The equation is $$ \Delta u+cu=0 $$
on the $\mathbb{R}^2$ plane, where $c$ is a constant. My purpose is to find a suitable constant to get a solution of this PDE.
My idea is to let $u(x,y)=g(x^2+y^2)$, then the equation turns into the following ODE:
$$ 4rg''(r)+4g'(r)+cg(r)=0 $$
So I set $c=4$ and try to solve the ODE
$$ rg''(r)+g'(r)+g(r)=0 $$
However, I failed to solve it.
Can anyone help me?