A practice question says:
Find and sketch the characteristics for $2u_t + (1+t^2)u_x = 0$
So I found that the vector field for where $u$ is constant is $(1 + t^2, 2)$ and so I'm looking for a set of curves such that $\frac{d}{d\tau}(x(\tau), t(\tau)) = (1+(t(\tau))^2, 2)$ and I got that $t(\tau) = 2\tau + t_0$ and therefore $x(\tau) = \tau + \frac{1}{6}(2\tau + t_0)^3 + x_0$ but this seems awfully complicated to sketch?! And so I think I may have done something wrong?