= Solution
The conditions of regularity at $r=0$ and $R(a)=0$ select the <radial vibrations of a circular membrane> as an <eigenvalue problem>. Multiply $-(rR')'=\lambda rR$ by $\overline R$ and integrate. <Integration by parts> gives
$$
\lambda\int_0^ar|R|^2dr=\int_0^ar|R'|^2dr-[r\overline RR']_0^a=\int_0^ar|R'|^2dr.
$$
The boundary term vanishes by regularity and $R(a)=0$. For a nonzero eigenfunction the denominator is positive; the numerator could vanish only for a constant function, and the outer condition would then make that function zero. Consequently
$$
\boxed{\lambda=\frac{\int_0^ar|R'|^2dr}{\int_0^ar|R|^2dr}>0,\qquad \nu=\frac{c\sqrt\lambda}{2\pi}\in\mathbb R.}
$$
This also proves reality without presuming real-valued eigenfunctions. Explicitly, regular solutions are <Bessel functions of the first kind> $R=J_0(\sqrt\lambda r)$, and the positive zeros of $J_0(\sqrt\lambda a)$ give the discrete spectrum.
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