 Solution

For an exact differential form, equality of the mixed partial derivatives requires $P_y=Q_x$. HereEquality for all $x>0$ and real $y$ forces $1/a=1/2$, so $\boxed{a=2}$. To construct the potential, integrate $P$ with respect to $x$:Differentiation with respect to $y$ gives $v_y=Q+C^{\prime }(y)$, so $C^{\prime }(y)=0$. Thus the most general potential on the connected domain $x>0$ isIts explicit partial derivatives verify sufficiency as well as necessity.