Am I going about this question the right way?
Solve $u_t + x^2tu_x = 0 $ with initial condition $u_0(x) = \cos x$
I first started by finding the vector field for where $u$ is constant which is $(x^2t,1)$ and so I'm looking for a set of curves such that $\frac{d}{d\tau}(x(\tau), t(\tau)) = (x(\tau)^2t(\tau), 1)$ and so I got that $t(\tau) = \tau$ and so after inverting $t(\tau) = \tau$ we're looking for a function $x(t)$ such that $\dot{x}(t) = x(t)^2t$ but I can't think of such a function as this requires solving a non-linear ODE, am I on completely the wrong track?
EDIT
I have thought of a function! It wasn't too complicated after all, the function is $x(t) = -2t^{-2}$. I will attempt the solution now and make another edit.
EDIT 2: On second thoughts I'm missing the constant $c$, but when I make $x(t) = -2t^{-2} + c$ it makes the function really complicated when squaring etc.. does this matter?