= Solution
For $k^{(p)}\ll k_J^{(p)}$, neglect <pressure> support. The growing and decaying <Jeans instability> modes obey
$$
\ddot\delta_{\mathbf k}+2H\dot\delta_{\mathbf k}-4\pi G\bar\rho\,\delta_{\mathbf k}=0.
$$
In an <Einstein-de Sitter universe>, $a\propto t^{2/3}$ gives $H=2/(3t)$, and the matter-density relation $8\pi G\bar\rho=3H^2$ gives $4\pi G\bar\rho=2/(3t^2)$. Thus
$$
\ddot\delta_{\mathbf k}+\frac4{3t}\dot\delta_{\mathbf k}-\frac2{3t^2}\delta_{\mathbf k}=0.
$$
Try a power law $\delta_{\mathbf k}\propto t^p$. Its <characteristic equation> is
$$
p(p-1)+\frac43p-\frac23=0,
\qquad (p-2/3)(p+1)=0.
$$
Therefore the <Einstein-de Sitter density-growth modes> are
$$
\boxed{\delta_{\mathbf k}(t)=C_+t^{2/3}+C_-t^{-1}
=D_+a+D_-a^{-3/2}.}
$$
This is a linear result while $|\delta|\ll1$; a growing mode eventually leaves that regime.
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