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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.

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?

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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 $\color{red}{(x^y - y^x)/x} - y = z^2$?2012-03-22
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    @J.D., from the solutions given, I made a guess, and edited accordingly.2012-03-22
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    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?2012-03-22
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    Oh, and what does Diophantine have to do with it? The solutions you've written have nothing to do with the questions you are asking about drawing a graph and about finding maxima and minima.2012-03-22
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    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.2012-03-22
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    I don't know what you mean, "How can one prove these solutions, without actually guessing?" We don't *prove* solutions. We *find* solutions; we *verify* solutions; we don't *prove* solutions. Also, $x=a$, $y=n$, $z=a$ is *not* (in general) a solution to $z^2=(x^y+y^x)/(x+y)$, so I don't know why you say it is. Say, $a=2$, $n=3$, then $(2^3+3^2)/(2+3)\ne2^2$.2012-03-22
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    One thing still puzzles me: why is this tagged `(partial-derivatives)`2012-03-22
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    thank you for editing and the editing is correct. Plz find a solution for this post.2012-03-22
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    @pedja! Wow great! can you make graph for (x^y -y^x)/x-y = z^2?plz.2012-03-22
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    @Gerry Myerson! can you give proof or generalization to find solution of both the equations, without guessing?2012-03-22
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    @Pedja! thank you so much for preparing the graph?2012-03-22
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    @Pedja! can you expalin, why this function has infinity solutions? because, the grpah is broken. Still, how this function has infinity solutions?2012-03-22
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    @Gerry Myerson! Please excuse me. don't angry. I don't wnat the graph. my question is for you, without guessing, can you generalize the solutions of my both equations? once again I am so sorry, if I am troubling you.2012-03-22
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    @prema: I have merged your accounts. Please register your account to avoid problems with duplicate accounts in the future.2012-03-22

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