There exists a popular model in the Physics of heavy quark bound systems, called the Cornell potential model, in which the inter-quark potential is modeled to vary with radial distance $r$ as
$$V(r) = - \frac{\kappa}{r} + \frac{r}{a^2}$$
The mathematical problem is reduced to solving the radial part of the Schrodinger equation
$$-\frac{\hbar^2}{2m} \frac{d^2 u}{dr^2} + \left[ V(r) + \frac{l(l+1)}{2mr^2}\right]u = Eu$$
for the above potential, hence obtaining the reduced wavefunction $u(r)$.
The reduced wavefunction $u(r)$ is subject to the boundary conditions $u(r=0) = 0$, and $u'(r=0) = R(0) =$ some number $C$.
Each of the constants $\hbar$, $m$, $\kappa$, and $a$ are positive numbers.
$l$ serves the role of azimuthal quantum number, and can take non-negative integer values, i.e. $l = 0, 1, 2, 3 \ldots$ However, this is generally fed in as a constant input, and choice of $l$ segregates solutions into various categories, e.g. $l=0$ states are the $s-$wave states, $l=1$ are the $p-$wave states, and so on.
The energy eigenvalue $E$ can, in general, be either positive (called a Scattering State solution), or negative (called a Bound State solution); however, in this case, we are concerned with the latter variety ($E<0$) since the model caters precisely to bound heavy-quark systems.
Scanning across the literature, one only finds (abundantly) numerical/iterative approaches applied to this mathematical problem, and Wolfram Alpha also succumbs before it.
My questions are:
Is this physical model exactly solvable, i.e. is there any general solution to this differential equation for $u(r)$, for any $E<0$?
If not, what is the exact problem with this potential which hinders reaching an exact solution. (I say that because the singularity at $r=0$ is due to inverse $r$ part, but if the second term was missing from the potential, then the pure inverse $r$ potential is known to be exactly solvable, it is a textbook problem taught in all undergraduate Physics courses.)
Aside from numerical/iterative approaches, if there is any alternative method which can be used for this purpose, or can be used in circumventing this difficulty?