If I have an equation I want to solve, such as $tx''+t^2x'-3x=0$ using $x(0)=0$, how can I easily reason what the Laplace of the terms multiplied by $t$ would be?
Can I do the following:
$\mathcal{L}\{x''(t)\} = s^2X(s)+x'(0)$
$\mathcal{L}\{x'(t)\} = sX(s)$
So we have:
$-[s^2X(s)+x'(0)]' + [sX(s)]'' -3X(s) = 0$