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?
 A: 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.

A: Take $r = t^2$ and look up Bessel's equation.
A: Looks like you are trying to solve the homogeneous Helmholtz equation in 2 dimensions.  Following the method described here [ http://en.wikipedia.org/wiki/Helmholtz_equation#Solving_the_Helmholtz_equation_using_separation_of_variables ], you will get a solution
$$
u(r,\theta)= \sum_{n=1}^\infty J_n( \sqrt{c} r) \left[ c_{n} \sin n\theta + d_{n}\cos n\theta \right]
$$
