I'm having problems with finding all possible integers solutions of particular equations, like this one for example: $x^2 -xy + 2y^2 = 29$. What sets me off, is the term $xy$, I don't know how to deal with it.
I know methods for solving equations like $x^2 - 3y^2 = -3$ (then I work with the fundamental unit and with norm forms), but I'm stuck on a method for the first one. Substitutions don't seem to do the trick, but I could be wrong.
As always, any help would be dearly appreciated.
$$x=\dfrac{y\pm\sqrt{y^2-4(2y^2-29)}}2=\dfrac{y\pm\sqrt{116-7y^2}}2$$
We need $116-7y^2$ to be $\ge0$ and perfect sqaure
$y^2\le\dfrac{116}7<17$
So, we need to test for $1\le y\le4$