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Elementary Differential Equations and Boundary Value Problems S E V E N T H E D I T I O N Elementary Differential Equations and Boundary Value Problems William E. Boyce EdwardP.HamiltonProfessorEmeritus Richard C. DiPrima formerlyElizaRickettsFoundationProfessor DepartmentofMathematicalSciences RensselaerPolytechnicInstitute John Wiley & Sons, Inc. New York Chichester Weinheim Brisbane Toronto Singapore ASSOCIATEEDITOR MaryJohenk MARKETINGMANAGER JulieZ.Lindstrom PRODUCTIONEDITOR KenSantor COVERDESIGN MichaelJung INTERIORDESIGN FearnCutterDeVicqDeCumptich ILLUSTRATIONCOORDINATOR SigmundMalinowski ThisbookwassetinTimesRomanbyEigentypeCompositors,andprintedandboundby VonHoffmannPress,Inc.ThecoverwasprintedbyPhoenixColorCorporation. (cid:1) ∞ Thisbookisprintedonacid-freepaper. Thepaperinthisbookwasmanufacturedbyamillwhoseforestmanagementprogramsincludesustained yieldharvestingofitstimberlands.Sustainedyieldharvestingprinciplesensurethatthenumbersoftrees cuteachyeardoesnotexceedtheamountofnewgrowth. Copyright(cid:2)c 2001JohnWiley&Sons,Inc.Allrightsreserved. Nopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmitted inanyformorbyanymeans,electronic,mechanical,photocopying,recording,scanning orotherwise,exceptaspermittedunderSections107or108ofthe1976UnitedStates CopyrightAct,withouteitherthepriorwrittenpermissionofthePublisher,or authorizationthroughpaymentoftheappropriateper-copyfeetotheCopyright ClearanceCenter,222RosewoodDrive,Danvers,MA01923,(508)750-8400,fax (508)750-4470.RequeststothePublisherforpermissionshouldbeaddressedtothe PermissionsDepartment,JohnWiley&Sons,Inc.,605ThirdAvenue,NewYork,NY 10158-0012,(212)850-6011,fax(212)850-6008,E-Mail:[email protected]. LibraryofCongressCataloginginPublicationData: Boyce,WilliamE. Elementarydifferentialequationsandboundaryvalueproblems/WilliamE.Boyce, RichardC.DiPrima–7thed. p.cm. Includesindex. ISBN0-471-31999-6(cloth:alk.paper) 1.Differentialequations.2.Boundaryvalueproblems.I.DiPrima,RichardC.II.Title QA371.B7732000 515’.35–dc21 00-023752 PrintedintheUnitedStatesofAmerica 10987654321 ToElsa andMaureen ToSiobhan, James,Richard, Jr.,Carolyn, andAnn And tothe nextgeneration: Charles,Aidan, Stephanie, Veronica, andDeirdre The Authors WilliamE.BoycereceivedhisB.A.degreeinMathematicsfromRhodesCollege, andhisM.S.andPh.D.degreesinMathematicsfromCarnegie-MellonUniversity.He isamemberoftheAmericanMathematicalSociety,theMathematicalAssociation ofAmerica,andtheSocietyofIndustrialandAppliedMathematics.Heiscurrently theEdwardP.HamiltonDistinguishedProfessorEmeritusofScienceEducation (DepartmentofMathematicalSciences)atRensselaer.Heistheauthorofnumerous technicalpapersinboundaryvalueproblemsandrandomdifferentialequationsand theirapplications.Heistheauthorofseveraltextbooksincludingtwodifferential equationstexts,andisthecoauthor(withM.H.Holmes,J.G.Ecker,andW.L. Siegmann)ofatextonusingMapletoexploreCalculus.Heisalsocoauthor(with R.L.BorrelliandC.S.Coleman)ofDifferentialEquationsLaboratoryWorkbook (Wiley1992),whichreceivedtheEDUCOMBestMathematicsCurricularInnovation Awardin1993.ProfessorBoycewasamemberoftheNSF-sponsoredCODEE (ConsortiumforOrdinaryDifferentialEquationsExperiments)thatledtothe widely-acclaimedODEArchitect.Hehasalsobeenactiveincurriculuminnovation andreform.Amongotherthings,hewastheinitiatorofthe“ComputersinCalculus” projectatRensselaer,partiallysupportedbytheNSF.In1991hereceivedthe WilliamH.WileyDistinguishedFacultyAwardgivenbyRensselaer. RichardC.DiPrima(deceased)receivedhisB.S.,M.S.,andPh.D.degreesin MathematicsfromCarnegie-MellonUniversity.HejoinedthefacultyofRensselaer PolytechnicInstituteafterholdingresearchpositionsatMIT,Harvard,andHughes Aircraft.HeheldtheElizaRickettsFoundationProfessorshipofMathematicsat Rensselaer,wasafellowoftheAmericanSocietyofMechanicalEngineers,the AmericanAcademyofMechanics,andtheAmericanPhysicalSociety.Hewasalso amemberoftheAmericanMathematicalSociety,theMathematicalAssociationof America,andtheSocietyofIndustrialandAppliedMathematics.Heservedasthe ChairmanoftheDepartmentofMathematicalSciencesatRensselaer,asPresidentof theSocietyofIndustrialandAppliedMathematics,andasChairmanoftheExecutive CommitteeoftheAppliedMechanicsDivisionofASME.In1980,hewastherecip- ientoftheWilliamH.WileyDistinguishedFacultyAwardgivenbyRensselaer.He receivedFulbrightfellowshipsin1964–65and1983andaGuggenheimfellowshipin 1982–83.Hewastheauthorofnumeroustechnicalpapersinhydrodynamicstability andlubricationtheoryandtwotextsondifferentialequationsandboundaryvalue problems.ProfessorDiPrimadiedonSeptember10,1984. P R E F A C E Thisedition,likeitspredecessors,iswrittenfromtheviewpointoftheappliedmathe- matician,whoseinterestindifferentialequationsmaybehighlytheoretical,intensely practical,orsomewhereinbetween.Wehavesoughttocombineasoundandaccurate (but not abstract) exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have provedusefulinawidevarietyofapplications. The book is written primarily for undergraduate students of mathematics, science, orengineering,whotypicallytakeacourseondifferentialequationsduringtheirfirst or second year of study. The main prerequisite for reading the book is a working knowledgeofcalculus,gainedfromanormaltwo-orthree-semestercoursesequence oritsequivalent. A Changing Learning Environment Theenvironmentinwhichinstructorsteach,andstudentslearn,differentialequations haschangedenormouslyinthepastfewyearsandcontinuestoevolveatarapidpace. Computingequipmentofsomekind,whetheragraphingcalculator,anotebookcom- puter,oradesktopworkstationisavailabletomoststudentsofdifferentialequations. This equipment makes it relatively easy to execute extended numerical calculations, togenerategraphicaldisplaysofaveryhighquality,and,inmanycases,tocarryout complexsymbolicmanipulations.Ahigh-speedInternetconnectionoffersanenormous rangeoffurtherpossibilities. The fact that so many students now have these capabilities enables instructors, if they wish, to modify very substantially their presentation of the subject and their expectationsofstudentperformance.Notsurprisingly,instructorshavewidelyvarying opinions as to how a course on differential equations should be taught under these circumstances.Nevertheless,atmanycollegesanduniversitiescoursesondifferential equationsarebecomingmorevisual,morequantitative,moreproject-oriented,andless formula-centeredthaninthepast. vii viii Preface Mathematical Modeling The main reason for solving many differential equations is to try to learn something aboutanunderlyingphysicalprocessthattheequationisbelievedtomodel.Itisbasic totheimportanceofdifferentialequationsthateventhesimplestequationscorrespond tousefulphysicalmodels,suchasexponentialgrowthanddecay,spring-masssystems, orelectricalcircuits.Gaininganunderstandingofacomplexnaturalprocessisusually accomplished by combining or building upon simpler and more basic models. Thus a thorough knowledge of these models, the equations that describe them, and their solutions,isthefirstandindispensablesteptowardthesolutionofmorecomplexand realisticproblems. Moredifficultproblemsoftenrequiretheuseofavarietyoftools,bothanalyticaland numerical.Webelievestronglythatpencilandpapermethodsmustbecombinedwith effectiveuseofacomputer.Quantitativeresultsandgraphs,oftenproducedbyacom- puter,servetoillustrateandclarifyconclusionsthatmaybeobscuredbycomplicated analyticalexpressions.Ontheotherhand,theimplementationofanefficientnumerical procedure typically rests on a good deal of preliminary analysis – to determine the qualitativefeaturesofthesolutionasaguidetocomputation,toinvestigatelimitingor specialcases,ortodiscoverwhichranges ofthe variablesorparametersmay require ormeritspecialattention. Thus, a student should come to realize that investigating a difficult problem may well require both analysis and computation; that good judgment may be required to determine whichtool isbest-suitedfor a particulartask;and that resultscanoften be presentedinavarietyofforms. A Flexible Approach Tobewidelyusefulatextbookmustbeadaptabletoavarietyofinstructionalstrategies. Thisimpliesatleasttwothings.First,instructorsshouldhavemaximumflexibilityto choose both the particular topics that they wish to cover and also the order in which theywanttocoverthem.Second,thebookshouldbeusefultostudentshavingaccess toawiderangeoftechnologicalcapability. With respect to content, we provide this flexibility by making sure that, so far as possible,individualchaptersareindependentofeachother.Thus,afterthebasicparts ofthefirstthreechaptersarecompleted(roughlySections1.1through1.3,2.1through 2.5,and3.1through3.6)theselectionofadditionaltopics,andtheorderanddepthin whichtheyarecovered,isatthediscretionoftheinstructor.Forexample,whilethereis agooddealofmaterialonapplicationsofvariouskinds,especiallyinChapters2,3,9, and10,mostofthismaterialappearsinseparatesections,sothataninstructorcaneasily choose which applications to include and which to omit. Alternatively, an instructor who wishes to emphasize a systems approach to differential equations can take up Chapter 7 (Linear Systems) and perhaps even Chapter 9 (Nonlinear Autonomous Systems)immediatelyafterChapter2.Or,whilewepresentthebasictheoryoflinear equations first in the context of a single second order equation (Chapter 3), many instructors have combined this material with the corresponding treatment of higher orderequations(Chapter4)oroflinearsystems(Chapter7).Manyotherchoicesand