= Solution
Let $A=A_{21}$ and $r=g_2/g_1$. Before <stimulated emission> and <stimulated absorption> are added, the source and decay terms give
$$
\dot N_2=R_2-AN_2,\qquad
\dot N_1=AN_2-\frac{N_1}{\tau_1}.
$$
The upper level is fed at a specified rate of addition to the <number density>, its spontaneous decay feeds the lower level, and lower-level removal leads to states outside this <two-level system>. This is therefore an open pumped system, not a closed <two-level system> with conserved population.
At steady state,
$$
N_2^{(0)}=\frac{R_2}{A},\qquad N_1^{(0)}=R_2\tau_1,
\qquad\boxed{N_0^*=R_2\left(\frac1A-r\tau_1\right).}
$$
The degeneracy-weighted <population inversion> compares population per <magnetic sublevel>. It is positive only if $rA\tau_1<1$; a positive pump rate alone cannot guarantee <optical gain> if lower-level removal is too slow. A short lower-state lifetime is consequently helpful even when the upper-state pumping is unchanged.
The equations assume finite positive $A$ and $\tau_1$, approximately constant pumping and no other specified sources, sinks or branching channels. A finite steady state would not exist for continuous pumping if the lower level had no escape route. The fixed source rate also presumes access to an external reservoir; it does not imply that the two populations can grow beyond all available active atoms in a real finite medium.
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