Problem $11$: derive the isentropic flow relationships $($BONUS $+10$ pts $).$ In this process, we begin

with the differential equation

$\frac{dp}{p}=-\frac{dv}{v}$

and arrive at the final form

$\frac{p_2}{p_1}=(\frac{{}_2}{{}_1}{)}^{}=(\frac{T_2}{T_1}{)}^{\frac{}{-1}}$

Hints:

Integrate it between points $1$ and $2$

${\displaystyle \underset{x_1}{\overset{x_2}{}}}(\frac{1}{x})dx=ln{x}_2-ln{x}_1$

Use relationship between specific volume $(v)$ and density $()$:$v=\frac{1}{}$

Use the equation of state $p=RT$