= Solution
Let $\delta=\Sigma'/\Sigma_0$ and define the derivative following the background <Keplerian shear> by
$$
\mathcal D_0=\partial_t-\frac32\Omega x\partial_y.
$$
The linearized continuity and momentum equations are
$$
\boxed{\mathcal D_0\delta+\partial_xu_x'+\partial_yu_y'=0},
$$
$$
\boxed{\mathcal D_0u_x'-2\Omega u_y'=-c_s^2\partial_x\delta},
$$
$$
\boxed{\mathcal D_0u_y'+\frac12\Omega u_x'=-c_s^2\partial_y\delta}.
$$
The coefficient $\Omega/2$ combines the background shear with the <Coriolis acceleration>.
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