what is the solution of the differential equation ` (dx)/(dy) = ( x +y+1)/(x +y-1)` ?

Options

A. y -x +4 In (x+y) =c
B. y+z +c In (x +y) =c
C. y -x+In (x +y) =c
D. y +x+2 In (x +y) =c

Answer

C

Explanation

Let ` t=x +y Rightarrow (dt)/(dy) = (dx)/(dy) +1 `
So, ` (dt)/(dy) -1 , (t+1)(t-1)`
` Rightarrow (dt)/(dy)-1 = (t +1)/(t-1)`
`Rightarrow (dt)/(dy) = (t +1 + t-1)/(t-1) = (2t)/(t-1)`
` Rightarrow int (t-1)/t dt =2 int dy `
` Rightarrow t -log t =2y +C_(1)`
` Rightarrow x +y -log (x+y) -2y =C_(1)`
` Rightarrow y-x + log(x+y) = -C_(1) =C`