Definition of a transport equation: $u_t+b\cdot u_x=0$ on $\mathbb{R}\times (0,\infty)$ and given the initial value problem: \begin{cases} u_t+tu_x=0, & (x,t)\in\mathbb{R}\times(0,\infty) \\ u=\sin(x), & (x,t)\in\mathbb{R}\times\{t=0\} \end{cases}
Using the method of characteristics, this solves to be $u(x,t)=\sin(x-\frac{t^2}{2})$ according to https://www.math.toronto.edu/jko/APM346_summary_2_2020.pdf 1.3 Example Problems.
But when using the formula proven in Evans' PDE, we know $u(x,t)=g(x-bt)=\sin(x-bt)=\sin(x-t^2),\;(b=t)$, but now this is different by a half $t^2$, is this by construction the case when using this formula, we might be off by some constant? Or did I use any assumptions on this formula wrong? Or does the method of characteristics do this (I haven't yet properly studies these)