Integer solutions to ax^2 + bx - cy^2 - dy = 0

[Puma]

Homework Statement

I am a hobbyist looking for solutions to ax^2 + bx - cy^2 - dy = 0 where all variables are integers and are non-zero. Is there a method of doing this effectively?

Homework Equations

The Attempt at a Solution

I can look at the numbers produced by ax^2 + bx vs cy^2 + dy and see that they have a relationship: what I mean is if I manually find a pair of close numbers, difference = d, I find the next set of values is d+2 apart, then d + 4 and so on. So it looks as though there should be a method in algebraic terms for doing this.

 

[Svein]

Of course, the trivial solution is (0, 0). Otherwise, rewrite the equation as $x(ax+b)-y(cy+d)=0$.

 

[Puma]

Sorry I know the variables a, b, c, and d but I don't know x, y. It looks as though I might be able to do something with modular arithmetic given that both x(ax+b) and y(cy+d) now seem to both be integer multiples in other words either x or ax+b must necessarily contain some factors in common with y and cy+d. Is there a good way to find x,y? Thanks!

 

[profesrchaos]

http://www4a.wolframalpha.com/Calculate/MSP/MSP100420ag0a9de184i4e300006aa5e486371cg88e?MSPStoreType=image/gif&s=23&w=258.&h=46.
You can find solutions of y in the same way. Its a bit silly though as it requires you to know all but x. You can see intuitively the set of solutions from the form Svein put it in
x(ax+b)−y(cy+d)=0
Yeah you could do what your saying and then write out the set of solutions