I was wondering if perhaps someone here could give me a hand sorting out what's what between some different classes of diff. equations:

My basic understanding of a homogenous diff is that it can be written on a form of the type (for first order)

y'(x) + p(x)y(x) = 0

where being equal to zero was of prime concern. Recently though I've understood that this isn't a complete understanding at all. I recently head that the basic definition of a homogenous diff is in fact that you can express a dx/dy as a function of x/y; that is, the variables scale. And then you have the option of solving using a variable substitution, x/y = v for example.

So yea, currently I'm feeling a little bit lost in between linear, homogenous and linear homogenous. If anyone feel like giving a bit of an explanation, or perhaps link to some good source of disambiguation, as it were, I'd be very grateful!