if-y-x-x-y-and-y-and-x-are-both-not-equal-to-0-and-l-find-the-value-of-x-and-y-

Question Number 50064 by mondodotto@gmail.com last updated on 13/Dec/18

if y^{x} = x^{y}

and y and x are both not

equal to 0 and l

find the value of x and y

Answered by Necxx last updated on 13/Dec/18

x = y = 2 = 4 f r o m o m e g a f u n c t i o n

Answered by mr W last updated on 14/Dec/18

t h e r e a r e i n f i n i t e s o l u t i o n s .

x = y = a i s a s o l u t i o n w i t h a \in R \cup a > 0

x = a

y^{a} = a^{y}

y = a^{\frac{y}{a}}

y = e^{(\ln a) \frac{y}{a}}

{y e}^{- (\ln a) \frac{y}{a}} = 1

- (\ln a) \frac{y}{a} e^{- (\ln a) \frac{y}{a}} = - \frac{\ln a}{a}

- (\ln a) \frac{y}{a} = W (- \frac{\ln a}{a}) (L a m b e r t W f u n c t i o n)

\Rightarrow y = - \frac{a}{\ln a} W (- \frac{\ln a}{a})

\Rightarrow s o l u t i o n i s

{\begin{cases} x = a, a \in R a n d a > 0 \\ y = a \end{cases}

{\begin{cases} x = a, a \in R a n d a > 1 \\ y = - \frac{a}{\ln a} W (- \frac{\ln a}{a}) \end{cases}

e x a m p l e s :

x = \frac{1}{2} :

y = \frac{1}{2 \ln 2} W (2 \ln 2) = \frac{0.6931}{2 \ln 2} = 0.5

x = 2 :

y = - \frac{2}{\ln 2} W (- \frac{\ln 2}{2}) = {\begin{cases} - \frac{2}{\ln 2} \times (- 0.6931) = 2 \\ - \frac{2}{\ln 2} \times (- 1.3863) = 4 \end{cases}

x = 5 :

y = - \frac{5}{\ln 5} W (- \frac{\ln 5}{5}) = {\begin{cases} - \frac{5}{\ln 5} \times (- 0.5681) = 1.7649 \\ - \frac{5}{\ln 5} \times (- 1.6094) = 5 \end{cases}

Commented by Pk1167156@gmail.com last updated on 14/Dec/18

you are right sir.

Commented by Pk1167156@gmail.com last updated on 14/Dec/18

but only for x=y

Commented by mr W last updated on 14/Dec/18

w h y d i d y o u s a y o n l y f o r x = y ?

a s I h a v e p r o v e d :

f o r 0 < x ⩽ 1 t h e r e i s o n l y o n e v a l u e f o r y,

a n d i t i s y = x .

b u t f o r x > 1, t h e r e a r e {a l w a y s}^{*)} t w o v a l u e s

f o r y, t h e y a r e y = - \frac{x}{\ln x} W (- \frac{\ln x}{x}), t h i s

W - f u n c t i o n d e l i v e r s t w o r e s u l t s, i . e .

o n e v a l u e o f y i s x, t h e o t h e r o n e i s n o t x .

f o r e x a m p l e i f x = 2, y = - \frac{2}{\ln 2} W (- \frac{\ln 2}{2}) = {\begin{cases} 2 \\ 4 \end{cases}

s o w e h a v e t w o s o l u t i o n s : (2, 2) a n d

(2, 4) . y o u c a n c h e c k :

2^{2} = 2^{2} a n d 2^{4} = 4^{2} .

^{*)} e x c e p t i o n : w h e n \frac{\ln x}{x} = \frac{1}{e} o r x = e,

y = - \frac{x}{\ln x} W (- \frac{\ln x}{x}) h a s o n l y o n e v a l u e :

y = e = x .

Commented by mondodotto@gmail.com last updated on 14/Dec/18

thank you

Answered by mr W last updated on 15/Dec/18

a n o t h e r w a y w i t h o u t u s i n g L a m b e r t

f u n c t i o n

t h e s o l u t i o n x = y = a n y r e a l n u m b e r > 0

i s c l e a r . w e o n l y n e e d t o f i n d t h e

s o l u t i o n f o r x \neq y .

l e t y = λ x

{(λ x)}^{x} = x^{(λ x)}

λ x = x^{λ}

λ = x^{λ - 1}

\Rightarrow x = λ^{\frac{1}{λ - 1}}

y = λ \times λ^{\frac{1}{λ - 1}} = λ^{1 + \frac{1}{λ - 1}}

\Rightarrow y = λ^{\frac{λ}{λ - 1}}

i . e . t h e g e n e r a l s o l u t i o n i s

{\begin{cases} x = λ^{\frac{1}{λ - 1}} \\ y = λ^{\frac{λ}{λ - 1}} \end{cases} w i t h λ > 1

e x a m p l e s :

λ = 2 \Rightarrow x = 2, y = 4

λ = 3 \Rightarrow x = \sqrt{3}, y = 3 \sqrt{3}

λ = 4 \Rightarrow x =^{3} \sqrt{4}, y = 4^{3} \sqrt{4}

\dots \dots

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