= Solution
Seek a nonzero separated solution $\psi=T(t)R(r)\Theta(\theta)$. Substitution into the <heat equation> and division by the product gives
$$
\frac{T'}{\kappa T}=\frac{(r^2R')'}{r^2R}+\frac{1}{r^2\Theta\sin\theta}(\sin\theta\,\Theta')'.
$$
The left side depends only on time, so set it to $-k^2$. Multiplication by $r^2$ then separates the angular dependence, with separation constant $\ell(\ell+1)$. The three <ordinary differential equations> are
$$
\boxed{T'+\kappa k^2T=0,}
$$
$$
\boxed{\frac1{\sin\theta}(\sin\theta\,\Theta')'+\ell(\ell+1)\Theta=0,}
$$
$$
\boxed{(r^2R')'+[k^2r^2-\ell(\ell+1)]R=0.}
$$
The temporal solution is $T=T_0e^{-\kappa k^2t}$. With $q=\cos\theta$, the angular equation is the <Legendre differential equation>; regularity at both angular endpoints selects $\ell=0,1,2,\ldots$ and $\Theta\propto P_\ell(q)$, the <Legendre polynomial>. The radial equation has the <Spherical Bessel functions> $j_\ell(kr)$ and $y_\ell(kr)$ for $k>0$. Regularity at $r=0$, if imposed, excludes the singular second solution. For $k=0$ the radial powers are $r^\ell$ and $r^{-\ell-1}$. Without specified radial <boundary conditions>, no discrete set of $k$ is selected; $-k^2$ is a convention for the temporal separation constant.
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