The task was - to solve following system of ODE using Laplace transform at $[0,0.3]$:
$$ y' = e^{-(t^2+z^2)} + 2t; $$ $$ z' = 2y^2 + z; $$ $$ y(0) = 0.5; z(0) = 0$$
Usual algorithm is to apply Laplace transform on two equations and then solve the system for new variables. I stuck in the first step: how to apply Laplace transform on the first equation. I know that Laplace transform on $e^{-t^2}$ is equal to $1/2 \sqrt{\pi} e^{s^2/4} erfc(s/2)$. But how to deal with $e^{-z^2}$?