Prove quadratic diophantine has no solutions?

Question

I am trying to prove a quadratic diophantine equation has no integer solutions. Any input would be great, I am interested in the general method for this type of equation so any explanation / link to additional resources would help. Thanks a bunch! The equation always has this form:
$A^2x^2-C^2y^2=Ey-Dx-F,(\{A,C,D,E,F\}>0)$

Answer 1

There are several sources for solving quadratic Diophantine equations (a special case is Pell's equation). You can try your equation at this equation solver. It uses a nice algorithm (which exists). So one can always decide whether or not there is an integer solution.
The equation $$ a^2x^2-c^2y^2=ey-dx-f $$ is a special case of a quadratic Diophantine equation.