Here is the question : Suppose all solutions $u(x,t)$ to $au_x+bu_t=0$ satisfy $u(1,2)=u(3,6)$. What is $b/a$ ?
Along its characteristic curves $bx-at=c$, $c\in \mathbb{R}$, we know that $u$ is constant. These characteristics fill in the $xt-plane$ and if we suppose that $(1,2)$ and $(3,6)$ lie on the same characteristic, then the problem is easy. However, what if they don't ? This is, I think a possible case but can't figure out the solution. Thank you for any hint or advice !
The comment by @JJacquelin almost solves the problem. If all solutions satisfy $u(1,2) = u(3,6)$, then we deduce from $u(x,t) = F(bx - at)$ that $$ F(b - 2a) = F\big(3(b - 2a)\big) $$ must be satisfied for all $F$. This means that both arguments are equal, i.e. that $b/a = 2$.