Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2009 / B / 6 / d / i

Distinguish physical time $t$ from scaled time $T=t/\tau$. PutThen $K(P_s/\tau )p^{\prime }=M-[P_0+P_{s}p](M/P_0)v$, givingA steady state must be positive, since otherwise ventilation vanishes while production remains positive. It solves $\mu p_{\ast }^2+p_{\ast }-1=0$, soThe 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 ventilation balance.