Homogeneous 1st Order Differential Equations

Question

I have come across two different definitions of a homogeneous 1st order ODE; that the equation can be written in the form $y' = f(\frac{y}{x})$, and that in the form $M(x, y)dx + N(x, y)dy = 0$, it is the case that $\frac{M(tx, ty)}{M(x, y)} = \frac{N(tx, ty)}{N(x, y)} = t^n$. Is there an easy way to see why these criteria imply each other, if they do? Also, since there is a one-to-infinitely-many relationship between an equation $y' = f(\frac{y}{x})$ and corresponding $M(x, y)dx + N(x, y)dy = 0$, is the property that $\frac{M(tx, ty)}{M(x, y)} = \frac{N(tx, ty)}{N(x, y)} = t^n$ invariant under different $M(x, y)dx + N(x, y)dy = 0$ that correspond to the same $y' = f(\frac{y}{x})$?