= Solution
With negligible cold-dark-matter <velocity>, the <synchronous perfect-fluid density equation> gives $\delta_C'=-h'/2$. Differentiate and use the trace <Einstein equation>, taking the cold matter <sound speed> to vanish and its density to dominate:
$$
\delta_C''+\mathcal H\delta_C'=4\pi Ga^2\rho_C\delta_C.
$$
In a spatially flat <matter-dominated universe>, $a\propto\tau^2$, $\mathcal H=2/\tau$, and the conformal <Friedmann equation> gives $4\pi Ga^2\rho_C=3\mathcal H^2/2=6/\tau^2$. Hence
$$
\delta_C''+\frac2\tau\delta_C'-\frac6{\tau^2}\delta_C=0.
$$
Put $\delta_C\propto\tau^s$. The <indicial equation> is $s(s-1)+2s-6=(s-2)(s+3)=0$. These two independent modes span the solution space, so
$$
\boxed{\delta_C(\tau,x)=A(x)\tau^2+B(x)\tau^{-3}.}
$$
Equivalently the growing mode is proportional to $a$ and the decaying mode to $a^{-3/2}$. The coefficients are set by <initial data>. Since <synchronous gauge> has residual coordinate modes, interpreting the <density contrast> physically requires fixing its residual freedom consistently, for example by the cold-matter <rest frame> and <metric tensor> <integration> convention; the mode equation alone is not a gauge-invariant observable.
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