Can someone please help me solve the Poisson-Boltzmann equation in spherical coordinates over the domain $r\in(r_0, \infty)$ with Runge-Kutta:
$$ \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial\psi}{\partial r}\right) =a \sinh \left(\frac{\psi}{\psi_0}\right) $$
here $r$ is the radial coordinate, $a$ and $\psi_0$ are constants. The boundary conditions are $\psi=\psi_0$ and $\psi'=b$ at $r=r_0$.
I made an attempt to put the equation into a form suitable for Runge-Kutta integration with a $u$-substitution $u/r=\psi$, yielding
$$ \frac{\partial^2u}{\partial r^2}=a r\sinh\left(\frac{u}{r\psi_0}\right) $$ And, following from this the Runge-Kutta method can be applied $$ \frac{\partial}{\partial r}\begin{bmatrix} u \\ \partial u/\partial r \\ \end{bmatrix}=f\left(u, \frac{\partial u}{\partial r}, r\right)=\begin{bmatrix} \partial u/\partial r \\ ar\sinh\left(\frac{u}{r\psi_0}\right) \end{bmatrix} $$
This seems to work, but my question is: Is there a more direct way of applying Runge-Kutta without such a substitution?