The spherically symmetric bound state wavefunctions $\psi(r)$, where $r=|\mathbf{x}|$, for an electron orbiting in the Coulomb potential $V(r)=-e^{2} /\left(4 \pi \epsilon_{0} r\right)$ of a hydrogen atom nucleus, can be modelled as solutions to the equation

$$\frac{d^{2} \psi}{d r^{2}}+\frac{2}{r} \frac{d \psi}{d r}+\frac{a}{r} \psi(r)-b^{2} \psi(r)=0$$

for $r \geqslant 0$, where $a=e^{2} m /\left(2 \pi \epsilon_{0} \hbar^{2}\right), b=\sqrt{-2 m E} / \hbar$, and $E$ is the energy of the corresponding state. Show that there are normalisable and continuous wavefunctions $\psi(r)$ satisfying this equation with energies

$$E=-\frac{m e^{4}}{32 \pi^{2} \epsilon_{0}^{2} \hbar^{2} N^{2}}$$

for all integers $N \geqslant 1$.