= Solution
Mass conservation in the <Zeldovich approximation> gives $\rho\,d^3r=\bar\rho a^3\,d^3q$. Expanding the trajectory <Jacobian determinant> to first order therefore yields $\boxed{\delta=-\nabla_q\cdot\Psi}$. At the same order, $\nabla_r=a^{-1}\nabla_q$ and
$$
\nabla_rP=c_s^2\nabla_r\rho=-\frac{\bar\rho c_s^2}{a}\nabla_q(\nabla_q\cdot\Psi)=-\frac{\bar\rho c_s^2}{a}\nabla_q^2\Psi.
$$
The last equality uses $\nabla_q\times\Psi=0$ and the <vector calculus> $\nabla\operatorname{div}=\nabla^2+\nabla\times\nabla\times$.
Split the <gravitational potential> into its homogeneous-background and perturbed parts. The background has $\nabla_r\Phi_0=(4\pi G\bar\rho/3)r$. For the perturbation, <Poisson equation> and irrotationality give
$$
\nabla_r\phi=-4\pi G\bar\rho\,a\Psi,
$$
with a harmonic or uniform-acceleration contribution set to zero by the perturbation boundary conditions. Its divergence is $-4\pi G\bar\rho\nabla_q\cdot\Psi=4\pi G\bar\rho\delta$, as required. Insert this and the pressure gradient into $\ddot r=\ddot a(q+\Psi)+2\dot a\dot\Psi+a\ddot\Psi$. The background acceleration cancels using $\ddot a/a=-4\pi G\bar\rho/3$, leaving
$$
\boxed{\ddot\Psi+2\frac{\dot a}{a}\dot\Psi-4\pi G\bar\rho\Psi-\frac{c_s^2}{a^2}\nabla_q^2\Psi=0.}
$$
Take minus the comoving divergence, and expand $\delta(q,t)=\sum_k\delta_k(t)e^{ik\cdot q}$. Since $\nabla_q^2$ acts on a <Fourier mode> by $-k^2$, each mode satisfies
$$
\boxed{\ddot\delta_k+2\frac{\dot a}{a}\dot\delta_k-\left(4\pi G\bar\rho-\frac{c_s^2k^2}{a^2}\right)\delta_k=0.}
$$
Here the expansion coordinate is comoving; confusing it with the physical coordinate would lose the factor $a^{-2}$.
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