Find the solution to the following differential equation: $ \frac{dy}{dx} = \frac{x - y}{xy} $ The instructor in our Differential Equations class gave us the following to solve:
$$ \frac{dy}{dx} = \frac{x - y}{xy} $$
It was an item under separable differential equations. I have gotten as far as $ \frac{dy}{dx} = \frac{1}{y} - \frac{1}{x} $ which to me doesn't really seem much. I don't even know if it really is a separable equation.
I tried treating it as a homogeneous equation, multiplying both sides with $y$ to get (Do note that I just did the following for what it's worth)...
$$ y\frac{dy}{dx} = 1 - \frac{y}{x} $$
$$ vx (v + x \frac{dv}{dx}) = 1 - v $$
$$ v^2x + vx^2 \frac{dv}{dx} = 1 - v $$
$$ vx^2 \frac{dv}{dx} = 1 - v - v^2x$$
I am unsure how to proceed at this point.
What should I first do to solve the given differential equation?

I don't think it is a homogeneous ordinary differential equation. 
In order to be homogeneous, one should get the ODE in the form
$\frac{dy}{dx}=F\left(\frac{y}{x}\right)$, 
i.e the derivative of $y$ with respect to $x$ is some function of $\frac{y}{x}$. 
If yo try that for your equation, you get the form
$\frac{dy}{dx}=\frac{1-y/x}{y}$ 
which is clearly not a function of $y/x$ due to the presence of $y$ in the denominator.
