In cases where viscous stresses and thermal conduction can be neglected, the Euler equations describe the motion

Question:

In cases where viscous stresses and thermal conduction can be neglected, the Euler equations describe the motion of a fluid. The one dimensional form of the Euler equations (mass, momentum, and energy conservation accounting) is

where e = u + 1/2 ?²_x?is the specific total energy per unit mass and ?_x is the velocity of the fluid. This is a system of equations in the four unknowns ?, P, u, ?_x. Show that, if the fluid is a polytropic ideal gas, a fourth simple algebraic equation relating u to P and ? neatly closes the system of equations.