Utility of the Derivative of Laplace Transforms for ODE's

Question

Many texts discuss the derivative of Laplace transform $dF(s)/ds$. In general, differentiation of a Laplace is equivalent to multiplying the original function by $t$, and vice versa. So, if $\mathscr{L}\{f(t)\} = F(s)$, then: $$\mathscr{L}\{tf(t)\}=-\frac{dF(s)}{ds} $$ Similarly, if $\mathscr{L}\{f'(t)\} = sF(s) - f(0)$, then: $$\begin{split}\mathscr{L}\{{tf'(t)}\}&=-\frac{d}{ds}\left\{sF(s) - f(0)\right\}\\ &=-s\frac{dF(s)}{ds} - F(s)\end{split}$$ The problem (my problem) with the last equation is that a derivative with non-constant coefficient $(tf'(t))$ yielded a derivative with non-constant coefficient $(sF'(s))$. How then would this transform be useful? Are there any examples of the utility of this property of Laplace transforms.

Answer 1

This can be used to solve differential equations having a derivative term multiplied by $t$. e.g. : $$t\frac{\mathrm dy}{\mathrm dt}=t.$$