Consider the heat equation in a radially symmetric sphere of radius unity:

$\displaystyle u_t = u_{rr}+{2 \over r}u_r \ for \ (r,t) \in (0,1) x (0,\infty)$

with boundary conditions $\displaystyle \lim_{r \rightarrow 0}u(r,t) < \infty ; \ u(1,t)=0\ for \ t >0$

Now, using separation of variables $\displaystyle u=R(r)T(t)$ leads to the eigenvalue problem $\displaystyle rR''+2R'-\mu rR=0 \ \ with \ lim_{r \rightarrow 0}R(r) < \infty \ and \ R(1)=0$

Then using the change of variable $\displaystyle X(r)=rR(r) $this becomes $\displaystyle X''-\mu X = 0$

Now to find all the eigenvalues I need to know what the boundary conditions are, and from the info above, how do I determine the sets of boundary conditions to use to find the eigenvalues $\displaystyle \mu_n$??