I'm having a bit of trouble with a homogeneous ODE:

$\displaystyle \frac{dy}{dx} = \frac{x^2 + 3y^2}{2xy}$

In order to solve this by substituting v = y/x, I first need to manipulate the right side to get all variables in the form of y/x. However, when I simplify it, I get:

$\displaystyle \frac{dy}{dx} = \frac{x}{2y} + \frac{3y}{2x}$

...which leaves each fraction opposite of the other. What am I missing?