Homogeneous differential equations

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

Find the general solution of the equation .

So that'd get...

In my lecture notes I have written...

How do we solve this type of equation?

Given

, set

i.e.

, so

or

which is a separable equation.

where

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: and are derivatives with respect to x.