Q:

Equation development

I would like to know if it's possible to separate $x$ and $y$ variables in this equations:
$$
\frac{ax^2 + bxy}{cx+dy+ex^2+fy^2+gxy} = 0
$$
I would somethink like: $y = f(x)$. I tried many times, but without any results. Is it only possible?
Thanks

A:

No. Assume you could, and denote your expression (on the lhs) by $r(x,y)$. Then you would get $$f(x)+g(y) =r(x,y)$$ for some rational polynomials $f$ and $g$ depending only on $x$ and $y$. In particular, the product of the denominators of $f$ and $g$ must be a multiple of the denominator of $r(x,y)$. WLOG we can assume it is exactly it. But then, this gives us a factorization of $$k(x,y):=cx +dy +ex^2 +fy^2 +gxy,$$ which is irreducible, because it defines a general conic. Contradiction.

