Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 70 / 3 / c / Solution

A solution reaching arbitrarily small radii must obey $\sqrt{\lambda }h(\mathcal{M})=1+3x/2$ with $h\ge 2$. Taking $x\downarrow 0$ therefore requires $2\sqrt{\lambda }\le 1$, or $\lambda \le 1/4$. This upper bound is attained: at $\lambda =1/4$ the decreasing branch $0<\mathcal{M}<1$ has one solution for every $x>0$ and approaches Mach one as $x\to 0$. HenceThe argument establishes the maximum allowed by a globally continuing steady flow, without incorrectly claiming a finite-radius sonic point at $\gamma =5/3$.