The solution of the differential equation
`(dy)/(dx) = e^(x-y) (e^(x)-e^(y))` is

Options

A. ` e^(y)=(e^(x)+1)+Ce^(-e^(x))`
B. ` e^(y)=(e^(x)-1)+C`
C. ` e^(y) = (e^(x)-1)+Ce^(-e^(x))`
D. None of these

Answer

a

Explanation

Given , ` (dy)/(dx) = (e^(x))/(e^(y)) ( e^(x)-e^(y))`
` rArr e^(y) (dy)/(dx) + e^(x).e^(y) = e^(x) .e^(x)`
Let `e^(y) = t rArr e^(y) (dy)/(dx) = (dt)/(dx)`
Then , given equation reduces to
`(dt)/(dx) + e^(x)t = e^(2x)`
This is in the form of linear differential equation
On comparing with the form `(dy)/(dx) + Px = Q, ` we get `P = e^(x)`
and ` Q = e^(2x)`
` :. IF = e^(int Pdx)= e^(int e^(x)dx) = e^(e^(x))`
Required solution is
` t* e^(e^(x)) = int e^(2x) . e^(e^(x)) dx +C `
` rArr e^(y) = ( e^(x) -1) + Ce^(-e^(x))`