= Solution
In deep <matter domination>, the <scale factor> is $a\propto\tau^2$, so $a'/a=2/\tau$. Neglect radiation in the gravity source, including its backreaction, to obtain
$$
\delta_c''+\frac2\tau\delta_c'-\frac6{\tau^2}\delta_c=0.
$$
For a power $\tau^p$ the <Euler-Cauchy equation> becomes $p(p-1)+2p-6=(p-2)(p+3)=0$. Its two independent <matter-era growing and decaying density modes> give the full approximate solution
$$
\boxed{\delta_c=A(\mathbf k)(\tau/\tau_i)^2+
B(\mathbf k)(\tau/\tau_i)^{-3}.}
$$
There is no $k$-dependent pressure term for <cold dark matter>. Thus this leading equation is valid both outside and inside the horizon in the stated comoving <synchronous gauge>, as long as the perturbations remain linear and the omitted radiation gravity is negligible. A different time slicing would give a different density variable.
For $k>0$, put $\omega=k/\sqrt3$, $C_g=A/\tau_i^2$ and $C_d=B\tau_i^3$. The remaining radiation equation is a forced <harmonic oscillator>:
$$
\delta_r''+\omega^2\delta_r=\frac43\delta_c''
=\frac83C_g+16C_d\tau^{-5}.
$$
The <matter-era forced radiation response> can be written exactly within the matter-background approximation as
$$
\delta_r=C_1\cos(\omega\tau)+C_2\sin(\omega\tau)
+\frac{8C_g}{3\omega^2}
+\frac{16C_d}{\omega}\int_{\tau_*}^{\tau}s^{-5}
\sin[\omega(\tau-s)]\,ds.
$$
The last term is a <Green's function> particular solution; changing its lower limit only changes the two oscillatory constants. For the decaying source, a slowly varying particular solution obeys $\delta_{r,p}\simeq(4/3)\delta_c''/\omega^2$, since a second time derivative is smaller than $\omega^2\delta_{r,p}$ by order $(k\tau)^{-2}$. Consequently the subhorizon asymptotic form is
$$
\boxed{\delta_r=C_1\cos(k\tau/\sqrt3)+C_2\sin(k\tau/\sqrt3)
+\frac{8A}{k^2\tau_i^2}
+\frac{48B\tau_i^3}{k^2\tau^5}
+O\!\left(\frac{B\tau_i^3}{k^4\tau^7}\right).}
$$
For a pure growing mode, the constant offset and acoustic terms are an exact solution of this reduced radiation equation. In particular, the late radiation contrast does not follow the ever-growing $\delta_c$.
Before crossing, a regular growing <adiabatic cosmological perturbation> has both contrasts proportional to $\tau^2$, with $\delta_r=(4/3)\delta_c$ to leading order. Afterwards <cold dark matter> keeps this quadratic growth, whereas radiation undergoes <subhorizon radiation acoustic oscillations> about a constant forced offset. To make the requested sketch concrete, one regular leading matter-era solution on both sides is
$$
\delta_c=\varepsilon(\tau/\tau_h)^2,\qquad
\delta_r=\frac{8\varepsilon}{k^2\tau_h^2}
[1-\cos(k\tau/\sqrt3)],\qquad k\tau_h=2\pi.
$$
Its small-$k\tau$ expansion gives $\delta_r=(4/3)\delta_c+O((k\tau)^4)$, and its oscillation frequency is the radiation sound frequency. The sketch plots amplitudes in units of $\varepsilon$; $\varepsilon$ can be chosen small enough that the whole plotted interval remains linear. The line at $\tau_h$ uses the wavelength-to-particle-horizon convention in the question, rather than asserting that the smooth transition occurs at a sharp surface.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-67-matter-era-horizon.png]
{title=Growing cold-matter and forced radiation contrasts through horizon crossing}
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