When solving the hydrogen radial Schroedinger equation (with $r > 0$ the radial coordinate) for angular momentum $L=1$ and the modified radial wave function $P(r)=rR(r)$, $P(r)$ satisfies: $d^2P/dr^2=2(−1/r−E+1/r^2) P$, where $E$ is the eigenenergy to be found. One can re-write this as $1$st order system of equations (with $Q = dP/dr$):
\begin{align}
dP/dr &= Q \\ dQ/dr &= 2(−1/r−E+1/r^2)P \\ \end{align}
, with $P(0)=0$ and $Q(0)=0$.
Surprisingly, RK4 will output zero for all r > 0!
How to mend this? $P(r) \sim r^2$, for $r$ small. $E < 0$ is a parameter, usually of the order of unity.
$P( r ) = 0$ and $Q( r ) = 0$ are the solutions of the given equation for the initial values $P(0) = 0$ and $Q(0) = 0$. The problem is that the rate function is unbound for any solution starting anywhere but from zero (because of $1/r$ and $1/r^2$ terms). So, one cannot even invoke Cauchy theorem here, which means the problem is not well poised.