Q:

Differential Equation $ (2x^2 + y^2)\,dx - xy \, dy = 0 $

Solve $(2x^2 + y^2)\,dx - xy \, dy = 0$
Attempted :
The equation is not exact because $ M_y  \ne N_x $ for $ M = 2x^2 + y^2 $ and $ N = xy$
Or is it exact?  
The equation is also not separable. 
The equation is also not homogenous, I don't think. 
So.. what do I do? 
Thanks. 

A:

Hint: Divide by $xy$ and put $u = y/x$. The equation will then become separable. 

To elaborate, divide by $xy$ to get:
$$
y' = \frac{2x}{y} + \frac{y}{x}
$$
Put $u = y/x$, $y' = xu' + u$:
$$
xu' + u = \frac{2}{u} + u
$$
Rearrange to get:
$$
uu' = \frac{2}{x}
$$
Integrate both sides, solve for $u$ and put back $u = y/x$ to get the solution in terms of $y$.

