I am dealing with a series of quadratic diophantine equations that all have the same form: $$A^2x^2 - C^2y^2 + Dx - Ey + F = 0$$ ($A,C,D,E >0$ | $A$ and $C$ have a common factor (or $C=1$) | $D,E,F$ coprime)
Do you know if there's an analytical method (ie. one that doesn't require testing all factorizations of a large number) to find a single solution (one is enough) if it exists - or a way to show there are no solutions if that is the case?. Thanks!
Eg. $$324x^2 - 9y^2 + 101x - 13y + F =0$$ ($F= -229$: Sol ($x=1, y=4$))
($F= 50$: No solutions)