exact DE: [ 2x + y cos (x^2) - 2xy + 1 ] dx + [ sin (x^2) - x^2 ] dy = 0
[ 2x + y cos (x^2) - 2xy + 1 ] dx + [ sin (x^2) - x^2 ] dy = 0
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$(2x + y \cos x^2 - 2xy + 1)\
$(2x + y \cos x^2 - 2xy + 1)\,dx + (\sin x^2 - x^2)\,dy = 0$
$M = 2x + y \cos x^2 - 2xy + 1$
$\dfrac{\partial M}{\partial y} = \cos x^2 - 2x$
$N = \sin x^2 - x^2$
$\dfrac{\partial N}{\partial x} = 2x \cos x^2 - 2x$
$\dfrac{\partial M}{\partial y} \ne \dfrac{\partial N}{\partial x}$
The equation is not exact as you claimed it is. Maybe you mean this equation:
$(2xy \cos x^2 - 2xy + 1)\,dx + (\sin x^2 - x^2)\,dy = 0$
which is exact. I am not sure, of course, of what you really mean. It is just my speculation.