Past exam of the mathematical methods II course of the University of Oxford / 2022 / 3 / Solution

For an exact differential $Pdx+Qdy$, equality of mixed partial derivatives requires $P_y=Q_x$. The matching pair is $P=f_1$, $Q=f_3$, sinceThe other pairings do not give the same partial derivatives on an open region. Integrating $f_1$ with respect to $x$ gives $\ln x\sin y+e^x{\cos }^{2}y+C(y)$. Its $y$ derivative agrees with $f_3$ precisely when $C^{\prime }(y)=0$. HenceThis also proves existence, rather than relying only on the necessary mixed-partial derivative test.