dy/dx - 2y = (4x)/(y^1/2)
I assumed that u = y^1/2 would be the proper substitution, but it didn't seem like things were going in the right direction.
Note that $\displaystyle \frac{\,dy}{\,dx}-2y=\frac{4x}{y^{\frac{1}{2}}}\implies \frac{\,dy}{\,dx}-2y=4xy^{-\frac{1}{2}}$,
The substitution is of the form $\displaystyle z=y^{1-n}$, so we see in this case, the proper substitution would be $\displaystyle z=y^{\frac{3}{2}}$
Try to take it from here. This should lead you to the desired result.
See here for an example on how to solve a Bernoulli equation (see post #3)
I hope this helps!
--Chris