Determine the general solution of the differential equation
y′ = (y/2x) − (xy)^3 .
what method should i use for this
Rearranging, you get $\displaystyle \frac{\,dy}{\,dx}-\frac{1}{2x}y=-x^3y^3$
This can be recognized as a Bernoulli Equation.
Let $\displaystyle z=y^{1-3}=y^{-2}\implies z^{-\frac{1}{2}}=y$
Thus, $\displaystyle \frac{\,dy}{\,dx}=\frac{\,dy}{\,dz}\cdot\frac{\,dz }{\,dx}=-\tfrac{1}{2}z^{-\frac{3}{2}}\frac{\,dz}{\,dx}$
Making the appropriate substitutions, we end up with:
$\displaystyle -\tfrac{1}{2}z^{-\frac{3}{2}}\frac{\,dz}{\,dx}-\frac{1}{2x}z^{-\frac{1}{2}}=-x^{3}z^{-\frac{3}{2}}$ which becomes the linear equation $\displaystyle \frac{\,dz}{\,dx}+\frac{1}{x}z=2x^3$
Can you take it from here?