= Solution
The <equation of motion in a rotating frame> with constant <angular velocity> $\boldsymbol\omega$ is
$$
m\left[\ddot{\mathbf x}+2\boldsymbol\omega\times\dot{\mathbf x}
+\boldsymbol\omega\times(\boldsymbol\omega\times\mathbf x)\right]=\mathbf F.
$$
With $\mathbf F=-4m\omega^2\mathbf x$ and $\boldsymbol\omega=(0,0,\omega)$, this becomes
$$
\ddot x-2\omega\dot y+3\omega^2x=0,\qquad
\ddot y+2\omega\dot x+3\omega^2y=0,\qquad
\ddot z+4\omega^2z=0.
$$
The given initial data make $z=0$. For the <complex coordinate> $\zeta=x+iy$, the planar equations combine into
$$
\ddot\zeta+2i\omega\dot\zeta+3\omega^2\zeta=0.
$$
Its <characteristic roots> are $i\omega$ and $-3i\omega$. Applying $\zeta(0)=1$, $\dot\zeta(0)=0$ gives $\zeta=\tfrac34e^{i\omega t}+\tfrac14e^{-3i\omega t}$. The triple-angle identities reduce this <astroid motion of a harmonic oscillator in a rotating frame> to
$$
\boxed{(x,y,z)=(\cos^3\omega t,\sin^3\omega t,0).}
$$
It traces an <astroid>. Differentiating gives the relative <speed>
$$
|\dot{\mathbf x}|^2
=9\omega^2\sin^2\omega t\cos^2\omega t
=\frac94\omega^2\sin^2(2\omega t).
$$
For $\omega>0$, its maxima occur at $t=(2n+1)\pi/(4\omega)$, with
$$
\boxed{|\dot{\mathbf x}|_{\max}=\frac{3\omega}{2}.}
$$
This is speed measured in the specified <rotating frame>. The same physical motion has inertial complex coordinate $e^{i\omega t}\zeta=\cos(2\omega t)+(i/2)\sin(2\omega t)$, an <ellipse>; its initial <velocity> is the frame-rotation contribution even though the initial relative <velocity> is zero.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/ia/paper-4-rotating-oscillator.png]
{title=The oscillator's astroid in rotating coordinates, ellipse in inertial coordinates, and maxima of relative speed}
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