Question

(D^2-D'^2-3D+3D')z=e^x-2y

Answer 1

There's some typing mistake in question, the correct equation is

$(D^2-3D)y=e^x-2y$

$\Rightarrow(D^2-3D+2)y=e^x$

Auxiliary Equation (A.E.) is $D^2-3D+2=0$

$\Rightarrow D^2-2D-D+2=0$

$\Rightarrow D(D-2)-(D-2)=0$

$\Rightarrow (D-1)(D-2)=0$

$\therefore \space D=1,2$

Complimentary Function (C.F) is

$y=c_1e^x+c_2e^{2x}$

For Particular Integral (P.I)

P.I $=\dfrac{1}{D^2-3D+2}e^x$

$=\dfrac{1}{(D-1)(D-2)}e^x$

$=-1\bigg[\dfrac{1}{(D-1)}e^x\bigg]$

$=-1.x.\dfrac{1}{1}e^x$

$=-xe^x$

Complete Solution (C.S) = C.F + P.I

$y=c_1e^x+c_2e^{2x}-xe^x$