The solution of the differential equation `(x+y)(dx-dy)=dx+dy`, is

Options

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

Answer

C

Explanation

Differents equation is
(x +y) (dx -dy) =dx +dy
dividing by dx on both the sides
` (x+y) (1-(dy)/(dx)) =1 + (dy)/(dx)`
The equation changes to
` v {1-((dv)/(dx) -1)} = (dv)/(dx)`
` v ( 2- (dv)/(dx)) = (dv)/(dx)`
` 2v - v (dv)/(dx) = (dv)/(dx)`
` 2 (1+v) (dv)/(dx)`
` ((1+v)/v)dv =2dx`
` or, (1/v +1) dv = 2dx`
Integrating on both the sides,
` int(dv)/v + intdv = 2int dx +c`
`log v +v = 2x +c`
putting v= x+y
` log (X+y) +x+y = 2x +c`
`or, log (x+y) +y-x =c`
or , y -x + log (x+y) =c