For an equation $(x^y - y^x)/ (x-y) = z^2$ has infinitely many solutions $(1, n, 1)$ for $n \in N.$ Apart from this, $(n, 1, 1), (2, 4, 0), (4, 2, 0), (2, 1, 1), (2, 3, 1), \text{etc}$ are also solutions of the above equation.
Now, with little modification, $(x^y+y^x)/ (x+y) = z^2$ also has infinitely many solutions i.e., $(a, n, a)$ for $n \in N.$
I tried to see the graph of these two functions in MATLAB and MATCAD. Unfortunately, I could not find. Can you sketch the graph? And by using partial derivatives or any other method, how can we have max and min values of this function? Could you explain please.
I am looking for a graph of these function(s). So that I can make some important notes on this equation. Moreover, this a Diophantine equation and had (a,n, a) solutions. How one can prove these solutions, without actually guessing? Finally, what I want to say, I need graphs of both functions in 3d. As well as, how to prove the infinitly solutions mathematically?