= Solution
During <matter domination>, the radiation source can be neglected irrespective of whether the mode is inside or outside the horizon in the supplied <linear cosmological density perturbation> equation. The <Friedmann equation> gives $\mathcal H=2/\tau$ and $4\pi Ga^2\rho_C=6/\tau^2$, just as in part (a). Therefore
$$
\delta''+\frac2\tau\delta'-\frac6{\tau^2}\delta=0.
$$
Substituting $\tau^q$ gives $(q-2)(q+3)=0$, so for every <Fourier mode>
$$
\delta_{C,\mathbf k}=A_{\mathbf k}\tau^2+B_{\mathbf k}\tau^{-3}.
$$
Thus the <matter-era growing and decaying density modes> have \b[growing behaviour $\delta_{C,\mathbf k}\propto\tau^2\propto a$ for all $k$]. There is no pressure-gradient term introducing a $k$-dependent growth rate for <cold dark matter>. The coefficients retain their different earlier histories; common growth does not erase the spectral shape. A specially chosen pure decaying solution is of course possible, so the growing-mode statement does not assert that every initial condition grows.
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