Homogeneous differential equations

I have never understood this. The question I have is...

Find the general solution of the equation $\displaystyle (x-y)\frac{dy}{dx}=x+y$.

So that'd get...

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

In my lecture notes I have written...

How do we solve this type of equation?

Given $\displaystyle y'=F(\frac{y}{x})$, set $\displaystyle \frac{y(x)}{x}=v(x)$

i.e. $\displaystyle y(x)=x v(x)$, so $\displaystyle v+xv'=F(v)$

or $\displaystyle xv'=F(v)-v$ which is a separable equation.

$\displaystyle \int\frac{dv}{F(v)-v}=\int\frac{dx}{x}$ where $\displaystyle x\neq 0; F(v)-v\neq 0$

I understand that that equation is separable and stuff, but the stuff before I just don't get at all. Like where F(v) suddenly comes from and that. Can anyone help? (Crying)

Note: $\displaystyle y'$ and $\displaystyle v'$ are derivatives with respect to x.