How to solve the following non-linear system?
$$ x^{2}=ax+by \\y^{2}=cx+dy $$
I am guessing of a substitution here, as in solving one equation in $u$, where $u$ is some linear combination of $x$ and $y$, but the different coefficients make it hard to make a guess for $u$.
from your first equation we get $$y=\frac{x^2-ax}{b}$$ for $b\ne 0$ plugging this in the second equation $$\left(\frac{x^2-ax}{b}\right)^2=cx+\frac{d(x^2-ax)}{b}$$ can you simplify this? and we get for $x$: $$x \left( {a}^{2}x+dba-2\,a{x}^{2}-c{b}^{2}-dbx+{x}^{3} \right) =0$$