常微分方程课件-第11讲

Lecture11,1,2,2,5.2GeneralTheoryofLinearSystemsofDifferentialEquations,5.2.1HomogeneousLinearSystemsofDifferentialEquations(齐次线性微分方程组),5.2.2NonHomogeneousLinearSystemsofDifferentialEquations(非齐次线性微分方程组),3,3,DefinitionC/P206,ExampleCfE/P349,4,4,DefinitionC/P206,PropositionCfC/P125/Theorem6,5,5,6,6,DefinitionC/P207(Solutionmatrixandfundamentalsolutionmatrix解矩阵和基解矩阵),7,7,Theorem1*C/P208,8,8,Theorem2*C/P208,9,9,Example5.2.1C/P208,10,10,Corollary1*C/P209,Question,Proof:,11,11,Corollary2*C/P210,Proof:,12,12,12,5.2GeneralTheoryofLinearSystemsofDifferentialEquations,5.2.1HomogeneousLinearSystemsofDifferentialEquations(齐次线性微分方程组),5.2.2NonHomogeneousLinearSystemsofDifferentialEquations(非齐次线性微分方程组),13,13,5.2GeneralTheoryofLinearSystemsofDifferentialEquations,5.2.1HomogeneousLinearSystemsofDifferentialEquations(齐次线性微分方程组),5.2.2NonHomogeneousLinearSystemsofDifferentialEquations(非齐次线性微分方程组),14,14,5.1.2NonHomogeneousLinearSystemsofDifferentialEquations(非齐次线性微分方程组),Property1C/P211;CfC/P127/Property1,Property2C/P211;CfC/P127/Property2,15,15,Theorem7C/P211(Structureofsolution(解的结构),Note:Thistheoremtellsus,inordertofindanysolutionof(5.14),weneedonlytofindaparticularsolutionof(5.14)andafundamentalsolutionmatrixof(5.15)!C/P211;E/P353+354,16,16,Infact,asdoneinC/PP127-129forthehigher-orderlinearnonhomogenerousDE,ifwehaveknownafundamentalsolutionmatrix(orafundamentalsystemofsolutions)of(5.14),wemayalsousethemethodofvariationofconstantstofindaparticularsolutionof(5.14)!,17,17,18,18,(5.24),19,19,20,20,Theorem8C/P212,RemarkC/P212,21,21,22,22,Example5.2.2C/P213,Solution:,23,23,Corollary3C/P214(theformulaofvariationofconstantsfornthordernonhomogeneouslinearDEcf.C/PP127-129(n阶非齐次线性微分方程的常数变易公式),24,24,Example5.2.3C/P215;E/P155,Solution,25,25,Example5.2.3C/P215;E/P155,Solution,26,26,26,Question:,Fromtherelationshipbetweenthehigh-orderDEandthesystemofDesinC/P191,doyouunderstandtheequations(4.17)inC/PP128-129whichwetook?,Example5.2.3C/P215;E/P155,LinearSystemsofDifferentialEquations,(线性微分方程组),ChapterV(第五章),5.1ExistenceandUniquenessTheorem(存在唯一性定理),5.2GeneralTheoryofLinearSystemsofDifferentialEquations(线性微分方程组的一般理论),5.3LinearSystemsofDifferentialEquationswithconstantcoefficients(常系数线性微分方程组),27,LinearSystemsofDifferentialEquations,(线性微分方程组),ChapterV(第五章),5.1ExistenceandUniquenessTheorem(存在唯一性定理),5.2GeneralTheoryofLinearSystemsofDifferentialEquations(线性微分方程组的一般理论),5.3LinearSystemsofDifferentialEquationswithconstantcoefficients(常系数线性微分方程组),28,5.3.1DefinitionandpropertiesofexponentialmatrixexpA(矩阵指数expA的定义和性质),5.3.2Computationalformulaoffundamentalsolutionmatrix(基解矩阵的计算公式),5.3LinearSystemsofDifferentialEquationswithconstantcoefficients(常系数线性微分方程组),29,5.3.1DefinitionandpropertiesofexponentialmatrixexpA(矩阵指数expA的定义和性质),5.3.2Computationalformulaoffundamentalsolutionmatrix(基解矩阵的计算公式),5.3LinearSystemsofDifferentialEquationswithconstantcoefficients(常系数线性微分方程组),30,5.3.1DefinitionandpropertiesofexponentialmatrixexpA(矩阵指数expA的定义和性质),31,32,Example5.3.1E/P403,33,Solution:,34,ExerciseCfE/P404/Example3,35,Solution:,ExerciseCfE/P404/Example3,RemarkE/P404,36,ExerciseCfE/P404/Example3,Question,37,38,39,39,Exercise,40,40,40,41,42,43,Theorem9C/P221,44,Proof:,45,Theorem9C/P221,45,Corollary,Question1,46,47,Quest