= Solution
For constant inflow $\Phi$ and efficiency $\epsilon>0$, with no recycling or outflow, <mass> conservation gives the <constant-inflow gas regulator solution>:
$$
\dot M_g=\Phi-\epsilon M_g,\qquad\boxed{M_g(t)=\frac\Phi\epsilon+\left(M_0-\frac\Phi\epsilon\right)e^{-\epsilon t}.}
$$
The equilibrium $M_{g,\rm eq}=\Phi/\epsilon$ is stable: a <gas>-<mass> perturbation obeys $\delta\dot M_g=-\epsilon\delta M_g$. At late times the <star formation rate> approaches $\Psi=\Phi$, while the <gas> reservoir approaches a constant.
Integrating $\dot M_\star=\epsilon M_g$ with $M_\star(0)=0$ gives the <stellar mass in a constant-inflow gas regulator>:
$$
\boxed{M_\star(t)=\Phi t+\left(M_0-\frac\Phi\epsilon\right)(1-e^{-\epsilon t}).}
$$
It also follows immediately from $M_g+M_\star=M_0+\Phi t$. \b[The <gas> <mass> equilibrates; the stellar <mass> generally keeps growing.] With $\Phi=0$, these reduce to exponential <gas> depletion and $M_\star=M_0(1-e^{-\epsilon t})$.
A constant inflow is implicit in the requested steady asymptote, not automatic for an arbitrary prescribed rate. For time-dependent $\Phi(t)$ the exact integrating-factor result is
$$
M_g(t)=M_0e^{-\epsilon t}+\int_0^t e^{-\epsilon(t-s)}\Phi(s)\,ds;
$$
a constant late-time <gas> <mass> then requires a corresponding asymptotically steady supply. Both printed mark groups remain in this single unlettered continuation of part(a).
\Image[/past-exam-of-the-formation-of-structure-in-the-universe-course-of-the-university-of-cambridge/2015/paper-1-gas-regulator.png]
{title=Gas approaches an inflow-supported equilibrium while the stellar mass continues to grow}
{source=paper-1-gas-regulator.py}
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