Ordinary Differential Equations/Higher Degrees

$(\frac{dy}{dx}{)}^n+F_1(x,y)(\frac{dy}{dx}{)}^{n-1}+...+F_{n-1}(x,y)(\frac{dy}{dx})+F_n(x,y)$

can theoretically be factored into

$(\frac{dy}{dx}-r_1(x,y))(\frac{dy}{dx}-r_2(x,y))...(\frac{dy}{dx}-r_n(x,y))$

Then any solution for the individual factors will be a solution to the whole equation. The general equation can be found to be the product of the solutions.