1. Concept of Homogeneity

I'm reading through this proof that any homogeneous equation $y' = f(x,y)$ can be rewritten as $y' = g(y/x)$. It says that, by homogeneity, $f(x,y) = f(tx, ty)$ for all real $t$ (I think it should have the proviso that $t \ne 0$, but it doesn't.). It then claims--and here's the part I don't get--that an appropriate choice of $t$ may be $1/x$. I thought $t$ was some real number, not a real-valued function. Am I missing something important here, or should I just take homogeneity to mean something like $f(x, y) = f(tx, ty)$ for any function $t(x,y)$?

2. Nuthin?

3. I think you should have $t^{\alpha}f(x,y) \ \alpha\in\mathbb{R}$

4. The book starts with what I gave but then amends it later in a way similar to what you have. So no clue about exactly what homogeneity means?

5. This homogeneous refers to if both coefficient functions M and N are of the same degree.

$\displaystyle M(x,y)dx+N(x,y)dy=0$

Homogeneous means the same; hence, the degrees have to be the same. However, this isn't the same as a homogeneous equation though.