Wave propagation with variable wave speed

If you really want to solve $u_t+c(x,t)u_x=0$ :

By method of characteristics,

$\begin{cases}\dfrac{dt}{ds}=1\\\dfrac{dx}{ds}=c(x,t)\\\dfrac{du}{ds}=0\end{cases}$

With reference to http://en.wikipedia.org/wiki/Method_of_characteristics#Example,

For $\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$

$\therefore\dfrac{dx}{dt}=c(x,t)$

For $\dfrac{du}{ds}=0$ , letting $u(0)=f(x(0))$ , we have $u(x(t),t)=f(x(0))$ , where $x(t)$ satisflies $\dfrac{dx}{dt}=c(x,t)$

But $\dfrac{dx}{dt}=c(x,t)$ is a general first-order ODE, as no general method of solving first-order ODE are known, we can only express like this.