Find the particular solution of the differential equation `xy(dy)/(dx) =(x+2) (y+2), ` it being given that y = -1 when x = 1.

Explanation

The given differential equation is
`xy(dy)/(dx) =(x+2) (y+2)`
`rArr (y)/((y+2))dy=((x+2))/(x)dx`
`rArr int(y)/((y+2))dy = int ((x+2))/(x)dy +C,` where C is an arbitrary constant
`rArr int{1-(2)/((y+2))}dy =int(1+(2)/(x))dx+C`
`rArr y-2log|y+2|=x +2log|x|+C. " " `...(i)
Putting x = 1 and y = -1 in (i), we get C = -2.
Hence, the required solution is
`y-2log|y+2|=x+2log|x|-2.`