= Solution
To construct the regular <open dust linear growth mode>, use <reduction of order>, writing $\delta=\delta_1u$ and denoting the coefficient of $\delta'$ by $p(x)$. Since $\delta_1$ solves the <homogeneous linear differential equation>, its terms cancel and leave
$$
u''+\left(2\frac{\delta_1'}{\delta_1}+p\right)u'=0,\qquad p=\frac3{2x}+\frac1{2(1+x)}.
$$
The <integrating factor> for this first-order equation in $u'$ is $\delta_1^2e^{\int p\,dx}$, so
$$
u'=C\frac{e^{-\int p\,dx}}{\delta_1^2}
=C\frac{x^{-3/2}(1+x)^{-1/2}}{(1+x)x^{-3}}
=C\left(\frac{x}{1+x}\right)^{3/2}.
$$
Choose $C=1$ and the lower <integration> endpoint zero. This selects the mode regular at early times without adding a multiple of $\delta_1$:
$$
\boxed{\delta_2(x)=\sqrt{\frac{1+x}{x^3}}\int_0^x\left(\frac{y}{1+y}\right)^{3/2}\,dy.}
$$
Its <Wronskian> with $\delta_1$ is $\delta_1\delta_2'-\delta_1'\delta_2=x^{-3/2}(1+x)^{-1/2}>0$, proving <linear independence>.
No explicit <antiderivative> is needed for the limits. For $x\ll1$, the integrand is $y^{3/2}[1+O(y)]$, whence its <integral> is $(2/5)x^{5/2}[1+O(x)]$. For $x\gg1$, the integrand tends to one, so its <integral> divided by $x$ tends to one, while $x\delta_1\to1$. Consequently
$$
\boxed{\delta_2\sim\frac25x\quad(x\to0),\qquad \delta_2\to1\quad(x\to\infty).}
$$
Early <matter domination> permits <gravitational instability> to amplify the <density contrast> in proportion to the <scale factor>. Later, the <matter density parameter> tends to zero and <Hubble friction> suppresses further amplification. Indeed, for $a\propto t$, neglecting the matter source in the <linear matter perturbation growth equation> gives $\ddot\delta+2\dot\delta/t\simeq0$, whose modes are a constant and $t^{-1}$. These agree with the late limits of the two exact modes.
\Image[/past-exam-of-the-cosmology-course-of-the-university-of-oxford/2025-open-universe-growth.png]
{title=Linear density growth saturates when spatial curvature dominates}
{description=The regular mode tends to $2x/5$ during <matter domination> and to one during <curvature-dominated universe>. Here $x=1$ marks equal contributions of matter and <spatial curvature of an FLRW universe> to the <Friedmann equation>. The normalization is the one in part (d).}
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