常微分方程课件-第07讲VIP

Lecture7,2,Higher-OrderDifferentialEquations(高阶微分方程),ChapterIV(第四章),4.1GeneralTheoryforLinearDifferentialEquations,线性微分方程的一般理论,3,Theorem6C/P125.,(StructureTheoremofGeneralSolutionofDE(4.2),微分方程(4.2)的通解结构定理,4,Theorem7C/P127.,(StructureTheoremofGeneralSolutionofDE(4.1),微分方程(4.1)的通解结构定理,5,6,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,themethodofvariationofconstantswilltellus:oncewehavefoundafundamentalsystemofsolutionsof(4.2),wecanfindone(particular)solutionof(4.1)!进一步，“常数变易法”告诉我们：一旦我们已经求得齐次线性微分方程(4.2)的一个基本解组，则我们就可以求得(4.1)的一个(特)解！,6,Question:,7,4.2MethodsofSolvingLinearDifferentialEquationswithConstantCoefficients常系数线性微分方程的解法,4.2.1Complex-ValuedFunctionsandComplex-ValuedSolutions(复值函数与复值解),4.2.2HomogeneousLinearDifferentialEquationswithConstantCoefficientsandEulerEquations(常系数齐次线性方程和欧拉方程),4.2.3NonhomogeneousLinearDifferentialEquationsMethodsofCoefficientComparison(非齐次线性方程比较系数法),8,4.2MethodsofSolvingLinearDifferentialEquationswithConstantCoefficients常系数线性微分方程的解法,4.2.1Complex-ValuedFunctionsandComplex-ValuedSolutions(复值函数与复值解),4.2.2HomogeneousLinearDifferentialEquationswithConstantCoefficientsandEulerEquations(常系数齐次线性方程和欧拉方程),4.2.3NonhomogeneousLinearDifferentialEquationsMethodsofCoefficientComparison(非齐次线性方程比较系数法),9,4.2.1Complex-ValuedFunctionsandComplex-ValuedSolutions(复值函数与复值解),I.Somebasicconcepts(基本概念),II.Complex-valuedexponentalfunction(复值指数函数)-aparticularcomplex-valuedfunction,III.Complex-valuedsolutions(复值解),10,I.Somebasicconcepts(基本概念),DefinitionC/P133,11,Question,12,ResultsC/PP133-134,13,4.2.1Complex-ValuedFunctionsandComplex-ValuedSolutions(复值函数与复值解),I.Somebasicconcepts(基本概念),II.Complex-valuedexponentalfunction(复值指数函数)-aparticularcomplex-valuedfunction,III.Complex-valuedsolutions(复值解),14,4.2.1Complex-ValuedFunctionsandComplex-ValuedSolutions(复值函数与复值解),I.Somebasicconcepts(基本概念),II.Complex-valuedexponentalfunction(复值指数函数)-aparticularcomplex-valuedfunction,III.Complex-valuedsolutions(复值解),15,II.Complex-valuedexponentalfunction(复值指数函数)-aparticularcomplex-valuedfunction,16,Thatis,Therefore,wemayobtainthefollowingformula,EulersFormula(欧拉公式),Clearly,17,18,InSummary(总结)C/P135,(1)DerivationRulesforcomplex-valuedfunctionsaresimilartothatforreal-valuedfunctions.,(2)Thepropertiesofthecomplex-valuedexponentialfunctionsaresimilartothatofthereal-valuedexponentialfunctions.,(实变量的)复值函数的求导公式与(实变量的)实值函数的求导公式完全类似,(实变量的)复指数函数与(实变量的)实指数函数具有完全类似的性质,19,4.2.1Complex-ValuedFunctionsandComplex-ValuedSolutions(复值函数与复值解),I.Somebasicconcepts(基本概念),II.Complex-valuedexponentalfunction(复值指数函数)-aparticularcomplex-valuedfunction,III.Complex-valuedsolutions(复值解),20,4.2.1Complex-ValuedFunctionsandComplex-ValuedSolutions(复值函数与复值解),I.Somebasicconcepts(基本概念),II.Complex-valuedexponentalfunction(复值指数函数)-aparticularcomplex-valuedfunction,III.Complex-valuedsolutions(复值解),21,III.Complex-valuedsolutions(复值解),Weconsiderthefollowingnth-orderlinearDE,DefinitionC/P136,22,Theorem8C/P136,23,Theorem9C/P136,and,24,4.2MethodsofSolvingLinearDifferentialEquationswithConstantCoefficients常系数线性微分方程的解法,4.2.1Complex-ValuedFunctionsandComplex-ValuedSolutions(复值函数与复值解),4.2.2HomogeneousLinearDifferentialEquationswithConstantCoefficientsandEulerEquations(常系数齐次线性方程和欧拉方程),4.2.3NonhomogeneousLinearDifferentialEquationsMethodsofCoefficientComparison(非齐次线性方程比较系数法),25,4.2MethodsofSolvingLinearDifferentialEquationswithConstantCoefficients常系数线性微分方程的解法,4.2.1Complex-ValuedFunctionsandComplex-ValuedSolutions(复值函数与复值解),4.2.2HomogeneousLinearDifferentialEquationswithConstantCoefficientsandEulerEquations(常系数齐次线性方程和欧拉方程),4.2.3NonhomogeneousLinearDifferentialEquationsMethodsofCoefficientComparison(非齐次线性方程比较系数法),26,4.2.2HomogeneousLinearDifferentialEquationswithConstantCoefficientsandEulerEquations(常系数齐次线性方程和欧拉方程),I.HomogeneouslinearDEswithconstantcoefficients(常系数齐线性方程),II.Eulerequations(欧拉方程),27,4.2.2HomogeneousLinearDifferentialEquationswithConstantCoefficientsandEulerEquations(常系数齐次线性方程和欧拉方程),I.HomogeneouslinearDEswithconstantcoefficients(常系数齐线性方程),28,4.2.2HomogeneousLinearDifferentialEquationswithConstantCoefficientsandEulerEquations(常系数齐次线性方程和欧拉方程),I.HomogeneouslinearDEswithconstantcoefficients(常系数齐线性方程),29,Clearly,30,Howdowefindthegeneralsolutionof,Theorem6inC/P125tellsusthatweneedonlytofindnlinearlyindependentsolutionsof(4.19)!,31,32,PropositionC/P137,DifferentialEquation/微分方程,AlgebraicEquation/代数方程,Proof:,33,DefinitionC/P137;E/P121,Thealgebraicequation(4.21)iscalledthecharacteristicequation(特征方程)oftheDE(4.19).,Arootof(4.21)iscalledacharacteristicroot(特征根)of(4.19).,34,Example4.2.1,TrytowriteoutthecharacteristicequationsofthefollowingDEs,andfindthecharacteristicrootsofthem.,Answer:,Multiplicity3,35,代数基本定理意味着代数方程（即多项式方程）(4.21)有n个根，虽然这些根不必是不同的，也不必是实的！但是要求得这些根可能是困难的，甚至是不可能的.,36,(1)Distinctroots,(2)Repeatedroots/Equalroots,n个不同根(即无重根)的情形,有重根的情形,37,(1)DistinctrootsC/P137;E/P122-P127,38,Why?C/P138,Proof:,ResultI:,(1)DistinctrootsC/P137;E/P122-P127,39,Andhence,thegeneralsolutionof(4.19)is,Itisalsoafundamentalsystemofsolutionsof(4.19),40,Why?C/P138,ResultI:,(1)DistinctrootsC/P137;E/P122-P127,Example4.2.2E/P122,Findafundamentalsystemofsolutionsof,Solution,ThecharacteristicequationofthegivenDEis,Wesolvetheabovealgebraicequationbyfactoring(因式分解):,andsothecharacteristicequationhasthethreedistinctroots:,Therefore,thedesiredfundamentalsystemofsolutionsis,41,Example4.2.3,Findthegeneralsolutionof,Solution,ThecharacteristicequationofthegivenDEis,Bysolvingtheabovealgebraicequation,weobtainthethreedistinctroots:,Therefore,thedesiredgeneralsolutionis,Thisisthecomplex-valuedgeneralsolution!,Question:Howcanwefinditsreal-valuedgeneralsolution?,42,43,ResultI:,Exercises:,44,Example4.2.3,Findthegeneralsolutionof,Solution:,ThecharacteristicequationofthegivenDEis,Bysolvingtheabovealgebraicequation,weobtainthethreedistinctroots:,Therefore,thedesiredgeneralsolutionis,Thisisthecomplex-valuedgeneralsolution!,Howcanwefindthereal-valuedgeneralsolution?,45,Example4.2.3,Findthegeneralsolutionof,Solution:,ThecharacteristicequationofthegivenDEis,Bysolvingtheabovealgebraicequation,weobtainthethreedistinctroots:,Therefore,thedesired(real-valued)generalsolutionis,46,(1)Distinctroots,(2)Repeatedroots/Equalroots,Forexample,considertheDE,ThecharacteristicequationofthegivenDEis,47,(1)Distinctroots,(2)Repeatedroots/Equalroots,Ideaoffindingthegeneralsolutionof(4.19):,48,49,50,51,Let,isthecharacteristicequationof(4.23).,Why?C/P139,52,ResultII.,53,(4.26),Andwecanprovethattheyarelinearlyindependent!,Why?C/PP140-141,(4.26)isafundamentalsystemofsolutionsof(4.19)!,Note:somein(4.26)maybecomplex-valuedsolutionsof(4.19).,ResultII.,54,Example4.2.4C/P142,Findthegeneralsolutionof,Solution,ThecharacteristicequationofthegivenDEis,Bysolvingtheabovealgebraicequation,weobtainadistinctroot:,Therefore,thedesiredgeneralsolutionis,(multiplicity3/tripleroot),55,Example4.2.5C/P142,Findthegeneralsolutionof,Solution,ThecharacteristicequationofthegivenDEis,Bysolvingtheabovealgebraicequation,weobtaintwodistinctroots:,Therefore,thedesiredgeneralsolutionis,(multiplicity2/doubleroots),Thisisthecomplex-valuedgeneralsolution!,Howcanwefindthereal-valuedgeneralsolution?,56,Asdealingwithinthecase:ndistinctroots,wehavethefollowingresult,57,Re