= Solution
Distinguish physical <time> $t$ from scaled <time> $T=t/\tau$. Put
$$
P=P_0+P_s p,\qquad P_s=\frac M{GP_0},\qquad \dot V=\frac M{P_0}v,\qquad v(T)=[p(T-1)]_+.
$$
Then $K(P_s/\tau)p'=M-[P_0+P_sp](M/P_0)v$, giving
$$
\boxed{p'=\alpha[1-(1+\mu p)v],\quad\alpha=\frac{\tau GP_0}{K},\quad\mu=\frac M{GP_0^2},\quad p_1=p(T-1).}
$$
A <steady state> must be positive, since otherwise <pulmonary ventilation>[ventilation] vanishes while production remains positive. It solves $\mu p_*^2+p_*-1=0$, so
$$
\boxed{p_*=\frac{2}{1+\sqrt{1+4\mu}}=1-\mu+2\mu^2+O(\mu^3).}
$$
The value $1/(1+\mu)=1-\mu+\mu^2+O(\mu^3)$ printed in the question agrees only through first order. Thus it is a small-$\mu$ approximation with error of order $\mu^2$, not an exact <equilibrium> for nonzero $\mu$. The other quadratic root is negative and inadmissible for this steady <pulmonary ventilation>[ventilation] balance.
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