= Solution
For constant $a$ and constant $\rho_m>0$, <Hubble friction> vanishes. Put $\omega_J=\sqrt{4\pi G\rho_m}$. The constant-coefficient <ordinary differential equation> is $\ddot\delta-\omega_J^2\delta=0$, so its <characteristic polynomial> has the real <roots of a polynomial> $\pm\omega_J$:
$$
\boxed{\delta(t)=C_+e^{\omega_Jt}+C_-e^{-\omega_Jt}.}
$$
These are the pressureless special case of the <Static-universe Jeans modes>. A positive growing-mode coefficient gives exponential <gravitational instability>, with time scale $(4\pi G\rho_m)^{-1/2}$. For initial data $\delta(t_*)=\delta_*$ and $\dot\delta(t_*)=v_*$, the equivalent answer is $\delta_*\cosh[\omega_J(t-t_*)]+(v_*/\omega_J)\sinh[\omega_J(t-t_*)]$. This solves the stated perturbation model; a globally static dust-only <FRW metric> would require additional background support.
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