The solution of `(x+y) (dy)/(dx)=1` is

Options

A. ` x+y+1=Ce^(y)`
B. `x+y+1=Ce^(-y)`
C. `x+y-1=Ce^(y)`
D. `x+y-1=Ce^(-y)`

Answer

a

Explanation

Given equation can be converted into linear
differential equation by dividing `(dy)/(dx) ` and then solve it .
` (x+y) (dy)/(dx) = 1`
` rArr = (dx)/(dy) = (x+y) rArr (dx)/(dy) - x = y `
On comparing with the form `(dx)/(dy) + Px = Q` , we get
` P = - 1 , Q = y `
` :. IF = e^(int P dy)=e^(-int dy) rArr IF = e^(-y)" "` ...(i)
The general solution of the given differential equations is given by the relation ,
` x * IF int Q xx IF dy +C`
` rArr x* e^(-y) = int e^(-y)* y dy + C`
` rArr x* e^(-y) = int e^(-y) dy - int [d/(dy) y int e^(-y) dy ] dy + C `
[ integration by parts ]
` rArr xe^(y) = - y(e^(-y)) + int e^(-y) dy +C`
` rArr xe^(-y) = -ye^(-y) - e^(-y) + C rArr x = -y - 1 + e^(y)C`
` rArr x + y + 1 Ce^(y)`