Consider the nonuniform transport equation $$ u_t + (x^2 - 1) u_x = 0 $$ with initial condition $ u(0, x) = f(x)= e^{-x^2} $
I need to show that the solution is
$$ u(t, x) = \exp\left( -\left(\frac{x + 1 + (x - 1) e^{-2t}}{x + 1 - (x - 1) e^{-2t}}\right)^2 \right) $$ In this case, the characteristic curves are the solutions to $ \frac{dx}{dt} = x^2 - 1, $ and by seperation of variables we get the equation $ \beta(x) = \frac{1}{2} \log \left( \frac{x(t)-1}{x(t)+1} \right) = t +k $
I could not figure out how to proceed. I thought $ u(x,t) = f(\beta(x) -t) $ but this does not lead me to the desired result.