I'd like to try using a Bilateral laplace transform for a function where I know the solution is valid over some negative time $[-t,t].$
Just as practice, I start with $y'(t) = e^t y(t),$ then take the bilateral laplace transform $B$ of both sides to obtain $$sF(s) = F(s-1)$$ according to some rules I found on wikipedia.
But the solution to this functional equation is the gamma function, $F(s) = c/\Gamma(s+1)$ for some arbitrary constant $c.$ How do I apply the reverse bilateral transform here to arrive at a solution?