<s>[INST] Solve the equation $x^{x+y}=y^{y-x}$ in positive integers. 
Let $x,y$ are positive integers. Solve the equation
  $$x^{x+y}=y^{y-x}$$

smartplot(x^(x+y)=y^(y-x));

I used http://www.wolframalpha.com
 [/INST] Here I will show that $y= x^{a/b}$ where a and be are two consecutive odd or even numbers:
Clearly x and y have common factors, lets suppose $y= k x$ we may write:
$y = x ^ {\frac{x+y}{x-y}}= x^{\frac{(k+1)x}{(k-1)x}}=x^{\frac{k+1}{k-1}}$ 
$k+1 $ and $k -1$ are two consecutive odd or even numbers:
Examples:
$k=3 ⇒ a=k+1 =4 , b =k -1= 2 ⇒ y =x^2$; solution
x=3 and y=9 is a result  $
$ k=4 ⇒ a= 4+1=5, b=4-1=3 ⇒y=x^{5/3}$;solution
x=8 and y=32 is a result$
Hence equation $x^{x+y} =y^{y-x}$ is reduced to a simple form like $ y =x^{a/b}$ where a and b are two consecutive odd or even numbers. Clearly a and b can have certain values as showed in examples.
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