Problem 06
$y(y^2 + 1) \, dx + x(y^2 - 1) \, dy = 0$
Solution 06
$y(y^2 + 1) \, dx + x(y^2 - 1) \, dy = 0$
$y^3 \, dx + y \, dx + xy^2 \, dy - x \, dy = 0$
$(xy^2 \, dy + y^3 \, dx) + (y \, dx - x \, dy) = 0$
$y^2(x \, dy + y \, dx) + (y \, dx - x \, dy) = 0$
$(x \, dy + y \, dx) + \left( \dfrac{y \, dx - x \, dy}{y^2} \right) = 0$
$d(xy) + d\left( \dfrac{x}{y} \right) = 0$
$\displaystyle \int d(xy) + \int d\left( \dfrac{x}{y} \right) = 0$
$xy + \dfrac{x}{y} = c$
$xy^2 + x = cy$