Answer
$y(x)=c_1e^{2x}+c_2 e^{-x}-5+6x-2x^2$
Work Step by Step
The general solution for the given differential equation is: $y(x)=c_1e^{2x}+c_2 e^{-x}+(A_0+A_1x+A_2x^2)$
The trial solution for $y_p= A_0+A_1x+A_2x^2$ can be computed as by plugging back into the given differential equation.
So, we have: $(D-2)(D+1)y_p(x)=4(x^2-2x)\\ D^2-D-2(A_0+A_1x+A_2x^2)=4(x^2-2x)\\ (2A_2-A_12A_0)+(-2A_2-2A_1)x-2A_2x^2=4x^2-8x $
On comparing co-efficients, we get: $A_1=6; A_2=-2$ and $A_0=-5$
Therefore, the general solution for the given differential equation is: $y(x)=c_1e^{2x}+c_2 e^{-x}-5+6x-2x^2$
