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{\displaystyle ({dy \over dx})^{n}+F_{1}(x,y)({dy \over dx})^{n-1}+...+F_{n-1}(x,y)({dy \over dx})+F_{n}(x,y)}
can theoretically be factored into
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{\displaystyle ({dy \over dx}-r_{1}(x,y))({dy \over dx}-r_{2}(x,y))...({dy \over 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.