What is the Density Wave Equation and How Does it Describe Traffic Flow?

[Dustinsfl]

Traffic is moving with a uniform density of $\rho_0$.
$$
\frac{\partial\rho}{\partial t} + c(\rho)\frac{\partial\rho}{\partial x} = \beta_0
$$
where
$$
c(\rho) = u_{\text{max}}\left(1 - \frac{2\rho}{\rho_{\text{max}}}\right).
$$
Show that the variation of the initial density distribution is given by
$$
\rho = \beta_0t + \rho(x_0,0)
$$
along a characteristic emanating from $x = x_0$ described by
$$
x = x_0 + u_{\text{max}}\left(1 - \frac{2\rho(x_0,0)}{\rho_{\text{max}}}\right)t - \beta_0\frac{u_{\text{max}}}{\rho_{\text{max}}}t.
$$

So we have $\frac{dt}{ds} = 1$, $\frac{dx}{ds}=c(\rho)$ and $\frac{d\rho}{ds} = \beta_0$.
Then $t(s) = s + c$ where $t=s$ when $t(0) = 0$.
Not sure how to handle the other two though.

 

[Dustinsfl]

What I have done so far is:
$\frac{dt}{dr} = 1\Rightarrow t = r + c$ but when $t = 0$, we have $t = r$.

$\frac{dx}{dr} = c(\rho)\Rightarrow x = tu_{\text{max}}\left(1-\frac{2\rho}{\rho_{\text{max}}}\right)+c$ but when $t=0$, we have
$$
x = tu_{\text{max}}\left(1-\frac{2\rho}{\rho_{\text{max}}}\right) + x_0.
$$

$\frac{d\rho}{dr} = \beta_0\Rightarrow \rho = t\beta_0 + c$

How do I get to
$$
\rho(x,t) = t\beta_0 +\rho(x_0,0)
$$
and their characteristic?