= Solution
For $k_y=0$ and nonzero $k_x$, the <wavenumber> is constant. Continuity gives $\dot{\widetilde\Sigma}'=-ik_x\Sigma_0\widetilde v_x$, and the conserved <vortensity> relation gives
$$
ik_x\widetilde v_y=\Sigma_0\widetilde q'+\frac{(2\Omega-S)\widetilde\Sigma'}{\Sigma_0}.
$$
Differentiate continuity and substitute radial momentum and this identity. The <forced axisymmetric density mode with vortensity> obeys
$$
\boxed{\ddot{\widetilde\Sigma}'+\omega_k^2\widetilde\Sigma'=-2\Omega\Sigma_0^2\widetilde q',\qquad\omega_k^2=c_s^2k_x^2-2\pi G\Sigma_0|k_x|+\kappa^2,\quad\kappa^2=2\Omega(2\Omega-S).}
$$
The constant forcing represents the stationary vortical or balanced component, which is lost if every mode is assumed to have nonzero frequency. Let $k_*=\pi G\Sigma_0/c_s^2=\kappa/(Qc_s)$. Completing the square gives
$$
\omega_k^2=c_s^2(|k_x|-k_*)^2+\kappa^2(1-Q^{-2}).
$$
For $Q>1$, every nonzero radial <wavenumber> has positive $\omega_k^2$. The solution is a constant balanced offset $-2\Omega\Sigma_0^2\widetilde q'/\omega_k^2$ plus bounded sine and cosine oscillations. <Pressure> stabilizes short wavelengths, rotation stabilizes long wavelengths, and self-gravity is strongest at intermediate wavelengths.
For $Q<1$, the <Toomre stability criterion> fails in the band
$$
\boxed{k_*\left(1-\sqrt{1-Q^2}\right)<|k_x|<k_*\left(1+\sqrt{1-Q^2}\right).}
$$
There $\omega_k^2<0$ and the homogeneous solutions grow or decay exponentially. The fastest rate is $s_{\max}=\kappa\sqrt{Q^{-2}-1}$ at $|k_x|=k_*$. Outside the band the modes oscillate; at its edges the frequency vanishes.
At $Q=1$, $\omega_k^2\geq0$ and it vanishes only at $|k_x|=\kappa/c_s$. At that critical <wavenumber>,
$$
\boxed{\widetilde\Sigma'(t)=C_0+C_1t-\Omega\Sigma_0^2\widetilde q' t^2.}
$$
This <Toomre marginal algebraic growth> is spectrally marginal, not bounded for all data: a nonzero conserved <vortensity> gives quadratic secular growth, and even zero <vortensity> can allow the linear term. Other wavenumbers still oscillate about a balanced offset. \b[$Q>1$ gives bounded axisymmetric modes, $Q<1$ allows exponential instability, and $Q=1$ has a zero-frequency marginal mode with possible algebraic growth.] These statements are axisymmetric; nonaxisymmetric shearing disturbances can have transient swing behavior. For $k_x=k_y=0$, continuity instead fixes the uniform <mass density> shift, while the velocities execute epicyclic motion; no $1/k$ gravitational formula is used.
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