= Solution
Write proper position as $\mathbf r=a(t)\mathbf x$ and proper velocity as
$$
\mathbf u=\dot{\mathbf r}=H\mathbf r+\mathbf v,
\qquad
\mathbf v=a\dot{\mathbf x}.
$$
Substitute this decomposition into the proper-coordinate <Euler equations for an inviscid fluid>, use $\nabla_r=a^{-1}\nabla_x$, and subtract the homogeneous-background acceleration. With $\rho=\bar\rho(1+\delta)$, one obtains the comoving <peculiar-velocity equation>
$$
\boxed{\frac{\partial\mathbf v}{\partial t}+\frac{\dot a}{a}\mathbf v
+\frac1a(\mathbf v\mathbin\cdot\nabla)\mathbf v
=-\frac1a\nabla\Phi
-\frac1{a\bar\rho(1+\delta)}\nabla P.}
$$
The <peculiar gravitational potential> $\Phi$ is the total Newtonian potential with the potential of the exactly homogeneous expanding background subtracted. Its gradient therefore generates only accelerations relative to the <Hubble flow>.
For a pressureless fluid, <linearization> discards the quadratic advection term, leaving
$$
\dot{\mathbf v}+H\mathbf v=-\frac1a\nabla\Phi,
\qquad
\boxed{\frac d{dt}(a\mathbf v)=-\nabla\Phi.}
$$
In the early <Einstein-de Sitter universe>, the growing mode has $\Phi(\mathbf x,t)=[D(a)/a]\Phi_i(\mathbf x)$. Integration gives
$$
\boxed{\mathbf v=-\frac{\nabla\Phi_i}{a}\int^t\frac{D(a)}a\,dt.}
$$
The <linear growth factor> obeys
$$
\ddot D+2H\dot D=4\pi G\bar\rho D.
$$
Since $\bar\rho=\bar\rho_i/a^3$ after choosing $a_i=1$, this is equivalent to
$$
\boxed{\frac{D(a)}a=\frac1{4\pi G\bar\rho_i}
\frac d{dt}\left(a^2\frac{dD}{dt}\right).}
$$
It follows that $a\mathbf v=-a^2\dot D\nabla\Phi_i/(4\pi G\bar\rho_i)$. Since $\dot{\mathbf x}=\mathbf v/a$, one final integration gives the <Zel'dovich approximation>
$$
\boxed{\mathbf x(t)\simeq\mathbf x_i
-\frac{D(a)}{4\pi G\bar\rho_i}\nabla\Phi_i,}
$$
where an additive initial displacement has been absorbed into $\mathbf x_i$.
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