= Solution
The <Bendixson-Dulac criterion> says that if a $C^1$ planar field admits a $C^1$ multiplier $B$ on a <simply connected> region such that $\nabla\cdot(Bf)$ has one strict sign there, no nonconstant periodic orbit lies wholly in that region. The <Poincaré-Bendixson theorem> says that a nonempty compact omega-limit set of a planar $C^1$ flow which contains no equilibria is a periodic orbit.
For $\mu<0$, the divergence of the given field is $\mu-2x^2-3y^2<0$ throughout the plane. Taking $B=1$ proves \b[there are no periodic orbits].
For $\mu>0$, use $E=(x^2+y^2)/2$. Along trajectories,
$$
\dot E=x\dot x+y\dot y=y^2(\mu-2x^2-y^2).
$$
On a circle $r=r_-$ with $2r_-^2<\mu$ the radial flow is outward or tangent, and on a circle $r=r_+$ with $r_+^2>\mu$ it is inward or tangent. Their intervening closed annulus is positively invariant. At the tangent points $y=0$, the field has $\dot y=-x\ne0$, so no equilibrium sits on its boundaries. Indeed the only equilibrium is the origin, excluded from the annulus. Every forward trajectory in the annulus has a nonempty compact omega-limit set without equilibria. The <Poincaré-Bendixson theorem> consequently proves \b[at least one periodic orbit exists for every $\mu>0$]. No claim of uniqueness follows from this trapping argument.
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