= Solution
At the end of the pulse, the <photon number density>[photon density] returns to negligible size. Put $u=N_f^*/N_i^*$ and $r=N_i^*/N_{\rm th}^*$. The result of <1/c/i> gives
$$
0=\ln u+r(1-u).
$$
Using $x=\beta\eta=1-u$, the nonzero-pulse solution obeys
$$
\boxed{rx=-\ln(1-x),\qquad
x=1-e^{-rx},\qquad x=\beta\eta.}
$$
For $r>1$, select $0<x<1$, corresponding to depletion below threshold. The root $x=0$ is also an algebraic solution for every $r$ but represents no macroscopic pulse. A positive nonzero root exists only above threshold.
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