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.
$$u(t(r1),x(r1) = u(t(r2),x(r2))$$