= Solution
The independent dimensionless variable is $S=s/s_\infty$. Since $gZ_i=s_\infty GZ_i$ and $dS/ds=1/s_\infty$, the chain rule gives $d(gZ_i)/ds=d(GZ_i)/dS$. The derivative in the normalized equation must consequently be with respect to $S$. Substituting the gas history in the <retained-ejecta chemical evolution> equation yields
$$
\frac{d(GZ_i)}{dS}+\frac{1-\beta f}{(1-\beta)S(1-S)}GZ_i=y_if.
$$
This step allows a varying retention fraction $f(S)$; no derivative of $f$ is needed because $f$ enters the instantaneous metal source and loss terms.
Set $H=GZ_i$ and now take $f=S$. The coefficient in this first-order <ordinary differential equation> decomposes as
$$
\frac{1-\beta S}{(1-\beta)S(1-S)}=\frac1{(1-\beta)S}+\frac1{1-S}.
$$
Hence an <integrating factor> is $\mu(S)=S^{1/(1-\beta)}/(1-S)$. With $q=1+1/(1-\beta)=(2-\beta)/(1-\beta)$, multiplication by this <integrating factor> gives
$$
\mu H=S^qZ_i,
\qquad
\frac d{dS}(S^qZ_i)=\frac{y_iS^q}{1-S}.
$$
The general solution has an additional term $C/S^q$. Finite initial <gas-phase metallicity> rules it out. Consequently, for $0<S<1$,
$$
\boxed{Z_i(S)=\frac{y_i}{S^q}\int_0^S\frac{u^q}{1-u}\,du.}
$$
The upper limit is the dimensionless $S$, not the unnormalized stellar mass. On each compact subinterval of $0\leq S<1$, the <geometric series> $1/(1-u)=\sum_{k\geq0}u^k$ converges uniformly, permitting integration term by term. Thus
$$
Z_i=\frac{y_i}{S^q}\sum_{k=0}^{\infty}\frac{S^{q+k+1}}{q+k+1}
=\boxed{y_i\sum_{n=1}^{\infty}\frac{S^n}{q+n}.}
$$
In particular $Z_i\sim y_iS/(q+1)$ at small $S$. For positive <stellar yield>, the series is strictly increasing, so the inverse needed for the <metallicity distribution function> exists. Differentiating either representation gives $Z_i'=y_i/(1-S)-qZ_i/S>0$. It diverges logarithmically as gas exhaustion approaches; the trace-metal approximation eventually becomes inadequate.
The <quadratic gas-history enrichment model> is not a closed gas reservoir: its gas mass initially increases. This can be made consistent with the metal equation by pristine <galactic gas inflow>. Indeed gas accounting gives $dg/ds=-(1-\beta f)/(1-\beta)+dM_{\rm in}/ds$. Since $dg/ds=1-2S$ and $f=S$, the required inflow is $dM_{\rm in}/ds=q(1-S)\geq0$. It adds no metals and therefore leaves the preceding derivation intact.
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