= Solution
For a spherically symmetric function, the <Laplacian in spherical coordinates> reduces to
$$
\nabla^2\psi=\frac1{r^2}\frac{d}{dr}\left(r^2\frac{d\psi}{dr}\right).
$$
Multiplying the <Helmholtz equation> by $r^2$ gives
$$
\boxed{-\frac{d}{dr}(r^2\psi')=k^2r^2\psi,\qquad p(r)=r^2,\quad q(r)=0,\quad w(r)=r^2.}
$$
The endpoint at $r=0$ is singular in Sturm-Liouville terminology; bounded regular radial fields select the physical solution and eliminate the $1/r$ branch. At $r=a$ the field has the imposed zero Dirichlet value. For regular solutions and admissible trial functions, the endpoint term $r^2\psi\psi'$ vanishes at the origin and at the wall.
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