= Solution
Write $N_2=N_{\mathrm{th}}+u$ and $n=n_{\mathrm{ss}}+w$. Keeping terms linear in the perturbations gives
$$
\frac d{dt}\begin{pmatrix}u\\w\end{pmatrix}
=\begin{pmatrix}-r/\tau_2&-1/(f\tau_c)\\f(r-1)/\tau_2&0\end{pmatrix}
\begin{pmatrix}u\\w\end{pmatrix}.
$$
Equivalently,
$$
\ddot w+\frac r{\tau_2}\dot w+\frac{r-1}{\tau_c\tau_2}w=0.
$$
The characteristic exponents are
$$
s_\pm=-\frac r{2\tau_2}\pm i\sqrt{\frac{r-1}{\tau_c\tau_2}-\frac{r^2}{4\tau_2^2}}.
$$
Consequently the above-threshold <steady state> has damped <oscillations> precisely when
$$
\boxed{\tau_2>\frac{r^2}{4(r-1)}\tau_c,\qquad r>1.}
$$
Their ordinary <frequency> is
$$
\boxed{f_{\mathrm{ro}}=\frac1{2\pi}\sqrt{\frac{r-1}{\tau_c\tau_2}-\frac{r^2}{4\tau_2^2}}
\simeq\frac1{2\pi}\sqrt{\frac{r-1}{\tau_c\tau_2}}.}
$$
The approximation neglects the damping correction when $r^2\tau_c\ll4(r-1)\tau_2$. The exponential envelope decays on time $2\tau_2/r$. The doped-volume fraction cancels out of this linear <frequency> once the pump is expressed through $r$. https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/18f30fad63a62ef4dd894d3752b55a60_chapter7.pdf[MIT 6.974, Chapter 7, <laser> dynamics and relaxation <oscillations>]{ref}
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