Question:
Find the function $f(t)\neq 0$ such that for all $g(t)$ we have $L\{\frac{d}{dt}(f*g)\}=L\{f\}$
Note: $L$ stands for Laplace transform.
My try:
$F_1(s):=L\{f*g\}=L\{f\}\times L\{g\}$
$\implies L\{\frac{d}{dt}(f*g)\}=s\times L\{f*g\}-(f*g)(0)=s \times L\{f\} \times L\{g\}-(f*g)(0)=L\{f\}$
I'm not sure, but if $(f*g)(0)=0$, then we have:
$$L\{f\}(sL\{g\}-1)=0$$
At this point, It seems there is nothing to do!
Any idea? Thanks