I have the following differential equation I need to solve:
$x'(t) + 2x(t) = u(t)$
So here's how I proceed:
$F^{-1} (x'(t)) + 2 F^{-1} (x(t)) =F^{-1} ( u(t))$
$2\pi if \hat{x}(f) + 2\hat{x}(f) = \frac{1}{2} \delta(f) + \frac{1}{2 \pi if } $
$\hat{x}(f) (2 + 2 i \pi f) = \frac{1}{2} \delta(f) + \frac{1}{2 \pi if }$
$\hat{x}(f) = \frac{\delta(f)}{2 (2 + 2 i \pi f)} + \frac{1}{2\pi i f (2 + 2 i \pi f)}$
So now:
$x(t) = F^{-1} (\frac{\delta(f)}{2 (2 + 2 i \pi f)}) + F^{-1}(\frac{1}{2\pi i f (2 + 2 i \pi f)})$
And that's how far I got. I don't know how to proceed.