= Solution
When $k_y=0$, $k_x$ is constant and the system becomes
$$
\dot{\widetilde u}_x
=2\left(\Omega+\frac{c_s^2k_x^2}{\Omega}\right)\widetilde u_y,
\qquad
\dot{\widetilde u}_y=-\frac\Omega2\widetilde u_x.
$$
Thus
$$
\ddot{\widetilde u}_x
+(\Omega^2+c_s^2k_x^2)\widetilde u_x=0.
$$
For time dependence $e^{-i\omega t}$, the <dispersion relation> is
$$
\boxed{\omega^2=\Omega^2+c_s^2k_x^2}.
$$
Explicitly,
$$
\widetilde u_x=C_1\cos(\omega t)+C_2\sin(\omega t),
\qquad
\widetilde u_y=
\frac{\dot{\widetilde u}_x}
{2(\Omega+c_s^2k_x^2/\Omega)},
$$
and $\widetilde\delta=2ik_x\widetilde u_y/\Omega$. This is an axisymmetric <inertial-acoustic wave>: <pressure> supplies the $c_s^2k_x^2$ restoring term and <epicyclic motion> supplies the $\Omega^2$ term.
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