= Solution
For <pressureless matter>, $w=0$, the <constant-equation-of-state comoving density equation> reduces to $\Delta''+2\Delta'/\eta-6\Delta/\eta^2=0$. Substitution of $\Delta\propto\eta^s$ gives $s^2+s-6=(s-2)(s+3)=0$. Since $a\propto\eta^2$,
$$
\boxed{\Delta=C_+a+C_-a^{-3/2}.}
$$
In the growing mode, $a^2\bar\rho\Delta$ is constant, so the Poisson constraint makes $\phi$ constant for a nonzero Fourier mode. The momentum constraint and the flat <Friedmann equation> give
$$
\mathcal H\phi=-4\pi Ga^2\bar\rho v=-\frac32\mathcal H^2v,
\qquad\boxed{v=-\frac{2\phi}{3\mathcal H}=-\frac{\phi\eta}{3}\propto a^{1/2}.}
$$
This is the <growing-mode velocity potential in matter domination>, in the question's convention where the peculiar velocity is $\nabla v$.
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