= Solution
Change independent variable from <proper time> to the <scale factor>. Then $\dot\delta=\dot a\delta_a$ and $\ddot\delta=\dot a^2\delta_{aa}+\ddot a\delta_a$. Under <radiation domination>, $\ddot a=-\dot a^2/a$ and $H^2\simeq8\pi G\bar\rho_r/3$. Since $\bar\rho_m/\bar\rho_r=a/a_{\rm eq}$, the <linear matter perturbation growth equation> becomes
$$
\delta_{aa}+\frac1a\delta_a-\frac{3}{2aa_{\rm eq}}\delta=0.
$$
Putting $\delta=au$ gives
$$
\boxed{u_{aa}+\frac3a u_a+\frac1{a^2}\left(1-\frac32\frac a{a_{\rm eq}}\right)u=0}.
$$
This keeps the matter self-gravity term in a radiation-dominated background. It is not an exact background equation through radiation-matter equality.
At $a/a_{\rm eq}\ll1$, neglecting that small term gives $\delta_{aa}+\delta_a/a=0$. Integration yields
$$
\boxed{\delta_m=C_1+C_2\ln(a/a_*),\qquad u=[C_1+C_2\ln(a/a_*)]/a}.
$$
Thus matter has at most logarithmic growth during this leading radiation approximation, together with a constant independent mode. The constant is non-growing, not a mode that literally falls as $a^{-1}$; that power belongs to $u$ rather than $\delta_m$.
For an increasing/decreasing basis of the displayed equation with matter self-gravity retained, put $x=\sqrt{6a/a_{\rm eq}}$. Its equation becomes $x^2\delta_{xx}+x\delta_x-x^2\delta=0$. Hence
$$
\boxed{\delta_+=I_0(x),\qquad\delta_-=K_0(x)}.
$$
The <Modified Bessel function of the first kind> gives $I_0(x)=1+3a/(2a_{\rm eq})+O((a/a_{\rm eq})^2)$, which increases slowly. The <Modified Bessel function of the second kind> gives $K_0(x)=-\tfrac12\ln(a/a_{\rm eq})+\tfrac12\ln(2/3)-\gamma_E+O((a/a_{\rm eq})|\ln(a/a_{\rm eq})|)$, where $\gamma_E$ is <Euler's constant>; this mode decreases as $a$ increases. Their leading span is precisely the constant/logarithmic pair above. Mode labels depend on the chosen basis and normalization; no rapid matter-era growth $\delta\propto a$ occurs here. Neglected background corrections can change subleading terms, so the Bessel basis should not be extrapolated through equality.
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