .......................257
B.1 Maple...................................................258
B.2 MATLAB ...............................................260
Appendix C. Sample Examinations .............................265
Appendix D. Solutions and Hints to Selected Exercises .........271
Index...........................................................287
Preface
Therearemanyexcellenttextsonelementarydifferentialequationsdesignedfor
thestandardsophomorecourse.However,inspiteofthefactthatmostcourses
are one semester in length, the texts have evolved into calculus-like presen-
tations that include a large collection of methods and applications, packaged
with student manuals, and Web-based notes, projects, and supplements. All of
this comes in several hundred pages of text with busy formats. Most students
do not have the time or desire to read voluminous texts and explore internet
supplements. The format of this differential equations book is different; it is a
one-semester,brieftreatmentofthebasicideas,models,andsolutionmethods.
Itslimitedcoverageplacesitsomewherebetweenanoutlineandadetailedtext-
book.Ihavetriedtowriteconcisely,tothepoint,andinplainlanguage.Many
workedexamplesandexercisesareincluded.Astudentwhoworksthroughthis
primer will have the tools to go to the next level in applying differential equa-
tionstoproblemsinengineering,science,andappliedmathematics.Itcangive
some instructors, who want more concise coverage, an alternative to existing
texts.
The numerical solution of differential equations is a central activity in sci-
ence and engineering, and it is absolutely necessary to teach students some
aspects of scientific computation as early as possible. I tried to build in flex-
ibility regarding a computer environment. The text allows students to use a
calculator or a computer algebra system to solve some problems numerically
and symbolically, and templates of MATLAB and Maple programs and com-
mands are given in an appendix. The instructor can include as much of this,
or as little of this, as he or she desires.
For many years I have taught this material to students who have had a
standard three-semester calculus sequence. It was well received by those who
xii Preface
appreciated having a small, definitive parcel of material to learn. Moreover,
this text gives students the opportunity to start reading mathematics at a
slightly higher level than experienced in pre-calculus and calculus. Therefore
the book can be a bridge in their progress to study more advanced material
at the junior–senior level, where books leave a lot to the reader and are not
packaged in elementary formats.
Chapters 1, 2, 3, 5, and 6 should be covered in order. They provide a route
to geometric understanding, the phase plane, and the qualitative ideas that
are important in differential equations. Included are the usual treatments of
separable and linear first-order equations, along with second-order linear ho-
mogeneous and nonhomogeneous equations. There are many applications to
ecology, physics, engineering, and other areas. These topics will give students
key skills in the subject. Chapter 4, on Laplace transforms, can be covered at
any time after Chapter 3, or even omitted. Always an issue in teaching differ-
ential equations is how much linear algebra to cover. In two extended sections
inChapter5weintroduceamoderateamountofmatrixtheory,includingsolv-
ing linear systems, determinants, and the eigenvalue problem. In spite of the
book’s brevity, it still contains slightly more material than can be comfortably
covered in a single three-hour semester course. Generally, I assign most of the
exercises; hints and solutions for selected problems are given in Appendix D.
I welcome suggestions, comments, and corrections. Contact information is
on my Web site: http://www.math.unl.edu/˜dlogan, where additional items
may be found.
I would like to thank John Polking at Rice University for permitting me to
use his MATLAB program pplane7 to draw some of the phase plane diagrams
and Mark Spencer at Springer for his enthusiastic support of this project. Fi-
nally, I would like to thank Tess for her continual encouragement and support
for my work.
David Logan
Lincoln, Nebraska
To the Student
What is a course in differential equations about? Here are some informal,
preparatoryremarkstogiveyousomesenseofthesubjectbeforewetakeitup
seriously.
Youarefamiliarwithalgebraproblemsandsolvingalgebraicequations.For
example, the solutions to the quadratic equation
x2−x=0
are easily found to be x = 0 and x = 1, which are numbers. A differential
equation (sometimes abbreviated DE) is another type of equation where the
unknown is not a number, but a function. We will call it u(t) and think of it
as a function of time. A DE also contains derivatives of the unknown function,
which are also not known. So a DE is an equation that relates an unknown
function to some of its derivatives. A simple example of a DE is
u(cid:1)(t)=u(t),
whereu(cid:1)(t)denotesthederivativeofu(t).Weaskwhatfunctionu(t)solvesthis
equation. That is, what function u(t) has a derivative that is equal to itself?
From calculus you know that one such function is u(t) = et, the exponential
function. We say this function is a solution of the DE, or it solves the DE. Is
it the only one? If you try u(t) = Cet, where C is any constant whatsoever,
youwillalsofinditisasolution.Sodifferentialequationshavelotsofsolutions
(fortunatelywewillseetheyarequitesimilar,andthefactthattherearemany
allows some flexibility in imposing other desired conditions).
This DE was very simple and we could guess the answer from our calcu-
lus knowledge. But, unfortunately (or, fortunately!), differential equations are
usually more complicated. Consider, for example, the DE
u(cid:1)(cid:1)(t)+2u(cid:1)(t)+2u(t)=0.
xiv To the Student
This equation involves the unknown function and both its first and second
derivatives. We seek a function for which its second derivative, plus twice its
firstderivative,plustwicethefunctionitself,iszero.Nowcanyouquicklyguess
a function u(t) that solves this equation? It is not likely. An answer is
u(t)=e−tcost.
And,
u(t)=e−tsint
worksaswell.Let’scheckthislastonebyusingtheproductruleandcalculating
its derivatives:
u(t) = e−tsint,
u(cid:1)(t) = e−tcost−e−tsint,
u(cid:1)(cid:1)(t) = −e−tsint−2e−tcost+e−tsint.
Then
u(cid:1)(cid:1)(t)+2u(cid:1)(t)+2u(t)
= −e−tsint−2e−tcost+e−tsint+2(e−tcost−e−tsint)+2e−tsint
= 0.
So it works! The function u(t) = e−tsint solves the equation u(cid:1)(cid:1)(t)+2u(cid:1)(t)+
2u(t)=0. In fact,
u(t)=Ae−tsint+Be−tcost
is a solution regardless of the values of the constants A and B. So, again,
differential equations have lots of solutions.
Partly,thesubjectofdifferentialequationsisaboutdevelopingmethodsfor
finding solutions.
Why differential equations? Why are they so important to deserve a course
of study? Well, differential equations arise naturally as models in areas of sci-
ence, engineering, economics, and lots of other subjects. Physical systems, bi-
ological systems, economic systems—all these are marked by change. Differen-
tial equations model real-world systems by describing how they change. The
unknownfunctionu(t)couldbethecurrentinanelectricalcircuit,theconcen-
tration of a chemical undergoing reaction, the population of an animal species
in an ecosystem, or the demand for a commodity in a micro-economy. Differ-
ential equations are laws that dictate change, and the unknown u(t), for which
we solve, describes exactly how the changes occur. In fact, much of the reason
that the calculus was developed by Isaac Newton was to describe motion and
to solve differential equations.