I'm having trouble solving the following exercise:
If $f(x,y)=e^{-r^{2}}$ with $r=\sqrt{x^2+y^2}$, show that it exists a function $g(r)$ such that:
$$\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}=g(r)f(x,y)$$ and determine the function $g(r)$.
When I saw this, because of the $r=\sqrt{x^2+y^2}$, the first thing that came to my mind was that maybe this would be easier if we use polar coordinates, But I don't know how to approach the problem. How can I solve this?