# Non homogeneous differential equations

What is the differential equation for $$y=c_1e^{2x}\sin x + c_4 e^{2x} \cos x - xe^{-x}.$$ I'm not sure how to incorporate the last part of the solution into solving this problem?

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– Mhenni Benghorbal Feb 28 '14 at 3:19

Question: find differential equation for which $y = c_1+c_2x+c_3e^{2x}\cos(x)+c_4e^{2x}\sin(x)-xe^{-x}$ forms the general solution.
Solution Sketch: Well, it is fourth order with $\lambda = 0,0,2\pm i$ and so $L=D^2((D-2)^2+1)$ and $L[y]=g$ is the differential equation. Now the question is what is $g$ if $L[-xe^{-x}]=g$? I think you can figure that out if you can differentiate. Yes?
This answer refers to the original version of the question. For some reason, the OP changed the question (removed the first two terms in $y$). – Joel Reyes Noche Feb 28 '14 at 3:32