If someone can come up with a 3 variable equation, it might come out nicer (meaning you wouldn't need wolfram alpha to solve it). I'll check back if I think of anything.
Thanks, please check back if you figure it out. This is really bugging me, my friend who asked this is a teacher for 8th graders and this riddle is from their book. I'm somewhat older than 8th graders and I can't figure it out.
Another way to solve it is to put x^3=2+y^2 in to a graphical calculator and observe where both y and x are whole digits. Not exactly mathematical either though.