= Solution
Write the perturbation velocity as $(u,v)$, the density amplitude as $\sigma_1$, and $\gamma_d=\epsilon\Omega$. Axisymmetry removes advection by the background azimuthal flow, but the radial perturbation advects the <Keplerian shear>: $u\,\partial_xU_{g,y}=-3\Omega u/2$. Combining this with the <Coriolis force> gives the linearized <shearing sheet> equations
$$
s\sigma_1+ik\sigma_0u=0,\qquad
(s+\gamma_d)u-2\Omega v=-ik\left(\frac{c^2}{\sigma_0}-\frac{2\pi G}{|k|}\right)\sigma_1,\qquad
(s+\gamma_d)v+\frac\Omega2u=0.
$$
The <razor-thin disk Poisson kernel> supplies the self-gravity term. The $\Omega/2$ coefficient, rather than $2\Omega$, is essential: it includes the perturbed advection of the background velocity.
Put $A=c^2k^2-2\pi G\sigma_0|k|$ and $\omega^2=\Omega^2+A$. The determinant of the three amplitude equations is
$$
s\left[(s+\gamma_d)^2+\Omega^2\right]+A(s+\gamma_d)=0.
$$
Expanding it gives the <dust gravitational dispersion relation with gas drag>
$$
\boxed{s^3+2\epsilon\Omega s^2+(\omega^2+\epsilon^2\Omega^2)s
+\epsilon\Omega(\omega^2-\Omega^2)=0,\quad
\omega^2=\Omega^2-2\pi G\sigma_0|k|+c^2k^2.}
$$
Using a determinant avoids division by $s$ and retains the neutral/secular branch. The <radial epicyclic frequency> of this <Keplerian shearing sheet> is $\Omega$, so the three terms in $\omega^2$ represent rotational support, <self-gravity> and dust <pressure>.
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