$y(x^2 + y^2 - 1) \, dx + x(x^2 + y^2 + 1) \, dy = 0$

$y(x^2 + y^2) \, dx - y \, dx + x(x^2 + y^2) \, dy + x \, dy = 0$

$[ \, y(x^2 + y^2) \, dx + x(x^2 + y^2) \, dy \, ] - (y \, dx - x \, dy) = 0$

$(x^2 + y^2)(y \, dx + x \, dy) - (y \, dx - x \, dy) = 0$

$(y \, dx + x \, dy) - \dfrac{y \, dx - x \, dy}{x^2 + y^2} = 0$

$d(xy) - d [ \, \arctan (x/y) \, ] = 0$

$\displaystyle \int d(xy) - \int d[ \, \arctan (x/y) \, ] = 0$

$xy - \arctan (x/y) = c$ *answer*