常微分方程课件-第06讲

Lecture6,1,Higher-OrderDifferentialEquations(高阶微分方程),ChapterIV(第四章),2,4.1GeneralTheoryforLinearDifferentialEquations(线性微分方程的一般理论),4.1.1Introduction(引言),4.1.2ThepropertiesandstructureofsolutionsofhomogeneouslinearDEs(齐次线性微分方程的解的性质与结构),4.1.3NonhomogeneouslinearDEsandthemethodofvariationofconstants(非齐次线性微分方程与常数变易法),3,4.1GeneralTheoryforLinearDifferentialEquations(线性微分方程的一般理论),4.1.1Introduction(引言),4.1.3NonhomogeneouslinearDEsandthemethodofvariationofconstants(非齐次线性微分方程与常数变易法),4.1.2ThepropertiesandstructureofsolutionsofhomogeneouslinearDEs(齐次线性微分方程的解的性质与结构),4,4.1.1IntroductionC/P120;E/P96+P109,Weconsiderthefollowingnth-orderlinearDE,andiscalledthenth-orderhomogeneouslinearDE(n阶齐次线性微分方程).,Otherwise,(4.1)iscalledthenth-ordernonhomogeneouslinearDE(n阶非齐次线性微分方程).,I.Somebasicconcepts(一些基本概念),5,itsassociatedhomogeneouslinearDE(它所对应的齐线性微分方程)!,Result(结果):,ForanynonhomogeneouslinearDE,wemayalwayswriteoutitsassociatedhomogeneouslinearDE!,Forexample:,Remark(附注):,6,II.Existence-uniquenesstheoremoflinearDEs,Theorem1C/P121;E/P110,Considerthenth-orderlinearDE:,CfC/P85/Remark2+P201/Corollary,线性微分方程的存在唯一性定理,7,8,4.1GeneralTheoryforLinearDifferentialEquations(线性微分方程的一般理论),4.1.1Introduction(引言),4.1.3NonhomogeneouslinearDEsandthemethodofvariationofconstants(非齐次线性微分方程与常数变易法),4.1.2ThepropertiesandstructureofsolutionsofhomogeneouslinearDEs(齐次线性微分方程的解的性质与结构),9,4.1GeneralTheoryforLinearDifferentialEquations(线性微分方程的一般理论),4.1.1Introduction(引言),4.1.2ThepropertiesandstructureofsolutionsofhomogeneouslinearDEs(齐次线性微分方程的解的性质与结构),4.1.3NonhomogeneouslinearDEsandthemethodofvariationofconstants(非齐次线性微分方程与常数变易法),10,4.1.2ThepropertiesandstructureofsolutionsofhomogeneouslinearDEs(齐次线性微分方程的解的性质与结构),Theorem2(PrincipleofSuperposition叠加原理)C/P121;E/P109,ProofTheconclusionfollowsalmostimmediatelyfromthelinearityoftheoperationofdifferentiation.(SeeE/P98fork=2),11,Question:,Howcanwefindthegeneralsolutionof(4.2)?,howmany?,Whatisitsbasis?,DefinitionC/P122;E/PP112-113(Lineardependenceandindependenceoffunctions函数的线性相关性与线性无关性),ResultE/P112:,12,ForexampleC/P122,Question:,13,DefinitionC/P122;E/P113(Wronskian(determinant),朗斯基行列式,Exercise:,14,Theorem3C/P123;E/P113.,Proof:,15,Theorem3C/P123;E/P113.,RemarkC/P123.,Ingeneral,itsinversetheorem(逆定理)isnottrue.,ForexampleC/P123,Exercise:,Trytowriteoutitsinverse、negative、converse-negativeThs.,Why?,16,Theorem3C/P123;E/P113.,RemarkC/P123.,Ingeneral,itsinversetheorem(逆定理)isnottrue.,Exercise:,ButwhenthesefunctionsaresolutionsofsomelinearlyhomogeneousDE,theinversetheoremistrue!,Trytowriteoutitsinverse、negative、converse-negativeThs.,17,Theorem4C/P124;E/P115/THEOREM3(b).,Proof:,18,ResultC/P124;E/P115:,Thentherearejusttwopossibilities:,andoneandonlyoneofthesetwocasesholds!,Theorem4C/P124;E/P115/THEOREM3(b).,19,Corollary(推论).C/P124;E/P115,Thenwehave,Question:Isitpossiblethatthefollowingtwofunctionsaretwolinearlyindependentsolutionsofsomehomogeneouslineardifferentialequation?,20,Theorem5C/P125.,Proof:,21,Theorem5C/P125.,Remark:,22,Theorem6C/P125;E/P116.,(StructureTheoremofGeneralSolutionofDE(4.2)微分方程(4.2)的通解结构定理,Proof:,23,Theorem6C/P125;E/P116.,(StructureTheoremofGeneralSolutionofDE(4.2)微分方程(4.2)的通解结构定理,24,Corollary(推论)C/P126.,DefinitionC/P126.,Infact,afundamentalsystemofsolutionsof(4.2)isjustabasis(基)fortheabovelinearspaceV.,25,Note:afundamentalsystemofsolutionsof(4.2)isnotunique!,DefinitionC/P126(Standardfundamentalsystemofsolutions)标准基本解组,26,Example4.1Cf:C/P132/3(2),Ideaoftheproof:,StepI.,StepII.,27,4.1GeneralTheoryforLinearDifferentialEquations(线性微分方程的一般理论),4.1.1Introduction(引言),4.1.2ThepropertiesandstructureofsolutionsofhomogeneouslinearDEs(齐次线性微分方程的解的性质与结构),4.1.3NonhomogeneouslinearDEsandthemethodofvariationofconstants(非齐次线性微分方程与常数变易法),28,4.1GeneralTheoryforLinearDifferentialEquations(线性微分方程的一般理论),4.1.1Introduction(引言),4.1.3NonhomogeneouslinearDEsandthemethodofvariationofconstants(非齐次线性微分方程与常数变易法),4.1.2ThepropertiesandstructureofsolutionsofhomogeneouslinearDEs(齐次线性微分方程的解的性质与结构),29,4.1.3NonhomogeneouslinearDEsandthemethodofvariationofconstantsC/P126,I.Thestructureofsolutionsofnonhomogeneousnth-orderlinearDE(n阶非齐次线性微分方程的解的结构),TheassociatedlinearlyhomogeneousDEof(4.1)is,30,Property1C/P127;E/P117,Property2C/P127;E/P117,ThedifferenceofanytwosolutionsofthenonhomogeneouslinearDE(4.1)isasolutionoftheassociatedhomogeneouslinearDE(4.2).,31,Theorem7C/P127;E/P118,(StructureTheoremofGeneralSolutionofDE(4.1)微分方程(4.1)的通解结构定理,Proof:,32,Remark:,OneisthegeneralsolutionofitsassociatedhomogeneouslinearDE(4.2),theotherisoneparticularsolutionof(4.1)!,33,ResultC/P127,Inordertofindthegeneralsolutionof(4.1),weneedonlytofindafundamentalsystemofsolutionsofitsassociatedhomo-geneouslinearDE(4.2)andone(particular)solutionof(4.1)！为了求非齐次线性微分方程(4.1)的通解，我们只需求出它所对应的齐次线性微分方程(4.2)的一个基本解组及(4.1)的一个(特)解即可！,Furthermore,thefollowingmethodofvariationofconstantswilltellus:oncewehavefoundafundamentalsystemofsolutionsof(4.2),wec