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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.

Sir, 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? Thank you Sir.

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I did a body edit to enhance readability. Please make sure I did not change the question. When you say $(x^y - y^x)/ x-y = z^2$ do you mean $(x^y - y^x)/ \color{red}{(x-y)} = z^2$ or $\color{red}{(x^y - y^x)/x} - y = z^2$? – user2468 Mar 22 '12 at 4:28
@J.D., from the solutions given, I made a guess, and edited accordingly. – Gerry Myerson Mar 22 '12 at 5:08
thank you for editing and the editing is correct. Plz find a solution for this post. – prema Mar 22 '12 at 5:12
You've written equations, not functions, so I don't know what you mean by "the graph of these two functions". Perhaps you just mean the graphs of the equations $z^2=(x^y-y^x)/(x-y)$ and $z^2=(x^y+y^x)/(x+y)$. But then you ask for maxima and minima, so you must actually have a function in mind. Are you thinking of $z$ as a function of $x$ and $y$? It isn't a function, since for many $x,y$ pairs there are two values of $z$. So, what do you really mean? – Gerry Myerson Mar 22 '12 at 5:12
I guess I wasn't clear. YOU HAVEN'T GIVEN ANY FUNCTIONS! $z^2=(x^y-y^x)/(x-y)$ does NOT give a function! Look at something simpler: $y^2=x^2$ does NOT give a function (do you understand why?), so it would be nonsense to write $y^2=x^2$ and then ask people for a graph of this function. – Gerry Myerson Mar 22 '12 at 5:41

To sketch graphs you can use Maple command : implicitplot3d

For example :

implicitplot3d((x^y+y^x)/(x+y) = z^2, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10)

enter image description here

implicitplot3d((x^y-y^x)/(x-y) = z^2, x = 0 .. 5, y = 0 .. 5, z = 0 .. 5)

enter image description here

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