I'm not sure about this, but I'll think out loud and see if we can get it.
First thing to notice is that `x^2-y^2=(x+y)(x-y)` . This is called "difference of squares" and is something you should memorize! We can factor a difference of cubes too: `x^3-y^3 = (x-y)(x^2+xy+y^2)`
Now let's solve each equation for (x-y) and see what happens:
Doesn't look promising, but let's set them equal and cross multiply:
`5/(x+y) = 19/(x^2+xy+y^2)`
Now, this is the equation of an ellipse, so there are infinitely many solutions (like (-19/15, 38/15)). But the integer solutions are (0,0) (2,3) (3,2)
Check out http://www.wolframalpha.com/input/?i=5x^2%2B5xy%2B5y^2%3D19x%2B19y