I have a task to solve the following quasilinear PDE (find $c(x,t)$): $$ c_x v + c_t = - v_x c $$ $c \in (0,20) , t \in (0, \infty)$
where I know function $v(x)$ to be $v(x) = \frac{3}{40}(1+\cos(\frac{3}{20}x -12))$ and the initial value for $c(x,0)=c_0=0,02$.
Using method of characteristics, I formulated (what we here call) characteristic system: $$x'=v$$ $$t'=1$$ $$z'=-v_xz$$ (at this forum often noted as: $\frac{dx}{v} = \frac{dt}{1} = \frac{dz}{-v_xz}$)
but I have trouble solving it - finding the characteristic curves. Since $x,t,z$ are now parametrized and actually stand for $x(s),t(s),z(s)$, the fuction $v(x)$ now stands for $v(x(s))$. This leads to rather nasty integration with whitch I am struggling for a long time. To find the two characteristic curves, I need to solve:
$\int x'(s) + v(x(s))t'(s) ds = const.$
$\int t'(s)v_x(x(s)) + \frac{z'(s)}{z(s)} ds = const.$
Is this even the right approach? And if so, could someone help me with the integration? So far our school projects involved only "nice" functions, no need for the nested $v(x(s))$, so this is quite confusing...
You are on the right track, but the integrals involving $v(x)$ and $v_x(x)$ have to be expressed explicitly to go further :