How to find a solution to this PDE? 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?

What kind of solution are you looking for? It is possible to find many explicit solutions for any given value of the constant $c$.


*

*Solutions that depend only on $x$. Depending on the sign of $c$ this gives $u=A\,\sin(\sqrt{c}\,x)+B\,\cos(\sqrt{c\,}x)$ (if $c>0$), $u=A\,\sinh(\sqrt{-c}\,x)+B\,\cosh(\sqrt{-c\,}x)$ (if $c<0$), $u=A\,x+B$ (if $c=0$).

*Solutions that depend only on $y$: change $x$ by $y$ in the above.

*Solutions of the form $X(x)Y(y)$. This leads to
$$
-\frac{X''}{X}=\frac{Y''}{Y}+c=\text{constant}
$$
Form here you can get solutions like $u=\cos(\lambda\,x)\sin(\sqrt{c-\lambda^2}\,y)$ if $0<\lambda^2<c$.

*Radial solutions, which leads to Bessel functions.

*Linear combinations of the above.

