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Higher Order Linear Differential Equations

Higher OrderLinearDifferentialEquationsMath 240 Linear DELineardifferentialoperatorsFamiliar stuffExampleHomogeneousequationsHigher Order Linear Differential EquationsMath 240 Calculus IIIS ummer 2015, Session IITuesday, July 28, 2015 Higher OrderLinearDifferentialEquationsMath 240 Linear DELineardifferentialoperatorsFamiliar stuffExampleHomogeneousequationsAgenda1. Linear Differential Equations of ordernLinear Differential operatorsFamiliar stuffAn example2. homogeneous constant-coefficient Linear differentialequationsHigher OrderLinearDifferentialEquationsMath 240 Linear DELineardifferentialoperatorsFamiliar stuffExampleHomogeneousequationsIntroduc tionWe now turn our attention to solvinglinear differentialequations of general form of such an equation isa0(x)y(n)+a1(x)y(n 1)+ +an 1(x)y +an(x)y=F(x),wherea0, a1, .. , an,andFare functions defined on general strategy is to reformulate the above equation asLy=F,whereLis an appropriate Linear fact,Lwillbe alinear Differential OrderLinearDifferentialEquationsMath 240 Linear DELineardifferentialoperatorsFamiliar stuffExampleHomogeneousequationsLinear Differential operatorsRecall that the mappingD:Ck(I) Ck 1(I)defined byD(f) =f is a Linear transformation.

Homogeneous equations The general solution If we have a homogeneous linear di erential equation Ly = 0; its solution set will coincide with Ker(L). In particular, the kernel of a linear transformation is a subspace of its domain. Theorem The set of solutions to a linear di erential equation of order n is a subspace of Cn(I). It is called the ...

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