Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 14 / a

An exact differential form must have $P_y=Q_x$. Direct partial differentiation givesThus $(1-a)xy\cos (ax)=0$ must hold identically. Since $xy\cos (ax)$ is not identically zero for a fixed nonzero $a$, the only possibility isThe explicit potential in part (ii) verifies sufficiency.

For $a=1$, integrate $Q=2xy\sin x$ with respect to $y$ to obtainIts other partial derivative is $f_x=y^2\sin x+x{y}^2\cos x+C^{\prime }(x)$. Comparing with $P$ gives $C^{\prime }(x)=0$. Thus the most general potential on the plane is