= Solution
The initial-value problem is a <Riemann problem>. Both the one-dimensional <Euler equations> and the two constant initial states are unchanged under the simultaneous scaling $(x,t)\mapsto(\lambda x,\lambda t)$ for $\lambda>0$. There is no imposed length or time scale. Rescaling a physically admissible <entropy solution> therefore gives another solution with exactly the same initial data. Uniqueness of the physical Riemann solution makes the two coincide, so
$$
\boxed{(\rho,u,p)(x,t)=(\rho,u,p)(x/t).}
$$
The smooth pieces are consequently those in part (a), while shocks and contacts lie on rays $x=st$. Across shocks one uses the conservative mass, momentum and total-<energy> equations and the <entropy> admissibility condition, rather than extending the smooth <entropy>-advection equation through the discontinuity. Scale invariance by itself, without selecting the unique admissible solution, would not justify the uniqueness step.
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