= Solution
Applying $\mathcal D_0$ to the <shearing-wave ansatz> produces an extra phase term
$$
i\left(\dot k_x-\frac32\Omega k_y\right)x.
$$
The perturbation equations have spatially uniform amplitude coefficients only when this term vanishes. Hence
$$
\boxed{\dot k_x=\frac32\Omega k_y},
\qquad
\boxed{k_x(t)=k_x(0)+\frac32\Omega k_yt}.
$$
For $k_y\ne0$, the radial wavenumber changes linearly. A leading disturbance with $k_x/k_y<0$ first opens until $k_x=0$, then becomes an increasingly tightly wound trailing disturbance with $k_x/k_y>0$. This is the geometric <swing of a shearing wave>.
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