First order partial different equation

\[\frac{\partial u}{\partial t} + c(x,t)\frac{\partial u}{\partial x} = 0\]
find a curve that the u is constant along it.

\[ du = \frac{\partial u}{\partial t}dt + \frac{ \partial u}{\partial x}dx \]
\[ t = t(r),\qquad x=x(r) \]
\[ dt = \frac{dt}{dr} dr,\qquad dx = \frac{dx}{dr} dr\]

\[\frac{du}{dr} = \frac{\partial u}{\partial t}\frac{dt}{dr} + \frac{ \partial u}{\partial x}\frac{dx}{dr} \]

we hope $du=0$,then along the curve, the u is constant.
$\frac{du}{dr} = 0 = \frac{\partial u}{\partial t} + c(x,t)\frac{\partial u}{\partial x}$
we get \[\frac{dt}{dr}=1\],and \[\frac{dx}{dr}=c(x,t)\].
they determine the characteristic curve.
\[u(t(r1),x(r1) = u(t(r2),x(r2))\]