For the forthcoming exercises consider the $1$D transport equation $$u$/$t$+($f$($u$,$x$,$t$))\_$x$=0$ where t is time, and x represents location. We will assume that f$($u$,$x$,$t$)=$au where a$=$a$($x$,$u$,$t$)$ in general.

$(10$ points$)$ Consider the case where a$($u$)=$u$/2$ so that f$($u$)=$u$^2/2.$ The initial condition is u$($x$,0)=$u$\_0($x$).$ This is Burgers$$ equation.

What is dx$/$dt$?$