= Solution
Set $\psi(r)=u(r)/r$ for $r>0$. Direct differentiation gives $(r^2\psi')'=ru''$, so the radial equation reduces to
$$
u''+k^2u=0.
$$
Boundedness of $\psi$ requires $u(0)=0$, and the wall condition requires $u(a)=0$. The nontrivial solutions are consequently
$$
\boxed{k_j=\frac{j\pi}{a},\qquad\psi_j(r)=C_j\frac{\sin(j\pi r/a)}r,\qquad j=1,2,\ldots.}
$$
Their origin values are the removable limits $C_jj\pi/a$. At $k=0$, the only bounded radial solution is constant and the wall forces it to vanish, so there is no zero mode. The lowest <eigenvalue> is
$$
\boxed{k_0^2=\frac{\pi^2}{a^2}\simeq\frac{9.86960}{a^2}.}
$$
It is below $10.5/a^2$, as required by the <Rayleigh-Ritz variational principle>. The trial value overestimates $k_0^2$ by approximately $\boxed{6.39\%}$.
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