常微分方程课件-第04讲

Lecture4,2,2.3ExactDifferentialEquationsandIntegratingFactors,I.ExactDE,II.IntegratingFactors,3,3,II.Integratingfactors,DefinitionC/P55;E/45,isexact,Why?,4,4,Example2.3.5,Solution,Howcanwefindit?,5,5,NoteC/P55:,Question:,ResultC/P56:,ButitisaPDE!,Integatingfactorisnotunique!,6,6,Generally,itisverydifficulttofindasolutionof(2.57)!,Buttherearesomespecialcaseswherewemayfindsomesolutionof(2.57).,Forexample,Butthisequationismeaninglessunlesstheexpression,7,7,ResultC/P56:,8,8,Example2.3.6C/P57,Solution,9,9,Example2.3.6C/P57,isthedesiredgeneralsolution.,cf.C/P45andE/P45,Solution,10,10,Similarly,Trytogiveitsintegratingfactor!,11,11,12,12,andsoonC/P61/4,5,6,7,8,9.,13,13,Example2.3.7C/P58,Solution,Ithasanintegratingfactor,Thusthedesiredgeneralsolutionis,or,wherecisanarbitraryconstant.,14,14,Example2.3.8C/P58,Solution,MethodI:,Thusthedesiredgeneralsolutionis,15,15,Example2.3.8C/P58,Solution,MethodII:,anditsrighthandsidedependsonyalone,16,16,Thusthedesiredgeneralsolutionis,Example2.3.8C/P58,Solution,MethodII:,17,17,ThisisahomogeneousDE.,weget,Thusthedesiredgeneralsolutionis,Example2.3.8C/P58,Solution,MethodIII:,18,18,ThisisalinearODE(Unknownfunctionisx,independentvariableisy).,Thusthedesiredgeneralsolutionis,Example2.3.8C/P58,Solution,MethodIV:,19,2.41st-orderImplicitDifferentialEquationsandParametricRepresentation,一阶隐式微分方程与参数表示,I.SolutionsinParametricForm,II.First-orderImplicitODEs,参数形式的解,一阶隐式常微分方程,20,I.SolutionsinParametricForm,Definition2.3.3,Forthe(implicit)ODE,Ifthereexistsasystemofparametricfunctions,suchthat,参数形式的解,21,Similarly,Forexample,WemaydefinethegeneralsolutionintheparametricformofF(x,y,y)=0.,参数形式的通解,22,II.First-orderImplicitODE,23,Methodofsolution:,Differentiatingbothsidesof(2.63)w.r.t.xgives,(2.62),(2.63),(2.64),ItsanODEw.r.t.xandp,(2.64*),24,Why?,ResultIC/P63:,(2.64*),25,Example2.4.1C/P63,Answer:,Solution:,26,Example2.4.2C/P64,Answer:,Solution:,27,Warning!,(2.64*),28,Methodofsolution:,(2.69),Differentiatingbothsidesof(2.70)w.r.t.ygives,(2.70),(2.71),(2.71*),ItsanODEw.r.t.yandp,Notx,why?,29,ResultIIcfp66:,30,Example2.4.3C/P66/cfC/P63,Answer:,cfC/P64Theresultsarethesame,Solution:,31,31,Example2.4.4,Answer:,Solution:,32,Methodofsolution:,(2.73),Clearly,itgeometricallyrepresentsacurvein(x,p)-plane.,Ifwearecleverenoughtofindaparametricrepresentationofthiscurve,thatis,thereexists,isthegeneralsolutionof(2.73).,(2.74),33,Example2.4.5C/P67,Answer:,Solution:,34,Methodofsolution:,(2.75),Clearlyagain,itgeometricallyrepresentsacurvein(y,p)-plane.,Ifwearecleverenoughtofindaparametricrepresentationofthiscurve:,isthegeneralsolutionof(2.75).,35,InadditionC/P69,36,Example2.4.6C/P69,Answer:,Solution:,37,ReviewofChapterII(第二章复习),C/PP70-72,38,对于一阶微分方程,的若干类型的初等解法，归纳起来就是：,1、若方程能就y解出，即方程取形式,或,主要介绍了五种类型的方程（变量分离方程、齐次方程、线性方程、Bernoulli方程及恰当方程）的初等解法.但实际上作为基础的只有变量分离方程与恰当方程，其他类型的方程可借助变量变换或积分因子化为这两种类型，这可简略地用下图表示：,39,坐标变换,或,40,2、若方程能就y或x解出，即方程可化为,y=f(x,y)或x=f(y,y),则令y=p后，把问题化为求解关于p与x或y之间的一阶微分方程：,它们是导数已解出的微分方程.如果用分离变量法或恰当方程法求得通解为,41,则它的参数形式的通解为：,42,它们是导数