Question

Find (D² - 6D+9) y =e^3x​

Answer 1

Answer:

Complimentary function

Auxiliary equation (m-3)(m-3) =0 m=3,3 repeated root.

yc=(C1x+C2)e3x

Particular Integral

yp=1(D−3)2x3e3x

By operator shift theorem

1f(D)(eaxg(x))=eax1f(D+a)g(x)

accordingly yp=e3x1D2x3

Since D means differentiation and 1D means integration

yp=∫∫x3dxdx=x520

y=yc+yp=(C1x+C2)e3x+x520