If x^y = y^x, where x≠y, then find the value of x and y.

Solution
Given
x^y = y^x      .....(i)
x≠y      .....(ii)
Let x = ky      .....(iii)
where k is a constant
Equation .....(i), .....(ii) and .....(iii) implies that k≠1.
Inserting x = ky in equation .....(i)
(ky)^y = y^ky
Or, k^y . y^y = y^ky
Or, k^y = y^ky . y^-y
Or, k^y = y^{ky - y}
Or, k^y = y^y(k-1)
Or, k^y = (y^(k-1))^y
Here power of both the sides are y so.
Or, k = y^(k-1)

Raising both sides by power of  
 1
 k-1   we get
 

Or, k
 ^1
 k-1 
  = y

y = k
 ^1
 k-1 
  .....(iv)

So, x = ky is given by

x = k (k
 ^1
 k-1 
 )

x = k
 ^{1
 k-1  +1}
 

x = k
  ^{1 + k - 1 
k-1}
 

x = k
 ^k
 k-1 
  .....(v)

Equation .....(v) and .....(iv) gives the values of x and y respectively for k ≠ 1

∴x = k
 ^k
 k-1 
 
∴y = k
 ^1
 k-1