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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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A related problem. –  Mhenni Benghorbal Feb 28 at 3:19
Please do not change the question after it has been answered. –  Joel Reyes Noche Feb 28 at 3:33

1 Answer 1

up vote 2 down vote accepted

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?

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Yes, thank you. –  Ali Mubashir Feb 28 at 2:19
Glad to help, interesting question. –  James S. Cook Feb 28 at 2:25
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 at 3:32
@JoelReyesNoche indeed, it is for your implicit concern that I added the first two lines to my answer. –  James S. Cook Feb 28 at 13:32

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