Differential Equations: An Introduction to Modern Methods and Applications

686 Pages·2015·17.9 MB·English

by  James R. Brannan

THIRD EDITION
James R. Brannan | William E. Boyce
D I F F E R E N T I A L E Q U A T I O N S
An Introduction to Modern Methods and Applications

DIFFERENTIAL EQUATIONS
An Introduction to Modern Methods and Applications
THIRD EDITION
James R. Brannan
ClemsonUniversity
William E. Boyce
RensselaerPolytechnicInstitute
withcontributionsby
Mark A. McKibben
WestChesterUniversity
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P R E F A C E
This is a textbook for a first course in differential equations. The book is intended for
science and engineering majors who have completed the calculus sequence, but not nec-
essarily a first course in linear algebra. It emphasizes a systems approach to the subject
and integrates the use of modern computing technology in the context of contemporary
applicationsfromengineeringandscience.
Our goal in writing this text is to provide these students with both an introduction to,
andasurveyof,modernmethods,applications,andtheoryofdifferentialequationsthatis
likelytoservethemwellintheirchosenfieldofstudy.Thesubjectmatterispresentedina
mannerconsistentwiththewaypractitionersusedifferentialequationsintheirwork;tech-
nologyisusedfreely,withmoreemphasisonmethods,modeling,graphicalrepresentation,
qualitativeconcepts,andgeometricintuitionthanontheory.
Notable Changes in the Third Edition
This edition is a substantial revision of the second edition. The most significant changes
are:
▶ EnhancedPageLayout Wehaveplacedimportantresults,theorems,definitions,and
tables in highlighted boxes and have put subheadings just before the most important
topicsineachsection.Thisshouldenhancereadabilityforbothstudentsandinstructors
andhelpstudentstoreviewmaterialforexams.
▶ IncreasedEmphasisonQualitativeMethods Qualitativemethodsareintroducedearly.
Throughoutthetext,newexamplesandproblemshavebeenaddedthatrequirethestu-
denttousequalitativemethodstoanalyzesolutionbehavioranddependenceofsolutions
onparameters.
▶ NewChapteronNumericalMethods Discussions on numerical methods, dispersed
overthreechaptersinthesecondedition,havebeenrevisedandreassembledasaunit
inChapter8.However,thefirstthreesectionsofChapter8canbestudiedbystudents
aftertheyhavestudiedChapter1andthefirsttwosectionsofChapter2.
▶ Chapter1:Introduction This chapter has been reduced to three sections. In Section
1.1wefollowuponintroductorymodelsandconceptswithadiscussionoftheartand
craftofmathematicalmodeling.Section1.2hasbeenreplacedbyanearlyintroduction
toqualitativemethods,inparticular,phaselinesanddirectionfields.Linearizationand
stabilitypropertiesofequilibriumsolutionsarealsodiscussed.InSection1.3wecover
definitions, classification, and terminology to help give the student an organizational
overviewofthesubjectofdifferentialequations.
▶ Chapter2:FirstOrderDifferentialEquations Newmathematicalmodelingproblems
havebeenaddedtoSection2.3,andanewSection2.7onsubsitutionmethodshasbeen
added.SectionsonnumericalmethodshavebeenmovedtoChapter8.
▶ Chapter3:SystemsofTwoFirstOrderEquations ThediscussionofWronskiansand
fundamental sets of solutions has been supplemented with the definition of, and rela-
tionshipto,linearlyindependentsolutionsoftwo-dimensionallinearsystems.
▶ Chapter4:SecondOrderLinearEquations Section4.6onforcedvibrations,frequency
response,andresonance has been rewrittentoimprove itsreadabilityforstudents and
instructors.
v
vi Preface
▶ Chapter 10: Orthogonal Functions, Fourier Series and Boundary-Value Problems
ThischaptergivesaunifiedtreatmentofclassicalandgeneralizedFourierseriesinthe
frameworkoforthogonalfamiliesin thespacePC[a,b].
▶ Chapter11:ElementaryPartialDifferentialEquations Material and projects on the
heat equation,waveequation,andLaplace’sequationthatappearedinChapters9and
10ofthesecondedition,havebeenmovedtoChapter11inthethirdedition.
▶ MiscellaneousChangesandAdditions Changeshavebeenmadeincurrentproblems,
and new problems have been added to many of the section problem sets. For ease in
assigninghomework,boldfaceheadingshavebeenaddedtopartitiontheproblemsinto
groupscorrespondingtomajortopicsdiscussedinthesection.
Major Features
▶ FlexibleOrganization. Chaptersarearranged,andsectionsandprojectsarestructured,
to facilitate choosing from a variety of possible course configurations depending on
desiredcoursegoals,topics,anddepthofcoverage.
▶ NumerousandVariedProblems. Throughoutthetext,sectionexercisesofvaryinglev-
elsofdifficultygivestudentshands-onexperienceinmodeling,analysis,andcomputer
experimentation.
▶ EmphasisonSystems. Systemsoffirstorderequations,acentralandunifyingtheme
ofthetext,areintroducedearly,inChapter3,andareusedfrequentlythereafter.
▶ LinearAlgebraandMatrixMethods. Two-dimensionallinearalgebrasufficientforthe
study of two first order equations, taken up in Chapter 3, is presented in Section 3.1.
Linearalgebraandmatrixmethodsrequiredforthestudyoflinearsystemsofdimension
n(Chapter6)aretreatedinAppendixA.
▶ Optional Computing Exercises. In most cases, problems requesting computer-
generatedsolutionsandgraphicsareoptional.
▶ VisualElements. Thetextcontainsalargenumberofillustrationsandgraphs.Inaddi-
tion,manyoftheproblemsaskthestudenttocomputeandplotsolutionsofdifferential
equations.
▶ ContemporaryProjectApplications. Optional projects at the end of all but one of
Chapters2through11integratesubjectmatterinthecontextofexciting,oftencontem-
porary,applicationsinscienceandengineering.
▶ LaplaceTransforms. AdetailedchapteronLaplacetransformsdiscussessystems,dis-
continuousandimpulsiveinputfunctions,transferfunctions,feedbackcontrolsystems,
poles,andstability.
▶ ControlTheory. Ideasandmethodsfromtheimportantapplicationareaofcontrolthe-
oryareintroducedinsomeexamples,someprojects,andinthelastsectiononLaplace
transforms.Allthismaterialisoptional.
▶ RecurringThemesandApplications. Important themes, methods, and applications,
suchasdynamicalsystemformulation,phaseportraits,linearization,stabilityofequilib-
riumsolutions,vibratingsystems,andfrequencyresponse,arerevisitedandreexamined
inavarietyofmathematicalmodelsunderdifferentmathematicalsettings.
▶ ChapterSummaries. A summary at the end of each chapter provides students and
instructorswithabird’s-eyeviewofthemostimportantideasinthechapter.
▶ AnswerstoProblems. Answers to selected odd-numbered problems are provided at
theendofthebook;manyofthemareaccompaniedbyafigure.
Problemsthatrequiretheuseofacomputeraremarkedwith .Whilewefeelthatstudents
willbenefitfromusingthecomputeronthoseproblemswherenumericalapproximations
Preface vii
or computer-generated graphics are requested, in most problems it is clear that use of a
computer,orevenagraphingcalculator,isoptional.Furthermoretherearealargenumber
ofproblemsthatdonotrequiretheuseofacomputer.Thusthebookcaneasilybeusedin
acoursewithoutusinganytechnology.
Relation of This Text to Boyce and DiPrima
BrannanandBoyceisanoffshootofthewell-knowntextbookbyBoyceandDiPrima.Read-
ersfamiliarwithBoyceandDiPrimawilldoubtlessrecognizeinthepresentbooksomeof
thehallmarkfeaturesthatdistinguishthattextbook.
Tohelpavoidconfusionamongpotentialusersofeithertext,theprimarydifferencesare
describedbelow:
▶ Brannan and Boyce is more sharply focused on the needs of students of engineering
and science, whereas Boyce and DiPrima targets a somewhat more general audience,
includingengineersandscientists.
▶ BrannanandBoyceisintendedtobemoreconsistentwiththewaycontemporaryscien-
tistsandengineersactuallyusedifferentialequationsintheworkplace.
▶ BrannanandBoyceemphasizessystemsoffirstorderequations,introducingthemear-
lier, and also examining them in more detail than Boyce and DiPrima. Brannan and
Boycehasanextensiveappendixonmatrixalgebratosupportthetreatmentofsystems
inndimensions.
▶ Brannan and Boyce integrates the use of computers more thoroughly than Boyce and
DiPrima, and assumes that most students will use computers to generate approximate
solutionsandgraphsthroughoutthebook.
▶ Brannan and Boyce emphasizes contemporary applications to a greater extent than
BoyceandDiPrima,primarilythroughend-of-chapterprojects.
▶ BrannanandBoycemakessomewhatmoreuseofgraphs,withmoreemphasisonphase
planedisplays,andusesengineeringlanguage(e.g.,statevariables,transferfunctions,
gainfunctions,andpoles)toagreaterextentthanBoyceandDiPrima.
Options for Course Structure
Chapterdependenciesareshowninthefollowingblockdiagram:
Chapter 6
Chapter 5
Appendix A Systems of
The
Matrix First Order
Laplace
Algebra Linear
Transform
Equations
Chapter 3 Chapter 4 Chapter 9
Chapter 2
Systems of Second Series
Chapter 1 First Order
Two Order Solutions of
Introduction Differential
First Order Linear Second Order
Equations
Equations Equations Equations
Chapter 7 Chapter 10 Chapter 11
Chapter 8 Nonlinear Orthogonal Elementary
Numerical Differential Functions, PDEs
Methods Equations Fourier Series,
and Stability and BVPs
viii Preface
Thebookhasmuchbuilt-inflexibilityandallowsinstructorstochoosefrommanyop-
tions. Depending on the course goals of the instructor and background of the students,
selectedsectionsmaybecoveredlightlyorevenomitted.
▶ Chapters5,6,and7areindependentofeachother,andChapters6and7arealsoinde-
pendentofChapter4.Itispossibletospendmuchclasstimeononeofthesechapters,
orclasstimecanbespreadovertwoormoreofthem.
▶ Theamountoftimedevotedtoprojectsisentirelyuptotheinstructor.
▶ For an honors class, a class consisting of students who have already had a course in
linearalgebra,oracourseinwhichlinearalgebraistobeemphasized,Chapter6may
betakenupimmediatelyfollowingChapter2.Inthiscase,materialfromAppendixA,
aswellassections,examples,andproblemsfromChapters3and4,maybeselectedas
neededordesired.ThisoffersthepossibilityofspendingmoreclasstimeonChapters5,
7,and/orselectedprojects.
Acknowledgments
Itisapleasuretoofferourgratefulappreciationtothemanypeoplewhohavegenerously
assistedinthepreparationofthisbook.
Totheindividualslistedbelowwhoreviewedpartsorallofthethirdeditionmanuscript
atvariousstagesofitsdevelopment:
Miklo´sBo´na,UniversityofFlorida
MarkW.Brittenham,UniversityofNebraska
YanzhaoCao,AuburnUniversity
DougCenzer,UniversityofFlorida
LeonardChastkofsky,UniversityofGeorgia
JonM.Collis,ColoradoSchoolofMines
DomenicoD’Alessandro,IowaStateUniversity
ShaozhongDeng,UniversityofNorthCarolinaatCharlotte
PatriciaJ.Diute,RochesterInstituteofTechnology
BehzadDjafariRouhani,UniversityofTexasatElPaso
AlinaN.Duca,NorthCarolinaStateUniversity
MarekZ.Elżanowski,PortlandStateUniversity
VincentGraziano,CaseWesternReserveUniversity
MansoorA.Haider,NorthCarolinaStateUniversity
M.D.Hendon,UniversityofGeorgia
MohamedAKhamsi,UniversityofTexasatElPaso
MarcusA.Khuri,StonyBrookUniversity
RichardC.LeBorne,TennesseeTechnologicalUniversity
GlennLedder,UniversityofNebraska-Lincoln
KristopherLee,IowaStateUniversity
JensLorenz,UniversityofNewMexico
AldoJ.Manfroi,UniversityofIllinois
MarcusMcGuff,AustinCommunityCollege
Preface ix
WilliamF.Moss,ClemsonUniversity
MikeNicholas,ColoradoSchoolofMines
MohamedAitNouh,UniversityofTexasatElPaso
FrancisJ.Poulin,UniversityofWaterloo
MaryJarrattSmith,BoiseStateUniversity
StephenJ.Summers,UniversityofFlorida
YiSun,UniversityofSouthCarolinaKyleThompson,NorthCarolinaStateUniversity
StellaThistlethwaite,UniversityofTennessee,Knoxville
VincentVatter,UniversityofFlorida
PuPatrickWang,UniversityofAlabama
DongmingWei,NazarbayevUniversity
LarissaWilliamson,UniversityofFlorida
HafizahYahya,UniversityofAlberta
KonstantinZuev,UniversityofSouthernCalifornia.
To Mark McKibben, West Chester University; Doug Meade, University of South
Carolina; Bill Siegmann, Rensselaer Polytechnic Institute, for their contributions to the
revision.
ToJenniferBlue,SUNYEmpireStateCollege,foraccuracycheckingpageproofs.
TotheeditorialandproductionstaffofJohnWileyandSons,Inc.,identifiedonpageiv,
whowereresponsibleforturningourmanuscriptintoafinishedbook.Intheprocess,they
maintainedthehigheststandardsofprofessionalism.
Wealsowishtoacknowledgethelesstangiblecontributionsofourfriendandcolleague,
thelateRichardDiPrima.Partsofthisbookdrawextensivelyonthebookondifferential
equationsbyBoyceandDiPrima.SinceDickDiPrimawasanequalpartnerincreatingthe
earlyeditionsofthatbook,hisinfluencelivesonmore thanthirtyyearsafterhisuntimely
death.
Finally,andmostimportantofall,wethankourwivesCherylandElsafortheirunder-
standing,encouragement,andpatiencethroughoutthewritingandproductionofthisbook.
Withouttheirsupportitwouldhavebeenmuchmoredifficult,ifnotimpossible,forusto
completethisproject.
JamesR.Brannan
Clemson,SouthCarolina
WilliamE.Boyce
Latham,NewYork
x Preface
Supplemental Resources for Instructors and Students
AnInstructor’sSolutionsManual,includessolutionsforallproblemsinthetext.
A Student Solutions Manual, ISBN 9781118981252, includes solutions for selected
problemsinthetext.
ACompanionwebsite,www.wiley.com/college/brannan,providesawealthofresources
forstudentsandinstructors,including:
▶ PowerPointslidesofimportantideasandgraphicsforstudyandnotetaking.
▶ Online Only Projects—these projects are like the end-of-chapter projects in the text.
They present contemporary problems that are not usually included among traditional
differentialequationstopics.Manyoftheprojectsinvolveapplicationsderivedfroma
variety of disciplines and integrate or extend theories and methods presented in core
material.
▶ Mathematica,Maple,andMATLABdatafilesareprovidedforselectedend-of-sectionor
end-of-chapterproblemsinthetextallowingforfurtherexplorationofimportantideas
inthecourseutilizingthesecomputeralgebraandnumericalanalysispackages.Students
willbenefitfromusingthecomputeronproblemswherenumericalapproximationsor
computergeneratedgraphicsarerequested.
▶ ReviewofIntegration—Anonlinereviewofintegrationtechniquesisprovidedforstu-
dentswhoneedarefresher.
WileyPLUS: Expect More from Your Classroom Technology
ThistextissupportedbyWileyPLUS—apowerfulandhighlyintegratedsuiteofteaching
andlearningresourcesdesignedtobridgethegapbetweenwhathappensintheclassroom
and what happens at home. WileyPLUS includes a complete online version of the text,
algorithmicallygeneratedexercises,allofthetextsupplements,pluscourseandhomework
managementtools,inoneeasy-to-usewebsite.
Organizedaroundtheeverydayactivitiesyouperforminclass,WileyPLUShelpsyou:
▶ PrepareandPresent: WileyPLUS letsyoucreateclasspresentationsquicklyandeasily
usingawealthofWiley-providedresources,includinganonlineversionofthetextbook,
PowerPointslides,andmore.Youcanadaptthiscontenttomeettheneedsofyourcourse.
▶ CreateAssignments: WileyPLUSenablesyoutoautomatetheprocessofassigningand
gradinghomeworkorquizzes.
▶ TrackStudentProgress: Aninstructor’sgradebookallowsyoutoanalyzeindividual
andoverallclassresultstodeterminestudents’progressandlevelofunderstanding.
▶ PromoteStrongProblem-SolvingSkills: WileyPLUScanlinkhomeworkproblemsto
the relevant section of the online text, providing students with context-sensitive help.
WileyPLUS also features mastery problems that promote conceptual understanding of
keytopicsandvideowalkthroughsofexampleproblems.
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videunlimitedself-practiceopportunitiesforstudents,aswellasproblemsforhomework
andtesting.
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