Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 2 / 8

Here $P_y=5=Q_x$, and direct integration constructs a global potential for this exact differential form. Integrate $P$ with respect to $x$ to obtain $f=6x^2+5xy-9x+g(y)$; matching $f_y=Q$ gives $g^{\prime }=3y-4$. ThusBoth derivatives recover the supplied coefficients.

The form is not exact, because $P_y=1$ and $Q_x=2$. For an integrating factor depending on one coordinate, the condition $(\mu P{)}_y=(\mu Q{)}_x$ with $\mu =\mu (x)$ gives ${\mu }^{\prime }/\mu =(P_y-Q_x)/Q=-1/(2x)$. ThusFor example, on $x>0$, the multiplied form is $(\sqrt{x}+y/\sqrt{x})dx+2\sqrt{x}dy$, with potential $\boxed{f=\frac{2}{3}{x}^{3/2}+2y\sqrt{x}+C}$. On $x<0$, a corresponding potential is $f=\frac{2}{3}(-x{)}^{3/2}-2y\sqrt{-x}+C$. The factor is local to these domains and is singular on the excluded line $x=0$.

The origin is excluded by the denominator. On the punctured plane, direct differentiation givesHence the form is exact on its whole stated domain, with $\boxed{f=-x/(x^2+y^2)+C}$. Although the punctured plane is not simply connected, the explicitly constructed single-valued potential establishes global exactness directly; a topological sufficiency criterion is unnecessary here.