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

I know the solution is supposed to be:


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


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

  • $\begingroup$ Have you ever encountered the method of undetermined coefficients? $\endgroup$
    – JohnK
    Oct 26, 2013 at 20:22

1 Answer 1


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 $E$.

  • 3
    $\begingroup$ Not quite - you're missing a $t^2$ term.. $\endgroup$
    – nbubis
    Oct 26, 2013 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, 2013 at 20:32

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