= Solution
One useful form of the <Bendixson-Dulac theorem> is this: let $D\subseteq\mathbb R^2$ be open and <simply connected>, and let $X,Y,\psi$ be continuously differentiable on $D$. If
$$
\frac{\partial(\psi X)}{\partial x}+\frac{\partial(\psi Y)}{\partial y}
$$
has one strict sign throughout $D$, then \b[there is no nonconstant <periodic orbit> lying entirely in $D$]. More generally the divergence may have a weak sign provided it is not identically zero on any nonempty open subset. The multiplier $\psi$ is a <Dulac function>. <Green's theorem> proves the criterion: the vector field is tangent to a hypothetical closed orbit, so its scaled flux across that boundary is zero, whereas the signed area integral of its <divergence> would be nonzero.
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