I'm working on a PDE problem and I came across a step I just can't figure out, would appreciate any help.
The question involves solving the general wave equation. I've got to here $$u_{t}+cu_x=f(x+ct)$$ The problem I'm having is what comes after.
(1) How do we derive $\frac{dx}{dt}=c$?
My best guess is that for any $au_x+bu_y=g$, we can deduce that $\frac{dy}{dx} = \frac{b}{a}$. How though?
(2) The solution defines paths $\frac{dx}{dt}=c$ and $x(h,0)=h$. What is the point of the latter?
(3) After defining the path, the biggest problem I have is how they jump to $$\frac{du}{dt}=f(x+ct)$$
Would appreciate any help understanding any of these. Thank you