Define what it means for a differential $P d x+Q d y$ to be exact, and derive a necessary condition on $P(x, y)$ and $Q(x, y)$ for this to hold. Show that one of the following two differentials is exact and the other is not:

$$\begin{aligned} &y^{2} d x+2 x y d y \\ &y^{2} d x+x y^{2} d y \end{aligned}$$

Show that the differential which is not exact can be written in the form $g d f$ for functions $f(x, y)$ and $g(y)$, to be determined.