常微分方程课件-第03讲

1,Lecture3,2,2.2LinearDifferentialEquationandMethodofVariationofConstant(线性微分方程与常数变易法),I.Linear(First-order)DifferentialEquations,II.MethodofVariationofConstant,III.BernoulliDE,伯努利微分方程,3,2.2LinearDifferentialEquationandMethodofVariationofConstant(线性微分方程与常数变易法),I.Linear(First-order)DifferentialEquations,II.MethodofVariationofConstant,III.BernoulliDE,伯努利微分方程,4,DefinitionC/P44;E/P45,issaidtobealinearfirst-orderdifferentialequation.,WealwayssupposethatthecoefficientfunctionsP(x)andQ(x)in(2.28)arebothcontinuousonsomeintervalI.,I.Linear(First-order)DifferentialEquations,5,(1),andwecall(2.3)ahomogeneouslinear1st-orderDE;,一阶齐次线性微分方程,(2),thenwecall(2.28)anonhomogeneouslinear1st-orderDE.,一阶非齐次线性微分方程,Meantime,wecall(2.3)tobeitscorrespondinghomogeneouslinearDE.,Forexample,Itisseparable!C/P33,6,2.2LinearDifferentialEquationandMethodofVariationofConstant,(线性微分方程与常数变易法),I.Linear(First-order)DifferentialEquations,II.MethodsofVariationofConstant,III.BernoulliDE,7,2.2LinearDifferentialEquationandMethodofVariationofConstant,(线性微分方程与常数变易法),I.Linear(First-order)DifferentialEquations,II.MethodsofVariationofConstant,III.BernoulliDE,8,1.ThesolutionofHomogeneouslinearDE(2.3),Example2.2.1,Solution,Thedesiredgeneralsolutionis,II.MethodofConstantVariation-themethodofsolvingfor(2.28),9,2.NonhomogeneousLinearDE(2.28)andMethodofVariationofConstant,非齐次线性微分方程(2.28)与常数变易法,10,Substitutingtheseinto(2.28)gives,Thusthegeneralsolutionof(2.28)is,wherecisanarbitraryconstant.,Variationofconstant,2.NonhomogeneousLinearDE(2.28)andMethodofVariationofConstant,Ifyoucanrememberit,itwillveryconvenience!,11,Example2.2.2C/P45,Solution,ItscorrespondinghomogeneouslinearDEis,anditsgeneralsolutionis,wherecisanarbitraryconstant.,(*),Findthegeneralsolutionof,Variationofconstant,12,12,Example2.2.2C/P45,Solution,ItscorrespondinghomogeneouslinearDEis,anditsgeneralsolutionis,wherecisanarbitraryconstant.,Substitutingitinto(2.32)gives,Thusthedesiredgeneralsolutionis,wherecisanarbitraryconstant.,(*),Findthegeneralsolutionof,13,Example2.2.3,Solution,Thusthedesiredparticularsolutionis,CfC/P49/1(6),14,Example2.2.4C/P46,Solution,Thusthedesiredgeneralsolutionis,i.e.,wherecisanarbitraryconstant.,15,2.2LinearDifferentialEquationandMethodofVariationofConstant,(线性微分方程与常数变易法),I.Linear(First-order)DifferentialEquations,II.MethodsofVariationofConstant,III.BernoulliDE,16,2.2LinearDifferentialEquationandMethodofVariationofConstant,(线性微分方程与常数变易法),I.Linear(First-order)DifferentialEquations,II.MethodsofVariationofConstant,III.BernoulliDE,17,III.BernoulliDEC/P47;E/P60,DefinitionC/P47;E/P60,Afirst-orderDEoftheform,iscalledaBernoulliDE,wherenisafixedconstant.,Otherwise,itisnotlinear.,伯努利微分方程,ItsOK!,18,ThisisalinearDE,19,Thusitsgeneralsolutionis,wherecisanarbitraryconstant.,Howabouty=0?,20,Example2.2.5C/P48,Solution,Thusthegeneralsolutionis,i.e.,wherecisanarbitraryconstant.,Inaddition,21,Example2.2.6C/P49/1(15),Solution,Thisisnotseparable,notahomogeneous,andnotaBernoulliEq.aboutunknownfunctionyforanyn!,22,Solution,ThisisaBernoulliEq.aboutunknownfunctionxofn=2.,Example2.2.6C/P49/1(15),23,Solution,or,Example2.2.6C/P49/1(15),24,2.3ExactDifferentialEquationsandIntegratingFactorsC/P50;E/P64,恰当方程与积分因子,I.ExactDE,II.IntegratingFactors,25,I.ExactDE,II.IntegratingFactors,2.3ExactDifferentialEquationsandIntegratingFactorsC/P50;E/P64,26,ItissometimesconvenienttorewriteEq.(*)inthefollowingsymmetricform(对称形式),andwealsocallittobeadifferentialform.,微分形式,27,DefinitionC/P51;E/P64,Considera1st-orderDEinthesymmetricform,then(2.42)issaidtobeanexactdifferentialequation.,I.ExactDE,28,Example2.3.1,ThefollowingODEareallexactequations:,29,Result(结果)C/P51,Proof:,30,30,Result(结果)C/P51,Forexample,31,QuestionC/P51,E/P64:,(1)HowcanwedeterminewhethertheODE(2.42)isexact?,(2)If(2.42)isexact,howcanwefindthefunctionu(x,y)?,32,TheoremC/PP51-53;E/P65(CriterionforExactness),恰当性准则,Furthermore,thedesiredfunctionuis,Proof:SeeE/PP65-66,33,Example2.3.2C/P53;E/P66,Solvethedifferentialequation,Solution,34,Therefore,thedesiredgeneralsolutionofthegivenDEisdefinedimplicitlybytheequation,Thus,wherecisanarbitraryconstant.,35,Remark:(MethodofDetachingTermsandRegroupC/P54),分项组合法,Wemustremember“totaldifferentials”asmanyaspossible!,ForexampleC/P54,36,Thus,DetachingTerms,Regroup,Totaldifferentialformulas,37,Example2.3.4C/P55,Solution:,Remark:,Forexample:,38,39,2.3ExactDifferentialEquationsandIntegratingFactors,I.ExactDE,II.IntegratingFactors,40,2.3ExactDifferentialEquationsandIntegratingFactors,I.ExactDE,II.IntegratingFactors,41,II.Integratingfactors,DefinitionC/P55;E/45,isexact,Why?,42,Example2.3.5,Solution,Howcanwefindit?,43,NoteC/P55:,Question:,ResultC/P56:,ButitisaPDE!,Integatingfactorisnotunique!,44,Generally,itisverydifficulttofindasolutionof(2.57)!,Buttherearesomespecialcaseswherewemayfindsomesolutionof(2.57).,Forexample,Butthisequationismeaninglessunlesstheexpression,45,ResultC/P56:,46,Example2.3.6C/P57,Solution,47,Example2.3.6C/P57,isthedesiredgeneralsolution.,cf.C/P45andE/P45,Solution,48,Similarly,Trytogiveitsintegratingfactor!,49,50,andsoonC/P61/4,5,6,7,8,9.,51,Example2.3.7C/P58,Solution,Ithasanintegratingfactor,Thusthedesiredgeneralsolutionis,or,wherecisanarbitraryconstant.,52,Example2.3.8C/P58,Solution,MethodI:,Thusthedesiredgeneralsolutionis,53,Example2.3.8(C/P58),Solution,MethodII:,anditsrighthandsidedependsonyalone,54,Thusthedesiredgeneralsolutionis,Example2.3.8(C/P58),Solution,MethodII:,55,Thisisahomoge