We introduce a dimensionless variable \( \rho = (1/\alpha) r \) where \( \alpha \) is a constant with dimension length and get $$ -\frac{\hbar^2}{2 m \alpha^2} \frac{d^2}{d\rho^2} u(\rho) + \left ( V(\rho) + \frac{l (l + 1)}{\rho^2} \frac{\hbar^2}{2 m\alpha^2} \right ) u(\rho) = E u(\rho) . $$ Let us specialize to \( l=0 \). Inserting \( V(\rho) = (1/2) k \alpha^2\rho^2 \) we end up with $$ -\frac{\hbar^2}{2 m \alpha^2} \frac{d^2}{d\rho^2} u(\rho) + \frac{k}{2} \alpha^2\rho^2u(\rho) = E u(\rho) . $$ We multiply thereafter with \( 2m\alpha^2/\hbar^2 \) on both sides and obtain $$ -\frac{d^2}{d\rho^2} u(\rho) + \frac{mk}{\hbar^2} \alpha^4\rho^2u(\rho) = \frac{2m\alpha^2}{\hbar^2}E u(\rho) . $$