Let $l\in\mathbb N\cup\{0\}$, let $\mu>0$, and let $V:(0,\infty)\to\mathbb R$ be locally integrable. Suppose $\psi(r,\theta,\phi)=R(r)Y_{l,m}(\theta,\phi)$ is a separated eigenfunction of the central Hamiltonian $H=-(\hbar^2/2\mu)\Delta+V(r)$ on a domain imposing regularity at the origin, with $R$ twice differentiable on $(0,\infty)$ and $u=U_l(R)\in L^2(0,\infty)$. Then $u$ satisfies \begin{align*} -\frac{\hbar^2}{2\mu}u''(r)+\left(V(r)+\frac{\hbar^2l(l+1)}{2\mu r^2}\right)u(r)=Eu(r) \end{align*} on $(0,\infty)$ in the classical sense wherever $V$ is regular, and in the distributional sense otherwise. For the regular radial domain inherited from the three-dimensional Hamiltonian, the boundary condition at the origin is $u(0)=0$.