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Suppose you have this nomhomogeneous ODE:

$$ Ay''+2By'+Cy= F\cos(Gx+H) $$

where $A, B, C, F, G ,H$ are real numbers.

The homogeneous part of it is trivial, but I can't figure out how to solve for the nonhomogeneous part.

I have the solution: $$Y_{NH}(x) = \frac{F(C-AG^2)}{(C-AG^2)^2+4B^2G^2}\cos(Gx+H)\nonumber \\ +\frac{2FBG}{(C-AG^2)^2+4B^2G^2}\sin(Gx+H)$$

But I cannot see how to reach this result, could anyone give a hand here? Thanks!

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  • $\begingroup$ You can use Laplace's transform to get to the result. I think it would be the most straightforward approach. $\endgroup$ – Jakobian Jul 8 '18 at 13:58
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The method of undetermined coefficients tells you that you can find a particular solution in the form $$ y_{NH}(x)=K\cos(Gx+H)+L\sin(Gx+H) $$ as long as $\pm iG$ is not a characteristic root of the left side, that is, not both of $B=0$ and $AG^2-C=0$ are true simultaneously.

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