Differential equations and linear algebra PDF
Preview Differential equations and linear algebra
FOURTH EDITION Differential Equations and Linear Algebra Stephen W. Goode and Scott A. Annin California State University, Fullerton Boston Columbus Indianapolis NewYork SanFrancisco Amsterdam CapeTown Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi MexicoCity SãoPaulo Sydney HongKong Seoul Singapore Taipei Tokyo EditorialDirector,Mathematics:ChristineHoag Editor-in-Chief:DeirdreLynch AcquisitionsEditor:WilliamHoffman ProjectTeamLead:ChristinaLepre ProjectManager:LaurenMorse EditorialAssistant:JenniferSnyder ProgramTeamLead:KarenWernholm ProgramManager:DanielleSimbajon CoverandIllustrationDesign:StudioMontage ProgramDesignLead:BethPaquin ProductMarketingManager:ClaireKozar ProductMarketingCoordiator:BrookeSmith FieldMarketingManager:EvanSt.Cyr SeniorAuthorSupport/TechnologySpecialist:JoeVetere SeniorProcurementSpecialist:CarolMelville InteriorDesign,ProductionManagement,AnswerArt,andComposition: iEnergizerAptara®,Ltd. CoverImage:LighttrailsonmodernbuildingbackgroundinShanghai,China–hxdyl/123RF Copyright©2017,2011,2005PearsonEducation,Inc.oritsaffiliates.AllRightsReserved.PrintedintheUnitedStatesof America.Thispublicationisprotectedbycopyright,andpermissionshouldbeobtainedfromthepublisherpriortoanyprohibited reproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying, recording,orotherwise.Forinformationregardingpermissions,requestformsandtheappropriatecontactswithinthePearson EducationGlobalRights&Permissionsdepartment,pleasevisitwww.pearsoned.com/permissions/. PEARSONandALWAYSLEARNINGareexclusivetrademarksintheU.S.and/orothercountriesownedbyPearsonEducation,Inc. oritsaffiliates. Unlessotherwiseindicatedherein,anythird-partytrademarksthatmayappearinthisworkarethepropertyoftheirrespectiveowners andanyreferencestothird-partytrademarks,logosorothertradedressarefordemonstrativeordescriptivepurposesonly.Such referencesarenotintendedtoimplyanysponsorship,endorsement,authorization,orpromotionofPearson’sproductsbytheowners ofsuchmarksoranyrelationshipbetweentheownerandPearsonEducation,Inc.oritsaffiliates,authors,licenseesordistributors. LibraryofCongressCataloging-in-PublicationData Goode,StephenW., Differentialequationsandlinearalgebra/StephenW.GoodeandScottA.Annin, CaliforniaStateUniversity,Fullerton.—4thedition. pagescm Includesindex ISBN978-0-321-96467-0—ISBN0-321-96467-5 1. Differentialequations. 2. Algebras,Linear. I. Annin,Scott. II.Title. QA371.G644 2015 515’.35—dc23 2014006015 12345678910—V031—1918171615 ISBN10:0-321-96467-5 www.pearsonhighered.com ISBN13:978-0-321-96467-0 Contents Preface vii 1 First-Order Differential Equations 1 1.1 DifferentialEquationsEverywhere 1 1.2 BasicIdeasandTerminology 13 1.3 TheGeometryofFirst-OrderDifferentialEquations 23 1.4 SeparableDifferentialEquations 34 1.5 SomeSimplePopulationModels 45 1.6 First-OrderLinearDifferentialEquations 53 1.7 ModelingProblemsUsingFirst-OrderLinear DifferentialEquations 61 1.8 ChangeofVariables 71 1.9 ExactDifferentialEquations 82 1.10 NumericalSolutiontoFirst-OrderDifferential Equations 93 1.11 SomeHigher-OrderDifferentialEquations 101 1.12 ChapterReview 106 2 Matrices and Systems of Linear Equations 114 2.1 Matrices:DefinitionsandNotation 115 2.2 MatrixAlgebra 122 2.3 TerminologyforSystemsofLinearEquations 138 2.4 Row-EchelonMatricesandElementaryRow Operations 146 2.5 GaussianElimination 156 2.6 TheInverseofaSquareMatrix 168 2.7 ElementaryMatricesandtheLUFactorization 179 2.8 TheInvertibleMatrixTheoremI 188 2.9 ChapterReview 190 3 Determinants 196 3.1 TheDefinitionoftheDeterminant 196 3.2 PropertiesofDeterminants 209 3.3 CofactorExpansions 222 3.4 SummaryofDeterminants 235 3.5 ChapterReview 242 iii iv Contents 4 Vector Spaces 246 4.1 VectorsinRn 248 4.2 DefinitionofaVectorSpace 252 4.3 Subspaces 263 4.4 SpanningSets 274 4.5 LinearDependenceandLinearIndependence 284 4.6 BasesandDimension 298 4.7 ChangeofBasis 311 4.8 RowSpaceandColumnSpace 319 4.9 TheRank-NullityTheorem 325 4.10 InvertibleMatrixTheoremII 331 4.11 ChapterReview 332 5 Inner Product Spaces 339 5.1 DefinitionofanInnerProductSpace 340 5.2 OrthogonalSetsofVectorsandOrthogonal Projections 352 5.3 TheGram-SchmidtProcess 362 5.4 LeastSquaresApproximation 366 5.5 ChapterReview 376 6 Linear Transformations 379 6.1 DefinitionofaLinearTransformation 380 6.2 TransformationsofR2 391 6.3 TheKernelandRangeofaLinearTransformation 397 6.4 AdditionalPropertiesofLinearTransformations 407 6.5 TheMatrixofaLinearTransformation 419 6.6 ChapterReview 428 7 Eigenvalues and Eigenvectors 433 7.1 TheEigenvalue/EigenvectorProblem 434 7.2 GeneralResultsforEigenvaluesandEigenvectors 446 7.3 Diagonalization 454 7.4 AnIntroductiontotheMatrixExponentialFunction 462 7.5 OrthogonalDiagonalizationandQuadraticForms 466 7.6 JordanCanonicalForms 475 7.7 ChapterReview 488 8 Linear Differential Equations of Order n 493 8.1 GeneralTheoryforLinearDifferentialEquations 495 8.2 ConstantCoefficientHomogeneousLinear DifferentialEquations 505 8.3 TheMethodofUndeterminedCoefficients: Annihilators 515 8.4 Complex-ValuedTrialSolutions 526 8.5 OscillationsofaMechanicalSystem 529 Contents v 8.6 RLCCircuits 542 8.7 TheVariationofParametersMethod 547 8.8 ADifferentialEquationwithNonconstantCoefficients 557 8.9 ReductionofOrder 568 8.10 ChapterReview 573 9 Systems of Differential Equations 580 9.1 First-OrderLinearSystems 582 9.2 VectorFormulation 588 9.3 GeneralResultsforFirst-OrderLinearDifferential Systems 593 9.4 VectorDifferentialEquations:Nondefective CoefficientMatrix 599 9.5 VectorDifferentialEquations:DefectiveCoefficient Matrix 608 9.6 Variation-of-ParametersforLinearSystems 620 9.7 SomeApplicationsofLinearSystemsofDifferential Equations 625 9.8 MatrixExponentialFunctionandSystemsof DifferentialEquations 635 9.9 ThePhasePlaneforLinearAutonomousSystems 643 9.10 NonlinearSystems 655 9.11 ChapterReview 663 10 The Laplace Transform and Some Elementary Applications 670 10.1 DefinitionoftheLaplaceTransform 670 10.2 TheExistenceoftheLaplaceTransformandthe InverseTransform 676 10.3 PeriodicFunctionsandtheLaplaceTransform 682 10.4 TheTransformofDerivativesandSolutionof Initial-ValueProblems 685 10.5 TheFirstShiftingTheorem 690 10.6 TheUnitStepFunction 695 10.7 TheSecondShiftingTheorem 699 10.8 ImpulsiveDrivingTerms:TheDiracDeltaFunction 706 10.9 TheConvolutionIntegral 711 10.10 ChapterReview 717 11 Series Solutions to Linear Differential Equations 722 11.1 AReviewofPowerSeries 723 11.2 SeriesSolutionsaboutanOrdinaryPoint 731 11.3 TheLegendreEquation 741 11.4 SeriesSolutionsaboutaRegularSingularPoint 750 11.5 FrobeniusTheory 759 11.6 Bessel’sEquationofOrderp 773 11.7 ChapterReview 785 vi Contents A Review of Complex Numbers 791 B Review of Partial Fractions 797 C Review of Integration Techniques 804 D Linearly Independent Solutions to x2y xp(x)y q(x)y 0 811 ′′ ′ + + = Answers to Odd-Numbered Exercises 814 Index 849 S. W. Goode dedicates this book to Megan and Tobi S. A. Annin dedicates this book to Arthur and Juliann, the best parents anyone could ask for Preface Like the first three editions of Differential Equations and Linear Algebra, this fourth editionisintendedforasophomorelevelcoursethatcoversmaterialinbothdifferential equationsandlinearalgebra.Inwritingthistextwehaveendeavoredtodevelopthestu- dent’sappreciationforthepowerofthegeneralvectorspaceframeworkinformulating andsolvinglinearproblems.Thematerialisaccessibletoscienceandengineeringstu- dentswhohavecompletedthreesemestersofcalculusandwhobringthematurityofthat successwiththemtothiscourse.Thistextiswrittenaswewouldnaturallyteach,blend- ing an abundance of examples and illustrations, but not at the expense of a deliberate andrigoroustreatment.Mostresultsareprovenindetail.However,manyofthesecanbe skippedinfavorofamoreproblem-solvingorientedapproachdependingonthereader’s objectives. Some readers may like to incorporate some form of technology (computer algebrasystem(CAS)orgraphingcalculator)andthereareseveralinstancesinthetext wherethepoweroftechnologyisillustratedusingtheCASMaple.Furthermore,many exercise sets have problems that require some form of technology for their solution. Theseproblemsaredesignatedwitha . ⋄ In developing the fourth edition we have once more kept maximum flexibility of the material in mind. In so doing, the text can effectively accommodate the different emphases that can be placed in a combined differential equations and linear algebra course,thevaryingbackgroundsofstudentswhoenrollinthistypeofcourse,andthe factthatdifferentinstitutionshavedifferentcreditvaluesforsuchacourse.Thewhole text can be covered in a five credit-hour course. For courses with a lower credit-hour value,someselectivitywillhavetobeexercised.Forexample,much(orall)ofChapter 1maybeomittedsincemoststudentswillhaveseenmanyofthesedifferentialequations topics in an earlier calculus course, and the remainder of the text does not depend on the techniques introduced in this chapter. Alternatively, while one of the major goals of the text is to interweave the material on differential equations with the tools from linearalgebrainasymbioticrelationshipasmuchaspossible,thecorematerialonlinear algebraisgiveninChapters2–7sothatitispossibletousethisbookforacoursethat focuses solely on the linear algebra presented in these six chapters. The material on differential equations is contained primarily in Chapters 1 and 8–11, and readers who havealreadytakenafirstcourseinlinearalgebracanchoosetoproceeddirectlytothese chapters. Thereareothermeansofeliminatingsectionstoreducetheamountofmaterialto be covered in a course. Section 2.7 contains material that is not required elsewhere in thetext,Chapter3canbecondensedtoasinglesection(Section3.4)forreadersneeding onlyacursoryoverviewofdeterminants,andSections4.7,5.4,andthelatersectionsof Chapters6and7couldallbereservedforasecondcourseinlinearalgebra.InChapter8, Sections8.4,8.8,and8.9canbeomitted,and,dependingonthegoalsofthecourse,Sec- tions8.5and8.6couldeitherbede-emphasizedoromittedcompletely.Similarremarks apply to Sections 9.7–9.10. At California State University, Fullerton we have a four credit-hourcourseforsophomoresthatisbasedaroundthematerialinChapters1–9. vii viii Preface Major Changes in the Fourth Edition Severalsectionsofthetexthavebeenmodifiedtoimprovetheclarityofthepresentation andtoprovidenewexamplesthatreflectinsightfulillustrationswehaveusedinourown courses at California State University, Fullerton. Other significant changes within the textarelistedbelow. 1. Thechapteronvectorspacesinthepreviouseditionhasbeensplitintotwochapters (Chapters 4and5)inthepresent edition, inorder tofocus separateattentionon vectorspacesandinnerproductspaces.Theshorterlengthofthesetwochapters isalsointendedtomakeeachofthemlessdaunting. 2. Thechapteroninnerproductspaces(Chapter5)includesanewsectionproviding anapplicationoflinearalgebratothesubjectofleastsquaresapproximation. 3. The chapter on linear transformations in the previous edition has been split into two chapters (Chapters 6 and 7) in the present edition. Chapter 6 is focused on linear transformations, while Chapter 7 places direct emphasis on the theory of eigenvaluesandeigenvectors.Oncemore,readersshouldfindtheshorterchapters coveringthesetopicsmoreapproachableandfocused. 4. Mostexercisesetshavebeenenlargedorrearranged.Over3,000problemsarenow containedwithinthetext,andmorethan600concept-orientedtrue/falseitemsare alsoincludedinthetext. 5. Every chapter of the book includes one or more optional projects that allow for morein-depthstudyandapplicationofthetopicsfoundinthetext. 6. ThebackofthebooknowincludestheanswertoeveryTrue-FalseReviewitem containedinthetext. Acknowledgments We would like to acknowledge the thoughtful input from the following reviewers of thefourthedition:JameyBassofCityCollegeofSanFrancisco,TamarFriedmannof UniversityofRochester,andLinghaiZhangofLehighUniversity. Alloftheircommentswereconsideredcarefullyinthepreparationofthetext. S.A.Annin:Ioncemorethankmyparents,ArthurandJuliannAnnin,fortheirlove andencouragementinallofmyprofessionalendeavors.Ialsogratefullyacknowledge themanystudentswhohavetakenthiscoursewithmeovertheyearsand,insodoing, haveenhancedmyloveforthesetopicsanddeeplyenrichedmycareerasaprofessor. 1 First-Order Differential Equations 1.1 Differential Equations Everywhere A differential equation is any equation that involves one or more derivatives of an unknownfunction.Forexample, d2y dy x2 y2 5sinx (1.1.1) dx2 + dx + = and dS e3t(S 1) (1.1.2) dt = − are differential equations. In the differential equation (1.1.1) the unknown function or dependentvariableisy,andxistheindependentvariable;inthedifferentialequation (1.1.2) the dependent and independent variables are S and t, respectively. Differential equations such as (1.1.1) and (1.1.2) in which the unknown function depends only on asingleindependentvariablearecalledordinarydifferentialequations.Bycontrast, thedifferentialequation(Laplace’sequation) ∂2u ∂2u 0 ∂x2 + ∂y2 = involvespartialderivativesoftheunknownfunctionu(x,y)oftwoindependentvariables x and y.Suchdifferentialequationsarecalledpartialdifferentialequations. Onewayinwhichdifferentialequationscanbecharacterizedisbytheorderofthe highestderivativethatoccursinthedifferentialequation.Thisnumberiscalledtheorder ofthedifferentialequation.Thus,(1.1.1)hasordertwo,whereas(1.1.2)isafirst-order differentialequation. 1
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