# How to solve an equation of the form $ax^2 - by^2 + cx - dy + e =0$?

I am trying to find out how to solve $ax^2 - by^2 + cx - dy + e = 0$ to get integer solutions, failing this the rational solutions.

Thanks!

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When write the equation, the solutions will be determined by the solutions of certain equations Pell. – individ Jun 10 '14 at 5:25

Solutions of the equation:

$ax^2-by^2+cx-dy+q=0$

you can record if the root of the whole: $k=\sqrt{(c-d)^2-4q(a-b)}$

Then using the solutions of the equation Pell: $p^2-abs^2=\pm1$

Then the formula of the solution, you can write:

$x=\frac{\pm1}{2(a-b)}(((d-c)\pm{k})p^2+2(bk\mp(bc-ad))ps+b(a(d+c)-2bc\pm{ak})s^2)$

$y=\frac{\pm1}{2(a-b)}(((d-c)\pm{k})p^2+2(ak\mp(bc-ad))ps-a(b(d+c)-2ad\mp{bk})s^2)$

If the root is a need to find out if this is equivalent to the quadratic form in which the root of the whole. This is usually accomplished this replacement: $x$ in such number $x+ty$

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Forgot to say. The characters inside the brackets do not depend on the sign of the Pell equation. It depends only before $\pm{1}$ – individ Jun 10 '14 at 13:21
I think it's harder than this. Can you tell me, for example, all the integer solutions to $x^2-37y^2=3$? – Jeroen Jan 5 at 19:17
@Jeroen it is necessary to investigate this equation using the General formula. artofproblemsolving.com/community/c3046h1048219 So the root was rational need to make the change. $x=x+6y$ Obtain the equation. $x^2+12xy-y^2=3$ Substitute in the formula and my solution is ready. – individ Jan 6 at 4:48
Why don't you answer the question rather than just theorising about how to answer it -- let me know the first few solutions. My point is I suspect that you are missing at least one of the subtleties involved in solving these equations: there is no simple formula for a generalised Pell equation because there are subtle arithmetic issues. – Jeroen Jan 6 at 7:25
@Jeroen I don't understand - the formula brought. Substitute and get the answer. You want me brought even arithmetic? I do laziness. Maybe I will if the mood is good. In this issue, the wording and formulation of the problem was different. I responded - wrote the General formula. The rest I'm not interested. – individ Jan 6 at 11:44