What-are-the-possible-solution-satisfying-x-y-y-x-

Question Number 7222 by Tawakalitu. last updated on 16/Aug/16

W h a t a r e t h e p o s s i b l e s o l u t i o n s a t i s f y i n g

x^{y} = y^{x}

Commented by Tawakalitu. last updated on 17/Aug/16

I a p p r e c i a t e y o u r e f f o r t . t h a n k s s i r .

Commented by Yozzia last updated on 17/Aug/16

x = y \in C c o n s t i t u t e s p a r t o f t h e s o l u t i o n

s e t .

- - - - - - - - - - - - - - - - - - - - - - -

x^{y} = y^{x}

y l n x = x l n y

- \frac{1}{x} l n x = - \frac{1}{y} l n y

x^{- 1} {l n x}^{- 1} = y^{- 1} {l n y}^{- 1}

e^{{l n x}^{- 1}} {l n x}^{- 1} = e^{{l n y}^{- 1}} {l n y}^{- 1}

W (e^{{l n x}^{- 1}} {l n x}^{- 1}) = W (e^{{l n y}^{- 1}} {l n y}^{- 1})

\Rightarrow {l n x}^{- 1} = {l n y}^{- 1}

\Rightarrow \frac{1}{x} = \frac{1}{y} \Rightarrow x = y \in C .

B y t h i s w o r k i n g, x^{y} = y^{x} \Leftrightarrow x = y .

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