Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 18 / b / Solution

For an exact differential $Pdx+Qdy$, equality of mixed partial derivatives requires $P_y=Q_x$. Here this givesOn an open two-dimensional domain the coefficient relation is $x{\mu }^{\prime }+3\mu =0$, so $\mu =C{x}^{-3}$ on a connected interval avoiding zero. The normalization gives $C=1$ on the positive interval through $x=1$. Integrating $f_x=2x^{-3}\sin y$ and checking $f_y$ yieldsThere is no smooth extension of this normalized integrating factor across $x=0$; an independently specified factor on $x<0$ is not fixed by data at one.