I'm having trouble solving the following equation:
$\ddot x + \omega^2 x = a \sin \Omega t$, where $t >0$, $\omega \neq \Omega$, $x(0+) = 0 = \dot x(0+)$.
It is asked to be done by Fourier transform. If I use
$\mathcal{F}\{f^{(2)} \} = (i \omega)^2 \mathcal{F}\{f\} $
on the interval $(0, \infty]$, I end up with zero on the left side of the equation. So I don't think I can use this since I don't think $x$ or $\dot x$ tends to $0$ as $t$ goes to infinity. Am I missing something about $x$ that is implied without saying? At least I think it should be reasonable to guess that $x$ should be a trigonometric function. Any hints are very appriciated.
Thanks.