In solving equations, we first look into whether solutions exist or not. If they do, we analyze if the solutions are finite or infinite in number. If infinite, we give solutions in general form. With Diophantine equations, we search for solutions in positive integers.
Now my question is how to proceed as above for the following problems:
$\displaystyle \frac{x^y - y^x}{x - y} = z^2$; I got one solution $(x, y, z) = (2, 3, 1)$.
$x^3 + y^2 - x^2y + x^2y^3 - y^2x = x + y$; I got one solution $(x, y) = (1, 1)$.
The above solutions I got by trial-and-error. I would like to know the following:
How to know if solutions exist or not for above equations?
If they do, how to know if there are finitely or infinitely many?
If finite how to find all solutions?
If infinite how to characterize all solutions by a general formula?
Is there any theory or particular method generalizing both the problems?
Thank you,