Solve (i) `xdx +ydy + (xdy - ydx)/(x^(2) + y^(2)) = 0`
(ii) `y(1+xy) dx - xdy = 0`

Explanation

(i) Identifying `(xdy - ydx)/(x^(2) + y^(2))` with `d(tan^(-1).(y)/(x))`, we have,
`xdx + ydy + d(tan^(-1).(y)/(x)) = 0`
Integrating we obtain,
`(x^(2))/(2) + (y^(2))/(2) + tan^(-1).(y)/(x) = k`, k being the constant of integration
`:. x^(2) + y^(2) + 2 tan^(-1).(y)/(x) = lambda, lambda` being another constant.
(ii) `y (1+xy)dx - xdy = 0`
`rArr (ydx - xdy) + xy^(2)dx = 0`
`rArr (ydx - xdy)/(y^(2)) + xdx = 0`
`rArr d((x)/(y)) + xdx = 0`
Integrating, we get
`(x^(2))/(y) + (y^(2))/(2) = k`, k being the constant of integration