I have $\begin{align} u_x + u_t &= 0\\(\rho u)_x + \rho_t &= 0 \\ \left(Eu\right)_{x}+E_{t}+pu_{x}&=0\end{align} $ satisfying these boundary conditions: $u\left(x,0\right)=x,\ \rho\left(x,0\right)=\left(x-1\right)^{2},\ E\left(x,0\right)=E_{0}\left(x\right)$ where $p$ is a constant and $E_0$ is arbitrary.
My attempt at a solution so far is as follows:
The first PDE I can solve easily enough by inspection to get $u=f(x-t)$. Substituting this into the second equation $u\rho_{x}+u_{x}\rho+\rho_{t}=0$, I get $(x-t)\rho_{x}+\rho+\rho_{t}=0.$
Apparently the integral curves of $(x-t)\dfrac{\partial}{\partial x}+\dfrac{\partial}{\partial t}$ are $x=x_0e^t+t$ but I'm lost and don't know what I'm doing. I feel like I need someone to walk me through how to solve it properly because I couldn't do it again on my own.
Sorry about this formatting, I tried to use the FAQ and learn some latex but yeah :p