I try to understand why according to my notes from some lecture the following equality is satisfied
$$ \mathcal{L}\left[ {\exp\left(\int_0^t ds \frac{\dot{x}(s)}{x(s)} \right)} \right] =\frac{1}{u-\frac{\mathcal{L}\left[\dot{x}\right](u)}{\mathcal{L}[x](u)}}, $$
where $\mathcal{L}[\cdot]$ denotes Laplace transform. Is this equation fulfilled for all $x(t)$, and how one can derive it?
The key point is to note, that
$\frac{d}{dt} \ln x(t)= \frac{\dot{x}(t)}{x(t)}$,
accordingly
$ \exp(\int\limits_0^t ds \frac{\dot{x}(s)}{x(s)})=\exp(\int\limits_0^t ds \frac{d}{ds} \ln x(s))=\frac{x(t)}{x(0)}$.
The rest can be obtained by using the formula for the Laplace Transform of the derivative.