= Solution
Let $u=\delta_m/H$. The supplied equation has integrating factor $(aH)^3$:
$$
\frac d{da}\bigl[(aH)^3u_a\bigr]=0.
$$
Consequently the <integral linear growth factor in a matter-Lambda universe> gives the two independent solutions
$$
\boxed{\delta_m=C_1H(a)+C_2H(a)\int_{a_i}^a\frac{d\widetilde a}{[\widetilde aH(\widetilde a)]^3}}.
$$
The first is the conventional matter-era decaying mode. A finite lower limit in the second fixes a decaying admixture; adding a multiple of $H$ can choose a pure growing normalization.
During <matter domination>, write $H=h\,a^{-3/2}$. The <integral> solution is
$$
H\int_{a_i}^a\frac{d\widetilde a}{(\widetilde aH)^3}
=\frac{2}{5h^2}\left(a-a_i^{5/2}a^{-3/2}\right).
$$
Removing the second term by the independent $H$ solution gives the <matter-era linear growth factor>
$$
\boxed{\delta_{\rm grow}\propto a,\qquad\delta_{\rm decay}\propto a^{-3/2}}.
$$
For positive <cosmological constant> at sufficiently late times, $H\to H_\Lambda>0$. The integrand approaches $H_\Lambda^{-3}a^{-3}$, whose tail is integrable. Therefore
$$
\boxed{\delta_{\rm grow}(a)\longrightarrow\text{a finite constant as }a\to\infty}.
$$
Its remaining approach has leading $a^{-2}$ behavior for the usual matter-plus-Lambda background. Growth of structure freezes rather than continuing as $a$. The basis function $H$ also approaches a constant; subtracting a suitable multiple of it isolates a genuinely decaying late-time solution $\propto a^{-2}$. The name “decaying mode” for $H$ refers to its matter-era behavior.
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