= Solution
Expand the <density contrast> in spatial <Fourier modes> using fixed <comoving coordinates>:
$$
\delta(\mathbf x,t)=\int\frac{d^3k}{(2\pi)^3}\,\delta_{\mathbf k}(t)e^{i\mathbf k\cdot\mathbf x}.
$$
The spatial <Laplacian> multiplies a mode by $-k^2$, while its comoving <wavenumber> $k=|\mathbf k|$ is time-independent. Writing $H=\dot a/a$, the <linear cosmological density perturbation> equation becomes
$$
\ddot\delta_{\mathbf k}+2H\dot\delta_{\mathbf k}
+\left(\frac{c_s^2k^2}{a^2}-4\pi G\bar\rho\right)\delta_{\mathbf k}=0.
$$
The proper <wavenumber> is $k^{(p)}=k/a$, since the proper <wavelength> is $2\pi a/k$. The proper <Jeans wavenumber> balances the <pressure> and gravity terms:
$$
\boxed{k_J^{(p)}=\frac{\sqrt{4\pi G\bar\rho}}{c_s},\qquad
\ddot\delta_{\mathbf k}+2H\dot\delta_{\mathbf k}
+c_s^2\left[(k^{(p)})^2-(k_J^{(p)})^2\right]\delta_{\mathbf k}=0.}
$$
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