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I know the basic graph and shifting the graphs. But I have a special equation. I will explain the specialty of the functions later. Okay...let me say the specialty. This function has infinitely many solutions. I would like to know the GRAPH of the function. I can't draw the graph of this function. Please use computer and show me the picture/ graph of the following functions. Once again thank you for this wonderful site and members of this site.

function is: $x^y$ - $z^2$x + $z^2$y - $y^x$ = 0

Also, I want to know that, what kind of function it is? I mean, is it elliptic curve? or something else....

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Not an elliptic curve. The equation determines a surface, and with appropriate software one can sketch that surface. Very little of a general nature is known about equations that mix variable exponents with polynomials. –  André Nicolas Mar 16 '12 at 5:30
    
Thank you so much! Can you draw the graph of this function please.... –  gopi_vC Mar 16 '12 at 7:47
    
If this is homework you are probably supposed to create a graph on your own, what software do you use? –  Listing Mar 16 '12 at 9:03
    
Sir, I am using MATLAB software. Unfortunately, I am not able to draw. Please could you draw this graph... –  gopi_vC Mar 17 '12 at 4:15
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1 Answer 1

$$\begin{align} &&x^y-z^2 x+z^2 y-y^x&=0 \\ &\implies&x^y-y^x-z^2(x-y)&=0 \\ &\implies&x^y-y^x&=z^2(x-y) \\ &\implies&\frac{x^y-y^x}{x-y}&=z^2 \\ &\implies&z=\pm\sqrt{\frac{x^y-y^x}{x-y}}&=\pm\frac{\sqrt{x^y-y^x}}{\sqrt{x-y}} \\ \end{align}$$

Applying WolframAlpha to $z=\frac{\sqrt{x^y-y^x}}{\sqrt{x-y}}$ (just the $+$ part, not the $-$ part) shows several visualizations.

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