高等数学英文版课件 15 Differential equations

Differentialequations,机动目录上页下页返回结束,15.2First-orderlinearequations,15.3Exactequations,15.4Strategyforsolvingfirst-orderequations,Chapter15,15.1Basicconcepts,separableandhomogeneousequations,Asecond-orderlineardifferentialequationhastheform(1)whereP,Q,R,andGarecontinuousfunctions.,15.5Second-OrderLinearEquations,IfG(x)=0forallx,suchequationsarecalledsecond-orderhomogeneouslinearequations.(ThisuseofthewordhomogeneoushasnothingtodowiththemeaninggiveninSection15.1.)(2),Ifforsomex,Equation1isnonhomogeneous.,(3)TheoremIfandarebothsolutionsofthelinearequation(2)andandareanyconstants,thenthefunctionisalsoasolutionofEquation2.,Thesecondtheoremsaysthatthegeneralsolutionofahomogeneouslinearequationisalinearcombinationoftwolinearlyindependentsolutions.,(4)TheoremIfandarelinearlyindependentsolutionsofEquation2,thenthegeneralsolutionisgivenbywhereandarearbitraryconstants.,Ingeneral,itisnoteasytodiscoverparticularsolutionstoasecond-orderlinearequation.ButitisalwayspossibletodosoifthecoefficientfunctionsP,QandRareconstantfunctions,thatis,ifthedifferentialequationhastheform(5),ItisnothardtothinkofsomelikelycandidatesforparticularsolutionsofEquation5.Forexample,theexponentialfunctionybecauseitsderivativesareconstantsmultipleofitself:.SubstitutetheseexpressionintoEquation5,Noticeisnever0soisasolutionofEquation5ifrisarootoftheequation(6)whichiscalledtheauxiliaryequation(orcharacteristicequation)ofEquation5.Usingthequadraticformula,therootandoftheauxiliaryequationcanbefound:(7),Wedistinguishthreecasesaccordingtothesignofthediscriminant.,Inthiscasetherootsandoftheauxiliaryequationarerealanddistinct,soandaretwolinearlyindependentsolutionsofEquation5.,(8)Iftherootsandoftheauxiliaryequationarerealandunequal,thenthegeneralsolutionofis,Example1Solvetheequation,Case1,Example2Solvetheequation,Inthiscase;thatis,therootoftheauxiliaryequationarerealandequal.Denoterasthecommonvalueofand,wehave(9),WeknowthatisonesolutionofEquation5.Wenowverifythatisalsoasolution:,Case2,(10)Iftheauxiliaryequationhasonlyonerealrootr,thenthegeneralsolutionofis,Sinceandarelinearlyindependentsolutions,Theorem4providesuswiththegeneralsolution:,Example3Solvetheequation,Inthiscasetherootsandoftheauxiliaryequationarecomplexnumbers,wecanwrite,Case3,UsingEulersequationwewritethesolutionofthedifferentialequationas,Wesummarizethediscussionasfollows:,(11)Iftherootsofauxiliaryequationarethecomplexnumbers,thenthegeneralsolutionofis,Example4Solvetheequation,Aninitial-valueproblemforthesecond-orderEquation1or2consistsoffindingasolutionyofthedifferentialequationthatalsosatisfiesinitialconditionsoftheformwhereandaregivenconstants.IfP,Q,R,andGarecontinuousonanintervalandthere,thenatheoremfoundinmoreadvancedbooksguaranteestheexistenceanduniquenessofasolutiontothisinitial-valueproblem.,Initial-valueandboundary-valueproblems,Aboundary-valueproblemforEquation1consistsoffindingasolutionofthedifferentialequationthatalsosatisfiesboundaryconditionsoftheformIncontrastwiththesituationforinitial-valueproblems,aboundary-valueproblemdoesnotalwayshaveasolution.,Example5Solvetheinitial-valueproblem,Example6Solvetheinitial-valueproblem,Example7Solvetheboundary-valueproblem,15.1Basicconcepts,separableandhomogeneousequations,机动目录上页下页返回结束,机动目录上页下页返回结束,10Twokindsofequations,Anordinarydifferentialequation:,Basicconcepts,Wehaveknowntheconceptofthedifferentialequations.,InSections8，,Suchas,involvesanunknownfunctionofasinglevariableandsomeofitsderivatives.,Def:,Apartialdifferentialequation:,involvesanunknownfunctionoftwoormorevariablesandsomeofitspartialderivatives.,Suchas,机动目录上页下页返回结束,Theorderofadifferentialequationistheorderofthehighestderivativethatappearsintheequation.,20Theorderoftheequation,Isanordinarydifferentialequationoforder1,Isathird-orderdifferentialequation,Isasecond-orderpartialdifferentialequation,机动目录上页下页返回结束,10Definetheseparableequations,12.1.2Separableequations,Note:Westudyonlyordinarydifferentialequationsmainly,Ingeneral,afirst-orderdifferentialequationhastheform:,whereFissomefunctionofthetwovariablesxandy,WhenFcanbefactoredasafunctionofxtimesafunction,Iscalledaseparableequation,then,机动目录上页下页返回结束,Form(1),Alsocanbewrittenas:,(2),Weintegratebothsides：,Thisdefinesyimplicitlyasafunctionofx.,thus,y=f(x)arecalledgeneralsolutionsoftheequation,Sometimes,wecansolveforyintermsofx，,机动目录上页下页返回结束,20Aninitial-valueproblem,Hasasolutionsatisfiesaninitialconditionoftheform,Theseparableequation,Wesaythisisaninitialproblem。,Ingeneral,thereisauniquesolutiontotheinitialproblemgivenbyequations(2),(3),(3),机动目录上页下页返回结束,Example1,Solution(a),Writeitas,Thisisthegeneralsolution,involvesanarbitraryconstantC,Solution(b),y(0)=1tellusx=o,y=1,So,(b)Solvetheinitial-valueproblem,(a)Solvethedifferentialequation,机动目录上页下页返回结束,Example2,Solution:,Writeitas,Weintegratebothsides,So,Solvethedifferentialequation,机动目录上页下页返回结束,Example3,Solution：,Weintegratebothsides,So,Writeitas,Solvethedifferentialequation,机动目录上页下页返回结束,Homogeneousequations,Afirst-orderdifferentialequation,IfF(x,y)canbewrittenas,Theform(4),iscalledhomogeneousequations,Forinstance:,Turnitas:,机动目录上页下页返回结束,Wesee,makethechangeofvariable,then,thus,Thisisaseparabledifferentialequation,So,theoriginaldifferentialequationhasthegeneralsolution,itsgeneralsolution,or,机动目录上页下页返回结束,Example5,Solution:,writeitas,Solvethedifferentialequation,机动目录上页下页返回结束,Example6,Solvethehomogeneousdifferentialequation,15.1Homework,P9551.3.5.,15.2First-orderlinearequations,机动目录上页下页返回结束,机动目录上页下页返回结束,Thegeneralsolutionsofthefirst-orderlinearequations,Itssolutionis,(1),(2),机动目录上页下页返回结束,Isthesolutionoftheequationform(3),Thisisaseparateequations,solveit,(3),WhenQ(x)=0,form(1)become,Thus,(4),Cisanarbitraryconstant,机动目录上页下页返回结束,Thesolutioniswrittenas,(3),Equation,LetusreplacetheconstantCinitbyarbitrary,Welookforthesolutionoftheequationoftheform,(5),机动目录上页下页返回结束,DifferentiatingEquation(5),weget,Putitinto(5),(6),Anothermethodseepage997,thus:Thesolutionoftheform(1)is,机动目录上页下页返回结束,Example1,Solution:,Bythegeneralsolutionform(6),Firststep:,Thesecondstep:,Laststep:,thisisalinearequation,Solvethedifferentialequation,机动目录上页下页返回结束,Example2,Solution:,First:,Second:,Last:,Sincey(1)=2,Therefore,Wemustfirstputthedifferentialequationintostandardform:,Wehave,thesolutiontotheinitial-valueproblemis,Findthesolutionoftheinitial-valueproblem,机动目录上页下页返回结束,Example3,findthesolutionoftheinitial-valueproblem,机动目录上页下页返回结束,ABernoullidifferentialequation,Observethat,ifn=0or1,itislinear.,Example1,transformstheBernoulliequationintothelinearequation,Solvethedifferentialequation,thesubstitution,15.2homework,15.3Exactequations,机动目录上页下页返回结束,Definitionoftheexact,Supposetheequation,Theny=f(x)satisfiesafirst-orderdifferentialequationobtainedbyusingtheChainRuletodifferentiatebothsidesofequationwithrespecttox:,(1),(2),AdifferentialequationoftheformofEquation(2)iscalledexact。,机动目录上页下页返回结束,Definition：,iscalledexactifthereisafunctionf(x,y)suchthat,Iff(x,y)isknown,thusthesolutionisgivenimplicitlyby,(3),(4),WemaybeabletosolveEquation4foryasanexplicitfunctionofx.,Afirstorderdifferentialequationoftheform,机动目录上页下页返回结束,Theorem,Wehavethefollowingconvenientmethodfortheexactnessofadifferentialequation.,(5),isexactifandonlyif,Thethedifferentialequation,havecontinuouspartialderivativesonasimplyconnecteddomain.,Theorem:,Suppose,and,机动目录上页下页返回结束,Example1Solvethedifferentialequation,Solution,Here,and,havecontinuouspartialderivativeson,Sothedifferentialequationisexactbytheorem.,(6),(7),Findsolutionsofexactequation,Thusthereexistsafunctionfsuchthat,Also,机动目录上页下页返回结束,Todeterminefwefirstintegrate6withrespecttox:,Nowwedifferentiate8withrespecttoy:,(8),(9),Comparing7and9,weseethat,Wedonotneedthearbitraryconstanthere,Thus,So,isthesolution.,andso,机动目录上页下页返回结束,Example2,Solution,Here,and,havecontinuouspartialderivativeson,Sothedifferentialequationisexactbytheorem.,thusthereexistsafunctionfsuchthat,Also,Solvethedifferentialequation,机动目录上页下页返回结束,Comparingthem,wesee,Wedonotneedthearbitraryconstanthere,Thus,So,isthesolution.,Todeterminefwefirstintegrate,Nowwedifferentiateitwithrespecttoy:,then,withrespecttox:,that,机动目录上页下页返回结束,Integratingfactors,Ifthedifferentialequation,Thedifferentialequation,(10),isnotexact.,theresultingequation,isexact.,Suchthat,aftermultiplicationbyI(x,y),welookforanintegratingfactorI(x,y),isnotexact,机动目录上页下页返回结束,Equation(10)isexactif,Thatis,or,(11),Ingeneral,itishardertosolvethispartialdifferentialequationthantheoriginaldifferentialequation.,ButitissometimespossibletofindIthatisafunctionofxoryalone.,机动目录上页下页返回结束,SupposeIisafunctionofxalone.,Then,Soequation11becomes,(12),if,ThenEquation12isafirstorderlinear(andseparable)ordinarydifferentialequation.,ItcanbesolvedforI(x),ThenEquation10isexactandcanbesolvedInExample1or2.,isafunctionofxalone,机动目录上页下页返回结束,Example3,Solution:,Here,since,thegivenequationisnotexact.,But,isafunctionofxalone.,SobyEquation12thereisanintegratingfactorIthatsatisfies,weget,Solvethedifferentialequation,机动目录上页下页返回结束,Multiplyingtheoriginalequationbyx,weget,Comparisonthengives,(13),so,Ifwelet,Then,SoEquation13isnowexact.thusthereisafunctionfsuchthat,Integratingthefirstoftheseequations,weget,(whichwecantaketobe0),sogisaconstant,机动目录上页下页返回结束,Thesolutionis,therefore,15.3homework,15.4Strategyforsolvingfirst-orderequations,机动目录上页下页返回结束,Insolvingfirst-orderdifferentialequationsweusedthetechniqueforseparableequationsinSections8.1andthemethodforlinearequationsinSection15.2.(onthetext).,Inthissectionwepresentamiscellaneouscollectionoffirst-orderdifferentialequationsandpartoftheproblemistorecognizewhichtechniqueshouldbeusedoneachequation.,WealsodevelopsmethodsforsolvinghomogeneousequationsinSection15.1(onthetext)andexactequationsinSection15.3(onthetext)。,机动目录上页下页返回结束,Here,however,theimportantthingisnotsomuchtheformofthefunctionsinvolvedeasitistheformoftheequationitself.,Aswiththestrategyofintegration(Section7.6)andthestrategyoftestingseries(Section10.7),themainideaistoclassifytheequationaccordingtoitsform.,机动目录上页下页返回结束,Alinearequationcanbeputintotheform,thatis,theexpressionfordy/dxcanbefactoredandaproductofafunctionofxandafunctionofy.,(1),Recallthataseparableequationcanbewrittenintheform,(2),机动目录上页下页返回结束,Ahomogeneousequationcanbeexpressedintheform,(3),Ifanequationhasnoneoftheseforms,wecan,asalastresort,attempttofindandintegratingfactorandthusequationexact.,where,(4),Anexactequationhastheform,机动目录上页下页返回结束,(ThisstepisanalogoustoSteplofthestrategyforintegration:algebraicimplification.),Ineachofthesecases,somepreliminaryalgebramayberequiredinordertoputagivenequationintooneoftheprecedingforms.,机动目录上页下页返回结束,(comparewithEquation2),Itisalsolinear,sincewecanwritetheequationas,(comparewithEquation1),isseparablebecausedy/dxcanbewrittenas,Forinstance,theequation,Itmayhappenthatagivenequationisofmorethanonetype.,机动目录上页下页返回结束,Insuchacasewecouldsolvetheequationusinganyoneofthecorrespondingmethods,althoughoneofthemethodsmightbeeasierthantheothers.,(comparewithEquation2).,Furthermore,itisalsohomogeneousbecausewecanwriteitas,机动目录上页下页返回结束,Example1,WenowrecognizetheequationadbeingseparableandwecansolveitusingthemethodsofSection15.1.,Initially,thisequationmaynotappeartobeinanyoftheformsofEquations1,2,3,or4,butobservethatwecanfactortherightsideandthereforewritetheequationas,Inthefollowingexamplesweidentifythetypeofeachequationwithoutworkingourthedetailsofthesolutions.,机动目录上页下页返回结束,Example2,Thechangeofvariablev=yxconvertstheequationintoaseparableequation.,(Wecouldhaveanticipatedthisbecausetheexpressionsx2,y2,and2xyareallofdegree2.),Whichshowsthatyisafunctionofy/xandtheequationishomogeneous(seeEquation3).,itisnotexact.,Since,Theequationisclearlymotseparable,norseparable,norisitlinear.,Butifwesolvefory,weget,机动目录上页下页返回结束,Example3,Therefore,theequationisindeedexactandcanbedolledbythemethodsofSection15.3.,then,If,Wesuspectismightbeexact,sowewriteitintheform,Thisequationisnotseparable,linear,orhomogeneous.,15.4homework,Itisthereforelinearandcanbesolvedusingtheintegratingfactor,WerecognizeitashavingtheformofEquation2.,Ifweputtheequationintheform,Example4,Inthissectionwedealwithsecond-ordernonhomogeneouslineardifferentialequationswithconstantcoefficient:(1)wherea,b,careconstantsandGisacontinuousfunction.,15.6NonhomogeneousLinearEquations,Therelatedhomogeneousequation(2)iscalledthecomplementaryequationandplaysanimportantroleinthesolutionoftheoriginalnonhomogeneousequation(1).,ProofWewanttoverifythatifyisanysolutionofEquation1,thenisasolutionofthecomplementaryEquation2.,Indeed,Therearetwomethodsforfindingaparticularsolution.ThemethodofundeterminedcoefficientsisstraightforwardbutworksonlyforarestrictedclassoffunctionsG.ThemethodofvariationofparametersworksforeveryfunctionGbutisusuallymoredifficulttoapplyinpractice.,Therefore,Theorem3saysthatweknowthegeneralsolutionofthenonhomogeneousequationassoonasweknowaparticularsolution.,ItisreasonabletoguessthattheparticularsolutionisapolynomialofthesamedegreeasGbecauseifyisapolynomial,thenisalsoapolynomial.Wethereforesubstituteapolynomial(ofthesamedegreeasG)intothedifferentialequationanddeterminethecoefficients.,Example1Solvetheequation,Themethodofundeterminedcoefficients,Wefirstillustratethemethodofundeterminedcoefficientsfortheequation,Case1G(x)isapolynomial,Case2G(x)isoftheform,whereCandkareconstants.,Wetakeasatrialsolutionafunctionofthesameform,becausethederivativesofareconstantmultiplesof.,Example2Solvetheequation,Becauseoftherulesfordifferentiatingthesineandcosinefunctions,wetakeasatrialparticularsolutionafunctionoftheform,Example3Solvetheequation,Case3G(x)iseitherCcoskxorCsinkx,Wetakethetrialsolutiontobeaproductoffunctionsofthesametype.Forinstance,insolvingthedifferentialequationwewouldtry,Weusetheeasilyverifiedprincipleofsuperposition,whichsaysifandaresolutionsofrespectively,thenisasolutionof,Case4G(x)isaproductoffunctionsoftheprecedingtypes,Case5G(x)isasumoffunctionsofthesetypes,Example4Solve,Example5Solve,Finallywenotethattherecommendedtrialsolutionsometimesturnsouttobeasolutionofthecomplementaryequationandthereforecannotbeasolutionofthenonhomogeneousequation.Insuchcaseswemultiplytherecommendedtrialsolutionbyx(orifnecessary)sothatnoterminisasolutionofthecomplementaryequation.,G(x)=Firsttry,Modification:Ifanytermofisasolutionofthecomplementaryequation,multiplybyx(orifnecessary).,Summaryofundeterminedcoefficient,Supposewehavealreadysolvedthehomogeneousequationandwrittenthesolutionas(4)whereandarelinearlyindependentsolutions.,LetusreplacetheconstantsandinEquation4byarbitraryfunctionsand.,Welookforaparticularsolutionofthenonhomogeneousequationoftheform(5),Themethodofvariationofparameters,Sinceandarearbitraryfunctions,wecanimposetwoconditionsonthem.Oneconditionisthatisasolutionofthedifferentialequation;wecanchoosetheotherconditionsoastosimplifyourcalculations.InviewoftheexpressioninEquation6,letusimposetheconditionthat(7),DifferentiatingEquation5,weget(6),Then,Substitutinginthedifferentialequation,wegetor(8),Butandaresolutionsofthecomplementaryequation,so,andEquation8simplifiesto(9),Equation7and9formasystemoftwoequationsintheunknownfunctionsand.AftersolvingthissystemwemaybeabletointegratetofindandthentheparticularsolutionisgivenbyEquation5.,Example6Solvetheequation,isnoteasytosolve.ButitisimportanttobeabletosolveequationssuchasEquation1sincetheyarisefromphysicalproblemsand,inparticular,inconnectionwiththeSchrdingerequation(薛定谔方程)inquantummechanics.,Evenasimple-lookingequationlike(1),15.8SeriesSolutions,Sofar,theonlysecond-orderdifferentialequationsthatwehavebeenabletosolvearelinearequationswithconstantcoefficients.,Insuchacase,weusethemethodofpowerseries;thatis,welookforasolutionoftheform,Themethodistosubstitutethisexpressionintothedifferentialequationanddeterminethevaluesofthecoefficients,Example1Usepowerseriestosolvetheequation,SolutionWeassumethereisasolutionoftheform(2),Rewriteas,Inordertocomparetheexpressionforyandmoreeasily,wesubstitutingtheseexpressionsintothediffe