homogeneous differential equations $y' = f(y/x)$ There is a weird Theorem that comes about when considering whether a function is homogeneous (in the sense of the title definition).
I was unable to prove it, or to find a proof to it.
Can any one help?
If an equation is of the form $$f(x,y) = \frac{x^{a_1}y^{b_1} + x^{a_2}y^{b_2} \cdots}{x^{c_1}y^{d_1} + x^{c_2}y^{d_2} \cdots}$$  
And we know that:
$$\forall i,j\in \Bbb N \colon \quad a_i+b_i = c_j + d_j $$
Then $$f(x,y) = \frac{(\frac{y}{x})^{b_1} + (\frac{y}{x})^{b_2} + \cdots}{(\frac{y}{x})^{d_1} + (\frac{y}{x})^{d_2} + \cdots}$$
Which is then easier to solve. Now as mentioned this came with no proof, I would like to see one, or at least to get a general idea of why this might be true. 

Rewrite
$$
f(x,y) = \frac{x^{a_1}y^{b_1} + x^{a_2}y^{b_2} \cdots}{x^{c_1}y^{d_1} + x^{c_2}y^{d_2} \cdots}
$$
as
$$
f(x,y) = \frac{x^{a_1} x^{b_1}(y/x)^{b_1} + x^{a_2}x^{b_2}(y/x)^{b_2} \cdots}{x^{c_1}x^{d_1}(y/x)^{d_1} + x^{c_2}x^{d_2}(y/x)^{d_2} \cdots}
$$
by multiplying monomials upstairs and downstairs by a suitable power of $x$,
and then simplify upstairs and downstairs (as $x^{a_1+b_1}=x^{c_1+d_1}=x^{c_2+d_2}=x^{a_2+b_2}$ etc).
