= Solution
The initial <angular velocity> cancels the flux term in the canonical <angular momentum>:
$$
J=mr_0^2\left(-\frac{qB}{2m}\right)+\frac{qBr_0^2}{2}=0.
$$
Define $\omega_c=qB/m$ and $\omega_s^2=4\pi^2k/m$. The radial equation becomes $\ddot r+\Omega^2r=\omega_s^2r_0$, so
$$
\boxed{\Omega^2=\omega_s^2+\frac{\omega_c^2}{4},\qquad
r_e=\frac{\omega_s^2}{\omega_s^2+\omega_c^2/4}r_0,\qquad
r(t)=r_e+(r_0-r_e)\cos\Omega t.}
$$
These identify the paper's $\omega$ with $\Omega$. For nonzero radius, $J=0$ also fixes
$$
\boxed{\phi(t)=-\frac{\omega_c}{2}t,\qquad\phi(0)=0.}
$$
A radius is nonnegative. The formal sinusoidal solution remains strictly positive for a full cycle only if $2r_e>r_0$, or $\omega_s^2>\omega_c^2/4$. At equality it touches zero; at stronger magnetic coupling it first reaches zero and its negative continuation is not a physical ring radius. No collision or re-expansion rule at collapse is supplied, so the circular-ring solution is then valid only up to the first zero. This domain qualification leaves the requested differential-equation solution intact.
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