= Solution
Put $\mathcal H=a'/a$ and choose the residual synchronous frame comoving with <cold dark matter>, so $\mathbf v_C=0$. Its Euler equation preserves this choice. The continuity equation then gives $\delta_C'=h'/2$. Neglect background <baryon> <pressure> and all unenhanced corrections of order $c_s^2$, but retain $c_s^2k^2$ because a short wavelength can make it important. The trace equation becomes
$$
h''+\mathcal Hh'=3\mathcal H^2(\Omega_C\delta_C+\Omega_B\delta_B).
$$
Using $h'=2\delta_C'$ proves
$$
\boxed{\delta_C''+\mathcal H\delta_C'-\frac32\mathcal H^2(\Omega_C\delta_C+\Omega_B\delta_B)=0}.
$$
For <baryons> let $\theta_B=i\mathbf k\cdot\mathbf v_B$. The retained continuity and Euler terms are $\delta_B'=-\theta_B+h'/2$ and $\theta_B'+\mathcal H\theta_B-c_s^2k^2\delta_B=0$. Differentiate the first and substitute the second:
$$
\delta_B''+\mathcal H\delta_B'+c_s^2k^2\delta_B=\frac12(h''+\mathcal Hh').
$$
Hence
$$
\boxed{\delta_B''+\mathcal H\delta_B'-\frac32\mathcal H^2(\Omega_C\delta_C+\Omega_B\delta_B)+c_s^2k^2\delta_B=0}.
$$
The <pressure> sign is stabilizing. The drag terms contain first derivatives of the <density> perturbations, as printed in the PDF; those primes are lost in the converted TeX. The approximation assumes slowly varying nonrelativistic background <baryon> <pressure> and the growing physical mode, not a residual <synchronous gauge> mode.
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