I am trying to solve the following DFQ: $y''-2y'-3y=-3te^{-t}$.
I solved the homogeneous equation to find that the general solution is $y(t)=c_1e^{3t}+c_2e^{-t}$, and therefore a good guess for the particular equation would be $Y_p(t)=Ate^{-t}$b because it is not proportional to $e^{3t}$ or $e^{-t}$. So, we get $Y_p'(t)=A e^{-t}-A e^{-t} t$ and $Y_p''(t)=A e^{-t} t-2 A e^{-t}$. But, when we plug $Y_p$ into $y''-2y'-3y=3te^{-t}$ we get:
\begin{align*}
Ate^{-t} -2Ae^{-t} +2Ate^{-t} -2Ae^{-t} -3Ate^{-t} &= -3te^{-t}\\
-4Ae^{-t} &= -3te^{-t}
\end{align*}
So, $Y_p$ turns out to be a bad guess. Why is it that $Y_p$ turns out to be a bad guess? I thought that it met all of the criteria, unless I am missing something.
Best Answer
I think the general rule (if you look up Stewart or some other "standard" book) is you need a higher order term than what appears on the right hand side. So in your case we should try $$At^{2}e^{-t}+Bte^{-t}$$I hope this helps as I do not have time to do the computation.