We have thus
$$
-\frac{d^2}{d\rho^2} u(\rho)
+ \frac{mk}{\hbar^2} \alpha^4\rho^2u(\rho) = \frac{2m\alpha^2}{\hbar^2}E u(\rho) .
$$
The constant \( \alpha \) can now be fixed
so that
$$
\frac{mk}{\hbar^2} \alpha^4 = 1,
$$
or
$$
\alpha = \left(\frac{\hbar^2}{mk}\right)^{1/4}.
$$