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I'm trying to find the solution to this non-homogenous third-order linear differential equation.

I know the solution is supposed to be:

$$c_1e^t+c_2te^t+c_3^{-2t}+\frac{e^tt^2}{6}-\frac{\sin(t)}{5}+\frac{\cos(t)}{10}$$

So far I've solved the left side of the equation to get the first half of the answer:

$$c_1e^t+c_2te^t+c_3^{-2t}$$

I don't know how to get the solutions from the right-hand side though. Thanks.

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  • $\begingroup$ Have you ever encountered the method of undetermined coefficients? $\endgroup$ – JohnK Oct 26 '13 at 20:22
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You can try the method of undetermined coefficients by setting $y_p= Ae^t+Bte^t+Ct^2e^t+D\sin t+E\cos t$. You take the derivatives of $y_p$ and substitute into the equation. By equating the coefficients of same terms you can determine constants $A, B, C, D$ and $E$.

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    $\begingroup$ Not quite - you're missing a $t^2$ term.. $\endgroup$ – nbubis Oct 26 '13 at 20:27
  • $\begingroup$ Thank you nbubis I edited it. I supposed $r=1$ is the single root of the caharacteristic equation. $\endgroup$ – daulomb Oct 26 '13 at 20:32

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