Table Of ContentINTRODUCTORY
DIFFERENTIAL
EQUATIONS
INTRODUCTORY
DIFFERENTIAL
EQUATIONS
FIFTH EDITION
Martha L. Abell
GeorgiaSouthernUniversity,Statesboro,GA,USA
James P. Braselton
GeorgiaSouthernUniversity,Statesboro,GA,USA
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Preface
Introductory Differential Equations began as computer algebra system so that they could be
a text originally called Introductory Differential exposed to the technology. However, we soon
EquationswithBoundaryValueProblems. realized that we were missing the great op-
When we were done with the revision, we portunity of allowing students to discover as-
no longer saw Introductory Differential Equations pects of the subject matter on their own. We
with Boundary Value Problems but rather a new revised our materials to include experimental
text that we have titled Introductory Differential problems and thought-provoking questions in
Equations. which students are asked to make conjectures
The first edition of this text, Modern Differen- and investigate supporting evidence. We also
tialEquationswas“modern”becauseitwasone developedapplicationprojectscalledDifferential
ofthefirsttextsthatrequiredaccesstoagraph- Equations at Work, not only to emphasize tech-
ing calculator, computer algebra system, or nu- nology,butalsotoimprovetheproblem-solving
mericalsoftwarepackage. and communication skills of our students. To
Computeralgebrasystemsandsophisticated preserve the “wow” aspects of technology, we
graphing calculators have changed the ways in continue to use it to observe solutions in class-
which we learn and teach ordinary differential roomdemonstrationsthroughsuchthingsasan-
equations. Instead of focusing students’ atten- imating the motion of springs and pendulums.
tion only on a sequence of solution methods, These demonstrations not only grab the atten-
wewantthemtousetheirmindstounderstand tionofstudents,butalsohelpthemtomakethe
whatsolutionsmeanandhowdifferentialequa- connection between a formula and what it rep-
tionscanbeusedtoanswerpertinentquestions. resents.
Nowtheiruseisexpectedinastandardcourse, This text is designed to serve as a text for
so the term “modern” no longer applies to the beginningcoursesinordinarydifferentialequa-
text. tions.Usually,introductoryordinarydifferential
Interestingly, this metamorphosis in the equations courses are taken by students who
teaching of differential equations described oc- have successfully completed a first-year calcu-
curredrelativelyquicklyandcoincidedwithour luscourse,abasiclinearalgebracourse,andthis
professional careers at Georgia Southern Uni- textiswrittenatalevelreadableforthem.
versity. Our interest in the use of technology
in the mathematics classroom began in 1990
when we started to use computer laboratories TECHNOLOGY
and demonstrations in our calculus, differen-
tialequations,andappliedmathematicscourses. Some advantages of incorporating technol-
Overthepastyearswehavelearnedsomeways ogy into mathematics courses include enhanc-
of how to and how not to use technology in ing the ability to solve a variety of problems,
themathematicscurriculum.Intheearlystages, helpingstudentsworkexamples,supportingin-
we simply wanted to show students how they terestingapplications,exploitingandimproving
could solve more difficult problems by using a geometric intuition, encouraging mathematical
vii
viii
PREFACE
experiments, and helping students understand data. In particular, obtaining closed form solu-
approximation.Inaddition,technologyisimple- tionsisnotnecessarily“easy”(oralwayspossi-
mentedthroughoutthistexttopromotethefol- ble).Theseapplications,evenifnotformallydis-
lowinggoalsinthelearningofdifferentialequa- cussed in class, show students that differential
tions: equations is an exciting and interesting subject
withextensiveapplicationsinmanyfields.
1. Solvingproblems:Usingdifferentmethodsto
solveproblemsandgeneralizesolutions;
2. Reasoning: Exploiting computer graphics to
STYLE
develop spatial reasoning through visualiza-
tion;
To keep the text as flexible as possible, ad-
3. Analyzing:Findingthemostreasonablesolu-
dressingtheneedsofbothaudienceswithdiffer-
tiontorealproblemsorobservingchangesin
ent mathematical backgrounds and instructors
thesolutionunderchangingconditions;
with varying preferences, Introductory Differen-
4. Communicating mathematics: Developing
tial Equations is written in an easy-to-read, yet
written, verbal, and visual skills to commu-
mathematicallyprecise,style.Itcontainsalltop-
nicatemathematicalideas;and
ics typically included in a standard first course
5. Synthesizing:Makinginferencesandgeneral-
in ordinary differential equations. Definitions,
izations,evaluatingoutcomes,classifyingob-
theorems, and proofs are concise but worded
jects,andcontrollingvariables.
precisely for mathematical accuracy Generally,
Students who develop these skills will suc- theorems are proved if the proof is instructive
ceed not only in differential equations, but also orhas“teachingvalue.”Ofcourse,discussionof
insubsequentcoursesandintheworkforce. such proofs is optional in the typical classroom
for which this text is written. In other cases,
The icon is used throughout the text to
proofs of theorems are developed in the exer-
indicate those examples in which technology is
cises or omitted. Theorems and definitions are
used in a nontrivial way to develop or visual-
boxed for easy reference; key terms are high-
izethesolutionortoindicatethesectionsofthe
lighted in boldface. Figures are used frequently
text, such as those discussing numerical meth-
to clarify material with a graphical interpreta-
ods,inwhichtheuseofappropriatetechnology
tion.
isessentialorinteresting.
FEATURES
APPLICATIONS
Introductory Differential Equations, Fifth edi-
Applicationsinthistextaretakenfromava-
tion, is an extensive revision of the Fourth edi-
rietyoffields,especiallybiology,physics,chem-
tion of Introductory Differential Equations. This
istry, engineering, and economics, and they are
editionincludesseveralofthefollowing:
documented by references. These applications
can be found in many of the examples and ex- • Because the mathematician’s who developed
ercises, in separate sections and chapters of the the mathematics discussed in this text were
text,andintheDifferentialEquationsatWorksub- (orare)stillinterestingintheirownright,we
sections at the end of each chapter. Many of have tried to include an image and interest-
theseapplicationsarewell-suitedtoexploration ingtidbitabouttheirliveswheneverpossible
with technology because they incorporate real hoping to help some students become more
ix
PEDAGOGICALFEATURES
interested in the course. When credit is not Technology
given for a photo, it is because we have rea-
Many students entering their first differen-
sonablereasontobelievethattheimageisin
tialcoursehavehadsubstantialexperiencewith
the public domain. If a copyright applies to
various sophisticated calculators and computer
animageandappropriatecredithasnotbeen
algebrasystems.
given, please alert us so that we can correct
Our first ordinary differential equations
thesituationpromptly.
course attempts to encourage students to use
• All graphics have been redone. In each case,
technology intelligently. We have italicized the
theintentofthegraphichasbeenquestioned.
words use technology intelligently because they
Insomecases,graphicshavebeeneliminated;
take on different meaning to different instruc-
in other cases, they have been redone to em-
tors because they depend on the instructor’s
phasize their purpose or new graphics have
philosophy, institution, and students. Students
beenintroducedtothetext.
also interpret the phrase differently depending
• Wehaverevisedtheexercisesetconsiderably.
upon their instructor and exposure to technol-
The total number of exercises remains about
ogy.
thesameasinpreviouseditionsbutwehave
In any case, many instructors have limited
deleted about 100 “outdated” exercises and
resources and would prefer that our students
replacedthemwithnewones.Wecontinueto
have a good grasp of the fundamentals rather
believethatastudentshouldbeabletosolve
than be “wowed!” by nonsense. We have tried
abasicnontrivialproblembyhand.
to use technology intelligently. We believe that
it should not be obtrusive so you should not
notice when we do. When required in an ex-
PEDAGOGICAL FEATURES
ample or exercise, it should be obvious to an
instructorandrelativelyeasytoconvinceastu-
Examples
dentthattherearetwowaystosolveaproblem:
the easy way and the hard way. We choose the
Throughoutthetext,numerousexamplesare
hard way when there is instructional value to
given, with thorough explanations and a sub-
stantialamountofdetail.Solutionstomoredif-
the approach. The icon is intended to alert
ficultexamplesareconstructedwiththehelpof
students that technology is intelligently (and
graphingcalculatorsoracomputeralgebrasys-
wisely) used to assist in solving the problem.
temandareindicatedbyanicon.
Typically,thetechnologywehaveusedisacom-
puteralgebrasystem,likeMathematicaorMaple.
“Think about it!” Technologyisusedthroughoutthetexttoex-
plore many of the applications and more diffi-
Many examples are followed by a question
cultexamples,especiallythosemarkedwith
indicatedbya icon.Generally,basicknowl- and the problems in the subsections Differential
edge about the behavior of functions is suffi- EquationsatWork.
cient to answer the question. Many of these Answers to most odd exercises are included
questionsencouragestudentstousetechnology. at the end of the text. More complete answers,
Others, focus on the graph of a solution. Thus, solutions, partial solutions, or hints to selected
“Thinkaboutit!”questionshelpstudentsdeter- exercises are available separately to students
minewhentousetechnologyandmakethistext and instructors. Differential Equations at Work
moreinteractive. subsections describe detailed economics, biol-
x
PREFACE
ogy, physics, chemistry, and engineering prob- basic techniques. Most sections also contain in-
lemsdocumentedbyreferences.Theseproblems terestingmathematicalandappliedproblemsto
include real data when available and require show that mathematics and its applications are
students to provide answers based on different both interesting and relevant. Instructors will
conditions. Students must analyze the problem findthattheycanassignalargenumberofprob-
andmakedecisionsaboutthebestwaytosolve lems,ifdesiredyetstillhaveplentyforreviewin
it, including the appropriate use of technology additiontothosefoundinthereviewsectionat
EachDifferentialEquationsatWorkprojectcanbe the end of each chapter. Answers to most odd-
assignedasaprojectrequiringawrittenreport, numbered exercises are included at the end of
forgroupwork,orfordiscussioninclass. the text; detailed solutions to selected exercises
Differential Equations at Work also illustrate areincludedintheStudentResourceManual.
how differential equations are used in the real
world. Students are often reluctant to believe
Chapter Summary and Review Exercises
that the subject matter in calculus, linear alge-
bra, and differential equations classes relates to
Each chapter ends with a chapter summary
subsequent courses and to their careers. Each
highlightingimportantconcepts,keytermsand
Differential Equations at Work subsection illus-
formulas, and theorems. The Review Exercises
trateshowthematerialdiscussedinthecourseis
followingthechaptersummaryofeachchapter
usedinreallife.WekeepeachDifferentialEqua-
offerstudentsextrapracticeonthetopicsinthat
tions at Work subsection short because nearly
chapter.Theseexercisesarearrangedbysection
allinstructorshaveenoughtroublecoveringthe
sothatstudentshavingdifficultycanturntothe
content expected of them. On the other hand,
appropriatematerialforreview.
when a student asks the question “When am I
going to use this?” or “How am I going to use
this?” these short subsections can give the in- Figures
structorideasastohowtohandlethequestion.
This text provides an abundance of figures
TheproblemsinDifferentialEquationsatWork
andgraphs,especiallyforsolutionstoexamples.
are not connected to a specific section of the
Inaddition,studentsareencouragedtodevelop
text; they require students to draw different
spatial visualization and reasoning skills, to in-
mathematical skills and concepts together to
terpretgraphs,andtodiscoverandexplorecon-
solve a problem. BecauseeachDifferential Equa-
tions at Work is cumulative in nature, students cepts from a graphical point of view. To ensure
must combine mathematical concepts, tech- accuracy,thefiguresandgraphshavebeencom-
niques,andexperiencesfrompreviouschapters pletely computer-generated. We hope you like
andmathcourses. theimprovementinthegraphicsinfifthedition
ofthetext.
Exercises
Historical Material
Numerous exercises, ranging in level from
easy to difficult, are included in each section of Nearly every topic is motivated by either an
thetext.Inparticular,theexercisesetsfortopics applicationoranappropriatehistoricalnote.We
thatstudentsfindmostdifficultarerichandvar- have also included images of paintings, draw-
ied.Forthefourtheditionofthetext,theabun- ings, or photographs of the many famous sci-
dant “routine” exercises have been completely entistsanddescriptionsofthemathematicsthey
revisedandtrytoencouragestudentstomaster discovered.
xi
CONTENT
CONTENT duced,whichincludesadiscussionofsev-
eral special equations and the properties
The highlights of each chapter are described of their solutions/equations important in
brieflybelow. many areas of applied mathematics and
Chapter1 After introducing preliminary def- physics.
initions, we discuss direction fields not Chapter5 Several applications of higher or-
only for first-order differential equations, der equations are presented. The distinc-
but also for systems of equations. In this tive presentation illustrates the motion
presentation, we establish a basic under- of spring-mass systems and pendulums
standingofsolutionsandtheirgraphs.We graphically to help students understand
give an overview of some of the applica- what solutions represent and to make the
tions covered later in the text to point out applicationsmoremeaningfultothem.
theusefulnessofthetopicandsomeofthe Chapter6 The study of systems of differential
reasons we have for studying differential equations is perhaps the most exciting of
equationsintheexercises. allthetopicscoveredinthetext.Although
Chapter2 In addition to discussing the stan- we direct most of our attention to solv-
dard techniques for solving several types ing systems of linear first-order equations
of first-order differential equations (sepa- with constant coefficients, technology al-
rable equations, homogeneous equations, lows us to investigate systems of nonlin-
exactequations,andlinearequations),we ear equations and observe phase planes.
introduceseveralnumericalmethods(Eu- Wealsoshowhowtouseeigenvaluesand
ler’s Method, Improved Euler’s Method, eigenvectorstounderstandthegeneralbe-
Runge-Kutta Method) and discuss the ex- havior of systems of linear and nonlinear
istenceanduniquenessofsolutionstofirst equations. We have added a section on
order initial-value problems. Throughout phaseportraitsinthisedition.
the chapter, we encourage students to Chapter7 Several applications discussed ear-
buildanintuitiveapproachtothesolution lier in the text are extended to more than
processbymatchingagraphtoasolution one dimension and solved using systems
withoutactuallysolvingtheequation. ofdifferentialequations,inanefforttore-
Chapter3 Not only do we cover most stan- inforce the understanding of these impor-
dard applications of first order equations tant problems.Numerous applications in-
inChapter3(orthogonaltrajectories,pop- volving nonlinear systems are discussed
ulation growth and decay, Newton’s law aswell.
of cooling, free-falling bodies), but we Chapter8 Laplacetransformsareimportantin
also present many that are not (due to many areasofengineering andexhibit in-
their computational difficulty) in Differen- triguing mathematical properties as well.
tialEquationsatWork. Throughout the chapter, we point out the
Chapter4 This chapter emphasizes the meth- importance of initial conditions and forc-
odsforsolvinghomogeneousandnonho- ingfunctionsoninitial-valueproblems.
mogeneoushigherorderdifferentialequa- For a one semester course introducing or-
tions.ItalsostressesthePrincipleofSuper- dinary differential equations, many instructors
position and the differences between the willchoosetocovertopicsfromChapters1to7
properties of solutions to linear and non- orfromChapters1to6andChapter8.Foratwo
linearequations.AfterdiscussingCauchy- semestercourse,theinstructorwilleasilybeable
Euler equations, series methods are intro- to cover the remaining chapters of the text. In
