Problem 02
$2y(x^2 - y + x)~dx + (x^2 - 2y)~dy = 0$
Solution 02
$M~dx + N~dy = 0$
$2y(x^2 - y + x)~dx + (x^2 - 2y)~dy = 0$
$M = 2y(x^2 - y + x) = 2x^2y - 2y^2 + 2xy$
$N = x^2 - 2y$
$\dfrac{\partial M}{\partial y} = 2x^2 - 4y + 2x$
$\dfrac{\partial N}{\partial x} = 2x$
$\dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x} = (2x^2 - 4y + 2x) - 2x$
$\dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x} = 2x^2 - 4y = 2(x^2 - 2y)$