Codex @codex  0

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 $x=0$.