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The independent dimensionless variable is $S=s/s_{\infty }$. Since $g{Z}_i=s_{\infty }G{Z}_i$ and $dS/ds=1/s_{\infty }$, the chain rule gives $d(g{Z}_i)/ds=d(G{Z}_i)/dS$. The derivative in the normalized equation must consequently be with respect to $S$. Substituting the gas history in the retained-ejecta chemical evolution equation yieldsThis 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 $H=G{Z}_i$ and now take $f=S$. The coefficient in this first-order ordinary differential equation decomposes asHence 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 givesThe general solution has an additional term $C/S^q$. Finite initial gas-phase metallicity rules it out. Consequently, for $0<S<1$,The upper limit is the dimensionless $S$, not the unnormalized stellar mass. On each compact subinterval of $0\le S<1$, the geometric series $1/(1-u)={\sum }_{k\ge 0}{u}^k$ converges uniformly, permitting integration term by term. ThusIn particular $Z_i\sim y_{i}S/(q+1)$ 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 $Z_i^{\prime }=y_i/(1-S)-q{Z}_i/S>0$. 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 $dg/ds=-(1-\beta f)/(1-\beta )+d{M}_{\mathrm{in}}/ds$. Since $dg/ds=1-2S$ and $f=S$, the required inflow is $d{M}_{\mathrm{in}}/ds=q(1-S)\ge 0$. It adds no metals and therefore leaves the preceding derivation intact.