Solution Manual for Elementary Differential Equations with Boundary Value Problems (Classic Version), 6th Edition

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SOLUTIONSMANUALElementaryDIFFERENTIALEQUATIONSWithBOUNDARYVALUEPROBLEMSSixthEditionC.HenryEdwards¢DavidE.Penney‘WiththeassistanceofDavidCalvis

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DownloadedfromStudyXY.com®+StudyXYSdYe.o>\|iF’prE\3SStudyAnythingThisContentHasbeenPostedOnStudyXY.comassupplementarylearningmaterial.StudyXYdoesnotendroseanyuniversity,collegeorpublisher.Allmaterialspostedareundertheliabilityofthecontributors.wv8)www.studyxy.com

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!I’CONTENTS1FIRST-ORDERDIFFERENTIALEQUATIONS1.1DifferentialEquationsandMathematicalModeling11.2IntegralsasGeneralandParticularSolutions91.3SlopeFieldsandSolutionCurves181.4 SeparableEquationsandApplications271.5LinearFirst-OrderEquations441.6SubstitutionMethodsandExactEquations521.7PopulationModels651.8 Acceleration-VelocityModels78Chapter1ReviewProblems872LINEAREQUATIONSOFHIGHERORDER2.1Introduction:Second-OrderLinearEquations912.2GeneralSolutionsofLinearEquations972.3HomogeneousEquationswithConstantCoefficients1052.4MechanicalVibrations1122.5 NonhomogeneousEquationsandtheMethodofUndeterminedCoefficients1222.6ForcedOscillationsandResonance1332.7ElectricalCircuits1472.8EndpointProblemsandEigenvalues1543POWERSERIESMETHODS3.1IntroductionandReviewofPowerSeries1623.2SeriesSolutionsNearOrdinaryPoints1683.3RegularSingularPoints1813.4MethodofFrobenius:TheExceptionalCases194||-

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iIBNh3.5 Bessel'sEquation2033.6ApplicationsofBesselFunctions2114LAPLACETRANSFORMMETHODS4.1LaplaceTransformsandInverseTransforms2164.2TransformationofInitialValueProblems2214.3TranslationandPartialFractions2304.4Derivatives,Integrals,andProductsofTransforms2384.5PeriodicandPiecewiseContinuousInputFunctions2454.6 ImpulsesandDeltaFunctions2585LINEARSYSTEMSOFDIFFERENTIALEQUATIONS5.1First-OrderSystemsandApplications2675.2TheMethodofElimination27653LinearSystemsandMatrices2975.4TheEigenvalueMethodforHomogeneousLinearSystems3055.5Second-OrderSystemsandMechanicalApplications3355.6MultipleEigenvalueSolutions3485.7MatrixExponentialsandLinearSystems3625.8 NonhomogeneousLinearSystems3716NUMERICALMETHODS6.1NumericalApproximation:Euler'sMethod3806.2ACloserLookattheEulerMethod3876.3TheRunge-KuttaMethod3976.4NumericalMethodsforSystems407||-

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7NONLINEARSYSTEMSANDPHENOMENA7.1EquilibriumSolutionsandStability4157.2StabilityandthePhasePlane4287.3LinearandAlmostLinearSystems4377.4EcologicalApplications:PredatorsandCompetitors4547.5NonlinearMechanicalSystems4697.6ChaosinDynamicalSystems4818FOURIERSERIESMETHODS8.1PeriodicFunctionsandTrigonometricSeries4868.2GeneralFourierSeriesandConvergence4968.3FourierSineandCosineSeries5108.4ApplicationsofFourierSeries5248.5HeatConductionandSeparationofVariables5308.6VibratingStringsandtheOne-DimensionalWaveEquation5368.7Steady-StateTemperatureandLaplace'sEquation5439EIGENVALUESANDBOUNDARYVALUEPROBLEMS9.1Sturm-LiouvilleProblemsandEigenfunctionExpansions5549.2ApplicationsofEigenfunctionSeries5639.3SteadyPeriodicSolutionsandNaturalFrequencies5759.4CylindricalCoordinateProblems5889.5 Higher-DimensionalPhenomena596APPENDIXExistenceandUniquenessofSolutions600|||©

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CHAPTER1FIRST-ORDERDIFFERENTIALEQUATIONSSECTION1.1DIFFERENTIALEQUATIONSANDMATHEMATICALMODELSThemainpurposeofSection1.1issimplytointroducethebasicnotationandterminologyofdifferentialequations,andtoshowthestudentwhatismeantbyasolutionofadifferentialequation.Also,theuseofdifferentialequationsinthemathematicalmodelingofreal-worldphenomenaisoutlined.Problems1-12areroutineverificationsbydirectsubstitutionofthesuggestedsolutionsintothegivendifferentialequations.Weincludeherejustsometypicalexamplesofsuchverifications.3.Ify,=cos2xandy,=sin2x,theny{=-2sin2xandy;=2cos2xsoYW=—4cos2x=—4y,andy;=—4sin2x=—4y,Thusy/+4y,=0andy)+4y,=0.4.Ify,=¢”andy,=e¢™,theny,=3¢andy,=-3e™soyy=9¢"=9yandpy=9e™=9y,5.Ify=e¢*—e™,theny'=e*+e*soy-y=(er+e)~(e*-e™)=2e”*.Thusy=y+2e”.6.Ifyy=e™andy,=xe™,theny]=-2¢™,yi=4e™,y,=¢™-2xe™,andVy=—4e+4xe™.HenceVirayay,=(46)+a(-2e7)+4(e>)=0andVi+dy,+dy,=(—4e>+hxe™)+4(e™-2xe™)+4(xe™)=0.mr+stay

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8.Ify,=cosx—cos2xandy,=sinx—cos2x,theny|=-sinx+2sin2x,y/=—cosx+4cos2x,andyj,=cosx+2sin2x,yj=—sinx+4cos2x.Hence+»=(-cosx+4cos2x)+(cosx—cos2x)=3cos2xand3+,=(-sinx+4cos2x)+(sinx—cos2x)=3cos2x.11.Ify=y=x7theny'=—-2x7and3"=6x",soX*Y"+5xy'+4y=x?(6x7*)+Sx(-2x7)+4(x?)=0.Ify=y,=x"Inxtheny'=x7-2x7Inxand3"=-5x"+6x*Inx,soXY"+5xy+4y=x?(-5x+6x7Inx)+Sx(x-2x7Inx)+4(xInx)=(-5x7+552)+(6x7~10x+42)Inx=0.13.Substitutionofy=e™into3y'=2ygivestheequation3re™=2¢™thatsimplifiesto3r=2.Thusr=2/3.14.Substitutionofy=e™into4y”"=ygivestheequation4r>e™=e™thatsimplifiesto472=1.Thusr=+1/2.15.Substitutionofy=e™intoy"+3'-2y=0givestheequationr’e™+re™—2¢™=0thatsimplifiesto72+7—-2=(r+2)(*—1)=0.Thusr=-2orr=1.16.Substitutionofy=e™into3y"+3y'—4y=0givestheequation3r%e™+3re™—4e™=0thatsimplifiesto37°+3r—4=0.Thequadraticformulathengivesthesolutions»=(-3+57)6.TheverificationsofthesuggestedsolutionsinProblems17-26aresimilartothoseinProblems1-12.WeillustratethedeterminationofthevalueofConlyinsometypicalcases.However,weillustratetypicalsolutioncurvesforeachoftheseproblems.2Chapter1—StudyXY111a

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17.C=218.C=30S—r:\|\\//03)©2)\///No))J1|\/||19.Ify(x)=Ce"~1theny(0)=5givesC~1=5,soC=6.Thefigureisontheleftbelow.|/||\ss//09\BREEPe/\7|JERI.20.Ify(x)=Ce™+x~1then»(0)=10givesC~1=10,soC=11.Thefigureisontherightabove.21.C=7.Thefigureisontheleftatthetopofthenextpage.istuancr111n

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oTsLN|\\————Jo=——22.Ify(x)=In(x+C)theny(0)=0givesInC=0,soC=1.Thefigureisontherightabove.23.Ify(x)=$x°+Cx7theny(2)=1givestheequation%-32+C-%=1withsolutionC=-56.Thefigureisontheleftbelow.—~|/©5/3DN24.C=17.Thefigureisontherightabove.4Chapter1|||iv

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25.Ify(x)=tan(x”+C)then¥(0)=1givestheequationtanC=1.HenceonevalueofCisC=m/4(asisthisvalueplusanyintegralmultipleof7).ht)“1[]1226.Substitutionofx=7andy=0intoy=(x+C)cosxyieldstheequation0=@=+C)-1),s0C=—7.:LAAN27.y=x+ySection1.15|1||.

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28.Theslopeofthelinethrough(x,y)and(x/2,0)is»'=(y=0)/(x—x/2)=2y/x,sothedifferentialequationisxy’=2y.29.Ifm="istheslopeofthetangentlineandnm’istheslopeofthenormallineat(x,),thentherelationmm'=—1yieldsm'=1/y"=(y~1)/(x~0).Solutionfory'thengivesthedifferentialequation(1-y)y'=x.30.Herem=)andm'=D,(x’+k)=2x,sotheorthogonalityrelationmm'=—1givesthedifferentialequation2x)’=—1.31.Theslopeofthelinethrough(x,y)and(-y,x)isy'=(x-y)/(=y—x),sothedifferentialequationis(x+y)y'=y—x.InProblems32-36wegetthedesireddifferentialequationwhenwereplacethe"timerateofchange"ofthedependentvariablewithitsderivative,theword"is"withthe=sign,thephrase"proportionalto"with%,andfinallytranslatetheremainderofthegivensentenceintosymbols.32.dP/dt=KP33.dv/dt=kV?34.dv/dt=k(250-v)35.dN/dt=k(P-N)36.dN/dt=kN(P-N)37.Thesecondderivativeofanylinearfunctioniszero,sowespotthetwosolutionsy(x)=1orp(x)=xofthedifferentialequationy"=0.38.Afunctionwhosederivativeequalsitself,andhenceasolutionofthedifferentialequationy'=yisy(x)=€*.39. Wereasonthatify=kx’,theneachterminthedifferentialequationisamultipleofx.Thechoicek=1balancestheequation,andprovidesthesolutiony(x)=x*.40.Ifyisaconstant,then»'=0sothedifferentialequationreducestoy?=1.Thisgivesthetwoconstant-valuedsolutionsy(x)=1andy(x)=—1.6Chapter1—StudyXY|||r1

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41.Wereasonthatify=e”,theneachterminthedifferentialequationisamultipleofe*.Thechoicek=1balancestheequation,andprovidesthesolutiony(x)=Le*.42.Twofunctions,eachequalingthenegativeofitsownsecondderivative,arethetwosolutionsy(x)=cosxandy(x)=sinxofthedifferentialequationy"=-y.43.(a)Weneedonlysubstitutex(t)=1/(C—kf)inbothsidesofthedifferentialequationx'=kx?foraroutineverification.(b)Thezero-valuedfunctionx(#)=0obviouslysatisfiestheinitialvalueproblemx’=kx?,x(0)=0.44.(a)Thefigureontheleftbelowshowstypicalgraphsofsolutionsofthedifferentialequationx'=1x’.-(b)Thefigureontherightaboveshowstypicalgraphsofsolutionsofthedifferentialequationx'=—1x’.Weseethat—whereasthegraphswithk=1appearto"divergetoinfinity"—eachsolutionwithk=—1appearstoapproach0as¢—co.Indeed,weseefromtheProblem43(a)solutionx(r)=1/(C—4r)thatx(f)>was—2C.However,withk=—1itisclearfromtheresultingsolutionx(t)=1/(C+11)thatx(#)remainsboundedonanyfiniteinterval,butx(f)=>0asf—>+o.45.SubstitutionofP'=1andP=10intothedifferentialequationP'=kP?givesk=<,soProblem43(a)yieldsasolutionoftheformP(f)=1/(C~#/100).TheinitialconditionP(0)=2nowyieldsC=1,sowegetthesolutionSection1.17—StudyXY|||iH

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1100P(t)=——=——,©1_1t 50-¢2100WenowfindreadilythatP=100whent=49,andthatP=1000when¢=49.9.ItappearsthatPgrowswithoutbound(andthus"explodes")as¢approaches50.46.Substitutionofv'=-1andv=5intothedifferentialequationv'=kv?givesk==,soProblem43(a)yieldsasolutionoftheformv(f)=1/(C+¢/25).Theinitialconditionv(0)=10nowyieldsC=,sowegetthesolution1501)=—-=——.0=TLTSex1025Wenowfindreadilythatv=1when#=22.5,andthatv=0.1when¢=247.5.Itappearsthatvapproaches0as¢increaseswithoutbound.Thustheboatgraduallyslows,butnevercomestoa"fullstop"inafiniteperiodoftime.47.(a)y(10)=10yields10=1/(C-10),soC=101/10.(b)ThereisnosuchvalueofC,buttheconstantfunctiony(x)=0satisfiestheconditionsy'=»?andy(0)=0.(©)Itisobviousvisually(inFig.1.1.8ofthetext)thatoneandonlyonesolutioncurvepassesthrougheachpoint(a,b)ofthexy-plane,soitfollowsthatthereexistsauniquesolutiontotheinitialvalueproblemy'=3?,y(a)=bh.48.(b)Obviouslythefunctionsu(x)=—x*andv(x)=+x*bothsatisfythedifferentialequationxy’=4.Buttheirderivatives#'(x)=-4x"andv(x)=+4x’matchatx=0,wherebotharezero.Hencethegivenpiecewise-definedfunctiony(x)isdifferentiable,andthereforesatisfiesthedifferentialequationbecauseu(x)andv(x)doso(forx<0andx20,respectively).©Ifa>0(forinstance),chooseC,fixedsothatC,a*=b.Thenthefunctionx)=Cxtifx<0,7Cx*ifx20satisfiesthegivendifferentialequationforeveryrealnumbervalueofC._.8Chapter1—StudyXY|||[i

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SECTION1.2INTEGRALSASGENERALANDPARTICULARSOLUTIONSThissectionintroducesgeneralsolutionsandparticularsolutionsintheverysimplestsituation—adifferentialequationoftheformy'=f(x)—whereonlydirectintegrationandevaluationoftheconstantofintegrationareinvolved.Studentsshouldreviewcarefullytheelementaryconceptsofvelocityandacceleration,aswellasthefpsandmksunitsystems.1.Integrationofy'=2x+1yieldsy(x)=Jex+1dx=x*+x+C.Thensubstitutionofx=0,y=3gives3=0+0+C=C,soy(x)=x+x+3.2.Integrationofy'=(x-2)*yieldsy(x)=fox—2)%dx=1(x-2)’+C.Thensubstitutionofx=2,y=1gives1=0+C=C,soy(x)=T(x-2)°+1.3.Integrationofy'=+xyieldsy(x)=|Vxdx=2x+C.Thensubstitutionofx=4,y=0gives0=%+C,soy(x)=3(x**-8).4.Integrationofy'=x7yieldsy(x)=fx?dx=—1/x+C.Thensubstitutionofx=1,y=5gives5=-1+C,soy(x)=-1/x+6.S.Integrationofy=(x+2)™?yieldsy(x)=fx+2)?dx=2Jx+2+C.Thensubstitutionofx=2,y=-1gives—1=2-2+C,sop(x)=2/x+2-5.6.Integrationofy'=x(x*+9)"?yieldsy(x)=fre?+9)dx=L(x?+9)?+C.Thensubstitutionofx=-4,y=0gives0=15’+C,soyx)=Hx?+9)y"2~125].7.Integrationof»'=10/(x"+1)yieldsp(x)=fron?+1)dx=10tan™x+C.Thensubstitutionofx=0,y=0gives0=10-0+C,sop(x)=10tan'x.8.Integrationofy'=cos2xyieldsy(x)=feos2xax=1sin2x+C.Thensubstitutionofx=0,y=1gives1=0+C,soy(x)=1sin2x+1.9.Integrationofy'=1/v1-xyieldsy(x)=furV1-x*dx=sin"x+C.Thensubstitutionofx=0,y=0gives0=0+C,sop(x)=sin”x.Section1.29I~StudyXY|||!i

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10.Integrationofy'=xe™yieldsy(x)=[xeax=Jue*au=(u-De"=—(x+De*+C(whenwesubstitute#=—xandapplyFormula#46insidethebackcoverofthetextbook).Thensubstitutionofx=0,y=1gives1=-1+C,soy(x)=—(x+1)e+2.11.Ifa(t)=50thenv(t)=Js0a=50t+v,=50¢+10.Hencex(t)=J(50¢+10)dr=2512+101+x,=257+101+20.12.Ifa(f)=-20thenv(f)=f(=20)ar=~20f+v,=—20r-15.HenceX(t)=[(2201-15)dr=~107~15t+x,=—10£~15¢+5.13.Ifa(t)=31thenv(¢)=[3tdt=$£+v,=3°+5.Hencex(t)=[Ge+5)dr=184514x,=1+5¢.4.Ifa(n)=2r+1thenv(t)=[Qr+1)dr=+1+v,=+17.Hence©)=[(+t-Ndt=LF+11-Tr+x,=LF+11-Tt+4.15.Ifaf)=4(+3).thenv(t)=[4(+3)dr=4(+3)°+C=4(1+3)=37(takingC=-37sothatv(0)=-1).Hencex(t)=[$437-37]dr=2(+3)'-37+C=L(t+3)'~37r-26.16.Ifa()=1/i+4thenv()=[1/i+ddi=2Ji+4+C=2Ji+4-5(takingC=-5sothatv(0)=-1).HenceX(t)=[@VI+A-5)dr=$0+4y7=51+C=$(t+4)?—51-2(takingC=-29/3sothatx(0)=1).10Chapter1|||[:

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17.Ifa(t)=(t+1)7thenv(r)=f+?dt=1+)?+C=L(+)?+4(takingC=1sothatv(0)=0).Hencex0)=[[-3e+D)?+4]dr=Le+)'+41+C=a++e-1](takingC=—1sothatx(0)=0).18.Ifa(t)=50sin5tthenv(t)=[50sinstdt=—10cos5t+C=—10cos5t(takingC=0sothatv(0)=-10).HenceX(t)=[(-10cos5t)dr=~2sinS5t+C=—2sin5t+10(takingC=-10sothatx(0)=8).19.Notethatv(r)=5for0<¢<5andthatv(r)=10—¢for5<¢<10.Hencex(@)=5t+Cfor0<r<5andx(r)=10r-4r*+C,for5<¢<10.NowC,=0becausex(0)=0,andcontinuityofx()requiresthatx(f)=>5¢andx(t)=106-1¢+C,agreewhen¢=35.ThisimpliesthatC,=-%,andwegetthefollowinggraph.a>y%246810tSection1.211—StudyXY|||!I

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