Question

the complementary function of (D²+2D+1) y=sinax ​

Answer 1

Homogeneous equation:

Homogeneous equation characteristic polynomial: λ3+a2λ=0⇒λ1=0,λ2,3=±ai

The general solution for the homogeneous equation y=C1+C2cos(ax)+C3sin(ax)

A particular solution of the base equation: as the right part corresponds with λ=±ai the resonance occurs and we find our solution as yp=x(α1cos(ax)+α2sin(ax))=x⋅u(x)

Dyp=α1cos(ax)+α2sin(ax)+x(−aα1sin(ax)+aα2cos(ax))

D3yp=D3(xu)=(D3x)u+3(D2x)(Du)+3(Dx)(D2u)+xD3u=0+0+3(−a2α1cos(ax)−a2α2sin(ax))+x(−a3α2cos(ax)+a3α1sin(ax))

(D3+a2D)yp=a2(−2α1cos(ax)−2α2sin(ax))=sin(ax)

Thus α1=0,α2=−12a2

Thus finally y=−xsin(ax)2a2+C1+C2cos(ax)+C3sin(ax)