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The governing equation of motion is then

$$

\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.

$$

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?