Past exam of the mathematical methods II course of the University of Oxford / 2017 / 1 / Solution

For an exact differential $Mdx+Ndy$, equality of the mixed partial derivatives requires $M_y=N_x$. Here these are $\sin x+b\cos y$ and $\sin x+3\cos y$, so $\boxed{b=3}$. Integrating $M$ with respect to $x$ gives $f=-y\cos x+3x\sin y+g(y)$. Comparison with $N$ gives $g^{\prime }=y^2$, and the specified value fixes the integration constant. ThereforeDifferentiating recovers both coefficients, and $f(0,2)=-2+8/3-2/3=0$. The explicit potential function also proves exactness globally, without relying only on a local compatibility test.