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


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

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?

This homogeneous refers to if both coefficient functions M and N are of the same degree.
$\displaystyle \displaystyle M(x,y)dx+N(x,y)dy=0$
Homogeneous means the same; hence, the degrees have to be the same. However, this isn't the same as a homogeneous equation though.