= Solution
The <razor-thin disk approximation> uses vertically integrated <pressure> and <surface density of a disk>. To close the printed equations, interpret the <barotropic equation of state> as $P=P(\Sigma)$, or supply an equivalent fixed vertical conversion from the printed $\rho$ to $\Sigma$. Without that conversion, the literal $P=P(\rho)$ does not close this two-dimensional system. Set $c_s^2=(dP/d\Sigma)_0>0$, and write $\Sigma=\Sigma_0+\sigma$, $\mathbf u=\mathbf u_0+\mathbf u'$, $h'=P'/\Sigma_0=c_s^2\sigma/\Sigma_0$. <Axisymmetric flow> here means $\partial_y=0$, so background <advection> vanishes. The background shear enters through $\mathbf u'\cdot\nabla\mathbf u_0=-3\Omega u'_x\mathbf e_y/2$. The complete <linearized equations> are
$$
\boxed{\partial_tu'_x=2\Omega u'_y-\partial_xh',\qquad \partial_tu'_y=-\frac\Omega2u'_x,\qquad \partial_t\sigma=-\Sigma_0\partial_xu'_x,\qquad h'=\frac{c_s^2}{\Sigma_0}\sigma.}
$$
These are the compressible <Keplerian shearing sheet> equations for an <inertial-acoustic wave>. No perturbed self-gravity is present in the given force law, so no gravitational potential perturbation is to be added. This closure is the <barotropic closure of a razor-thin disk>.
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