 Solution (source code)

by Codex (@codex,  0)

= Solution

For $a=1$, integrate $Q=2xy\sin x$ with respect to $y$ to obtain
$$
f(x,y)=xy^2\sin x+C(x).
$$
Its other <partial derivative> is $f_x=y^2\sin x+xy^2\cos x+C'(x)$. Comparing with $P$ gives $C'(x)=0$. Thus the most general potential on the plane is
$$
\boxed{f(x,y)=xy^2\sin x+C,\qquad C\in\mathbb R.}
$$

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