The system of PDEs below are a model for linearised compressible fluid flow

$\displaystyle u_x + u_t = 0$

$\displaystyle (\rho u)_x + \rho_t = 0$

$\displaystyle (E u)_x + E_t + p u_x = 0$

where $\displaystyle u=u(x,t) , \rho=\rho(x,t) , E= E(x,t)$ represent velocity, density and a measure of energy per unit volume of a fluid and assume p is constant.

Find a solution that satisfies the initial conditions

$\displaystyle u(x,0) = x$

$\displaystyle \rho(x,0) = (x-1)^2$

$\displaystyle E(x,0) = E_0 (x)$ where $\displaystyle E_0 (x)$ is a given function.

Any tips on where to start for this one, should I solve each equation independently or equate them and then solve?