This seemingly easy analysis is driving me up the wall.
$$\frac{\text{d}^{2}u}{\text{d}\rho^{2}}=\left[1-\frac{\rho_{0}}{\rho}+\frac{l(l+1)}{\rho^{2}}\right]u$$
why is it for
$$\rho\rightarrow0,$$ we get $$\frac{\text{d}^{2}u}{\text{d}\rho^{2}}=\frac{l(l+1)}{\rho^{2}}u$$
The whole dam thing obviously goes to infinity!
Comparing the three terms on the right hand side (for $\rho\to0$), you see that $$1 \ll \frac{\rho_0}{\rho} \ll \frac{l (l+1)}{\rho^2}$$ in the sense that their ratio goes to 0. So for $\rho\to0$, the centrifugal term $l(l+1)/\rho^2$ dominates and we can neglect the other two.