= Solution
The axisymmetric razor-thin <astrophysical disk> equations are
$$
\partial_t\Sigma+\frac1R\partial_R(R\Sigma u_R)=0,
$$
$$
\partial_tu_R+u_R\partial_Ru_R-\frac{u_\phi^2}{R}
=-\frac1\Sigma\partial_Rp-\partial_R\Phi,
$$
$$
\partial_tu_\phi+u_R\partial_Ru_\phi+\frac{u_Ru_\phi}{R}=0,
\qquad
\nabla^2\Phi=4\pi G\Sigma(R,t)\delta(z).
$$
For the stationary background, mass and azimuthal momentum conservation are automatic, while radial force balance and the <Poisson equation> give
$$
R\Omega^2=\frac1{\Sigma_0}\frac{dp_0}{dR}+\frac{d\Phi_0}{dR},
\qquad
\nabla^2\Phi_0=4\pi G\Sigma_0\delta(z).
$$
Here $dp_0/dR=0$ under the stated constant-pressure assumption.
Retaining first-order perturbations and using $c_s^2=dp/d\Sigma$ gives
$$
\partial_t\Sigma'+\frac{\Sigma_0}{R}\partial_R(Ru_R')=0,
$$
$$
\partial_tu_R'-2\Omega u_\phi'
=-\frac{c_s^2}{\Sigma_0}\partial_R\Sigma'-\partial_R\Phi',
$$
$$
\partial_tu_\phi'+\frac{\kappa^2}{2\Omega}u_R'=0,
\qquad
\nabla^2\Phi'=4\pi G\Sigma'\delta(z),
$$
where
$$
\kappa^2=4\Omega^2+2R\Omega\frac{d\Omega}{dR}
$$
is the squared <radial epicyclic frequency>. In the local $|k|R\gg1$ limit, a <Fourier mode> $e^{i(kR-\omega t)}$ obeys
$$
-i\omega\Sigma'+ik\Sigma_0u_R'=0,
$$
$$
-i\omega u_R'-2\Omega u_\phi'
=-ik\left(c_s^2\frac{\Sigma'}{\Sigma_0}+\Phi'\right),
$$
$$
-i\omega u_\phi'+\frac{\kappa^2}{2\Omega}u_R'=0,
\qquad
\Phi'=-\frac{2\pi G\Sigma'}{|k|}.
$$
Eliminating $\Sigma'$, $u_\phi'$, and $\Phi'$ yields the local <dispersion relation>
$$
\boxed{
\omega^2=\kappa^2-2\pi G\Sigma_0|k|+c_s^2k^2
}.
$$
For a <Keplerian orbit>, $\Omega\propto R^{-3/2}$, so
$$
\kappa^2=4\Omega^2-3\Omega^2=\Omega^2.
$$
Instability requires $\omega^2<0$, equivalently a mode with positive imaginary frequency. Treating the right-hand side as a quadratic in $x=|k|$, its two roots are
$$
x_\pm=\frac{\pi G\Sigma_0}{c_s^2}
\left(1\pm\sqrt{1-Q^2}\right),
\qquad
Q=\frac{c_s\Omega}{\pi G\Sigma_0}.
$$
Real distinct roots, and hence an unstable interval $x_-<|k|<x_+$, exist exactly when the <Toomre stability criterion> has $Q<1$.
With $h=c_s/\Omega$,
$$
\frac{\omega^2}{\Omega^2}
=1-\frac{2h|k|}{Q}+h^2k^2.
$$
The constant positive term is epicyclic restoration by rotation and stabilizes long wavelengths. The negative term is the razor-thin disk's self-gravity and drives collapse. The positive $k^2$ term is gas-pressure restoration and stabilizes short wavelengths. Gravitational instability can therefore survive only on an intermediate band of scales.
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