Differential equations. An introduction to modern methods and applications

717 Pages·2010·8.671 MB·English

by  Brannan J., Boyce W.

This page intentionally left blank
P1:OSO/OVY P2:OSO/OVY QC:OSO/OVY T1:OSO
JWCL373-FM JWCL373-Brannan-v1 October30,2010 18:19
ordable,
active learning
WileyPLUS
is an innovative, research-based, online environment for
effective teaching and learning.
WileyPLUS
...
…motivates students with …supports instructors with
conf idence-boosting reliable resources that
feedback and proof of reinforce course goals
progress, 24/7. inside and outside of
the classroom.
Includes
Interactive
Textbook &
Resources
WileyPLUS
... Learn More.
www.wileyplus.com
i
CONFIRMINGPAGES
P1:OSO/OVY P2:OSO/OVY QC:OSO/OVY T1:OSO
JWCL373-FM JWCL373-Brannan-v1 October30,2010 18:19
ALL THE HELP, RESOURCES, AND PERSONAL SUPPORT
YOU AND YOUR STUDENTS NEED!
www.wileyplus.com/resources
2-Minute Tutorials and all Student support from an Collaborate with your colleagues,
of the resources you & your experienced student user. find a mentor, attend virtual and live
students need to get started. events, and view resources.
www.WhereFacultyConnect.com
Pre-loaded, ready-to-use Technical Support 24/7 Your WileyPLUS Account Manager.
assignments and presentations. FAQs, online chat, Personal training and
Created by subject matter experts. and phone support. implementation support.
www.wileyplus.com/support
ii
CONFIRMINGPAGES
P1:OSO/OVY P2:OSO/OVY QC:OSO/OVY T1:OSO
JWCL373-FM JWCL373-Brannan-v1 October30,2010 18:19
DIFFERENTIAL
EQUATIONS
An Introduction to
Modern Methods and
Applications
iii
CONFIRMINGPAGES
This page intentionally left blank
P1:OSO/OVY P2:OSO/OVY QC:OSO/OVY T1:OSO
JWCL373-FM JWCL373-Brannan-v1 October30,2010 18:19
DIFFERENTIAL
EQUATIONS
An Introduction to
Modern Methods and
Applications
2nd Edition
James R. Brannan
ClemsonUniversity
William E. Boyce
RensselaerPolytechnicInstitute
John Wiley & Sons, Inc.
v
CONFIRMINGPAGES
P1:OSO/OVY P2:OSO/OVY QC:OSO/OVY T1:OSO
JWCL373-FM JWCL373-Brannan-v1 October30,2010 18:19
PUBLISHER LaurieRosatone
ACQUISITIONSEDITOR DavidDietz
ASSISTANTEDITOR JeffreyBenson
SENIOREDITORIALASSISTANT PamelaLashbrook
MARKETINGMANAGER SarahDavis
CREATIVEDIRECTOR HarryNolan
COVERDESIGN DianeGerard
COVERPHOTO JorgenSchytte/Photolibrary
SENIORPRODUCTIONEDITOR SujinHong
SENIORPHOTOEDITOR MaryAnnPrice
®
Thisbookwassetin10/12TimesbyAptara ,Inc.andprintedandboundbyDonnelley/VonHoffmann.The
coverwasprintedbyDonnelley/VonHoffmann.
Thisbookisprintedonacidfreepaper.∞
Foundedin1807,JohnWiley&Sons,Inc.hasbeenavaluedsourceofknowledgeandunderstandingformore
than200years,helpingpeoplearoundtheworldmeettheirneedsandfulfilltheiraspirations.Ourcompanyis
builtonafoundationofprinciplesthatincluderesponsibilitytothecommunitiesweserveandwherewelive
andwork.In2008,welaunchedaCorporateCitizenshipInitiative,aglobalefforttoaddresstheenvironmental,
social,economic,andethicalchallengeswefaceinourbusiness.Amongtheissuesweareaddressingarecarbon
impact,paperspecificationsandprocurement,ethicalconductwithinourbusinessandamongourvendors,and
communityandcharitablesupport.Formoreinformation,pleasevisitourwebsite:
www.wiley.com/go/citizenship.
Copyright(cid:2)c 2007,2011JohnWiley&Sons,Inc.Allrightsreserved.
Nopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmittedinanyformorbyany
means,electronic,mechanical,photocopying,recording,scanningorotherwise,exceptaspermittedunder
Sections107or108ofthe1976UnitedStatesCopyrightAct,withouteitherthepriorwrittenpermissionofthe
Publisher,orauthorizationthroughpaymentoftheappropriateper-copyfeetotheCopyrightClearanceCenter,
Inc.,222RosewoodDrive,Danvers,MA01923,websitewww.copyright.com.RequeststothePublisherfor
permissionshouldbeaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,
Hoboken,NJ07030-5774,[phone],fax[phone],websitewww.wiley.com/go/permissions.
Evaluationcopiesareprovidedtoqualifiedacademicsandprofessionalsforreviewpurposesonly,foruseintheir
coursesduringthenextacademicyear.Thesecopiesarelicencedandmaynotbesoldortransferredtoathird
party.Uponcompletionofthereviewperiod,pleasereturntheevaluationcopytoWiley.Returninstructions
andafreeofchargereturnshippinglabelareavailableatwww.wiley.com/go/returnlabel.Outsideofthe
UnitedStates,pleasecontactyourlocalrepresentative.
ISBN978-0-470-45824-2
ISBN978-1-118-01187-4(BRV)
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
vi
CONFIRMINGPAGES
P1:OSO/OVY P2:OSO/OVY QC:OSO/OVY T1:OSO
JWCL373-FM JWCL373-Brannan-v1 October30,2010 18:19
P R E F A C E
Thisisatextbookforafirstcourseindifferentialequations.Thebookisintendedforscience
andengineeringmajorswhohavecompletedthecalculussequence,butnotnecessarilya
firstcourseinlinearalgebra.Itemphasizesasystemsapproachtothesubjectandintegrates
theuseofmoderncomputingtechnologyinthecontextofcontemporaryapplicationsfrom
engineeringandscience.
Differential equations is an old and venerable subject that will always play a central
role in describing phenomena that change over time. Indeed, at least one course in dif-
ferential equations is part of the curriculum of most engineering and science students.
Our goal in writing this text is to provide these students with both an introduction to,
and a survey of, modern methods, applications, and theory of this beautiful and pow-
erful mathematical apparatus that is likely to serve them well in their chosen field of
study. The subject matter is presented in a manner consistent with the way practitioners
use differential equations in their work; technology is used freely, with more emphasis
on methods, modeling, graphical representation, qualitative concepts, and geometric in-
tuition than on theoretical issues. For example, some important theoretical results, such
astheoremsguaranteeingexistenceanduniquenessofsolutionstoinitialvalueproblems,
are not proved. Nevertheless, they are carefully stated, illustrated by examples, and used
frequently.
Anystudentwhostudiesmajorportionsofthisbook,doesareasonablenumberofthe
sectionexercises,andcompletessomeofthechapterprojectswillsurelydevelopanappre-
ciationforthepowerofdifferentialequationstoshedlightonissuesofsocietalimportance.
In addition, he or she will acquire skills in modeling, analysis, and computer simulation
thatwillbeusefulinawidevarietyofsituations.Itisforsuchstudentsthatwehavewritten
thisbook.
New to This Edition
In Rene´ Descartes’ Discourse on the Method, he reflected “that there is seldom so much
perfection in works composed of many separate parts, upon which different hands had
been employed, as in those completed by a single master. Thus it is observable that the
buildingswhichasinglearchitecthasplannedandexecuted,aregenerallymoreelegantand
commodiousthanthosewhichseveralhaveattemptedtoimprove,bymakingoldwallsserve
forpurposesforwhichtheywerenotoriginallybuilt.”InthefaceofDescartes’observations,
wepleadguiltytothefactthatmanyhandshavebeenemployedinthedevelopmentofthis
textbook.Inadditiontothethreeprincipalauthors(someofthewallsbuiltbyDickDiPrima
stillexist,andwebelievestillserveadmirablythepurposeforwhichtheywereintended),the
content and manner of presentation has been greatly influenced by numerous colleagues,
teachers, reviewers and students. Furthermore, these players “stand on the shoulders of
giants”whohavecontributedtothedevelopmentandapplicationofdifferentialequations
duringthelastthreehundredyears.
Computer technology and new applications of importance to engineers and scientists
continuetomakethesubjectofdifferentialequationsamovingtarget.Inaddition,ourideas
about how to best teach the subject continue to evolve. In this second edition, we have
followed the same pedagogical principles and pragmatic forces that guided us in writing
thefirstedition.Wehaveattemptedtoimprovetheexpositioninresponsetocommentsand
opinionsofstudentsandusersofthefirstedition.Ourbookoffersanalternativechoiceto
vii
CONFIRMINGPAGES
P1:OSO/OVY P2:OSO/OVY QC:OSO/OVY T1:OSO
JWCL373-FM JWCL373-Brannan-v1 October30,2010 18:19
viii Preface
many,ifnotmost,oftheconventionalapproachestakeninthecurrentcropofdifferential
equationstextbooks.
(cid:2) We have rewritten Section 1.1 to introduce fundamental concepts and issues early on:
notions of differential equations and where they come from, mathematical models,
solutions,anddirectionfields.
(cid:2) Using the groundwork laid in Section 1.1, in Section 1.2 we present the method of
integratingfactorsforsolvingfirstorderlinearequations.
(cid:2) Section 3.2 has been rewritten to introduce constant coefficient linear systems in the
contextofameaningfulrenewableenergyapplication,thatofarockpileusedtostore
heatenergyacquiredduringdaytimetohelpkeepagreenhousewarmduringnighttime.
Thediscussionisatwo-dimensionalparalleltothediscussioninSection1.1:thereader
isintroducedtosystemsofdifferentialequations,vectornotation,solutionsofsystems,
phaseportraitsanddirectionfields.
(cid:2) InSection3.3,weusetheeigenvaluemethodtosolvefirstorderlinearconstantcoefficient
systemsforwhichtheeigenvaluesofthematrixofcoefficientsarerealanddistinct.This
sectionhasbeenrewrittentoreflectthechangesmadeinSection3.2,inparticular,those
relatedtothegreenhouse/rockbedexample.Whileonepurposeoftheeigenvaluemethod
istofindsolutionsoflinearconstantcoefficientsystems,perhapsmoreimportantisthe
additional quantitative and qualitative information the approach provides: growth and
decayratesofsolutionsandcorrespondingtransientandlongtermbehavior,principal
directionsofmotioninthephaseplane,andnaturalfrequenciesinvibrationproblems
(treatedlaterinChapter4andinChapter6).Someoftheseideasareillustratedbythe
greenhouse/rockbedprobleminwhichtheeigenvaluesofthecoefficientmatrixcanbe
associated with two distinct time scales corresponding to two separate regimes in the
coolingprocess.
(cid:2) Section4.2hasbeenrewrittentohelpclarifythecloserelationshipbetweensecondorder
linearequationsandfirstorderlinearsystemsofdimensiontwo.Thediscussionoflinear
operators,whichhasbeenslightlyextended,alsobeginsatamoreelementarylevel.The
methodofreductionoforderappearsintheproblemsetattheendofthissection;itis
clearlyanoptionaltopicfortheinstructor.
(cid:2) Section4.3nowusesresultsfromChapter3andSection4.2toconstructgeneralsolu-
tions of linear homogeneous equations with constant coefficients and their equivalent
dynamicalsystemformulations.Subsequentdiscussionfocusesonscalarequations,but
qualitative issues such as phase portraits and stability properties of equilibrium solu-
tionsarereadilyaccessible.OptionalmaterialonCauchy–Eulerequationsappearsinthe
problemsetattheendofthissection.
(cid:2) Problems have been added, modified, or replaced in accordance with changes in the
expositionofChapters1through4.
(cid:2) WehaveaddedaproblemsettotheendofAppendixB:ComplexVariables.
Major Features
(cid:2) FlexibleOrganization. Chaptersarearranged,andsectionsandprojectsarestructured,
to facilitate choosing from a variety of possible course configurations depending on
desiredcoursegoals,topics,anddepthofcoverage.
(cid:2) NumerousandVariedProblems. Throughoutthetext,sectionexercisesofvarying
levelsofdifficultygivestudentshands-onexperienceinmodeling,analysis,andcomputer
experimentation.
CONFIRMINGPAGES