Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 6 / 1 / c / Solution

For this force, $\dot{V}=V$, hence $V(t)=e^{t}v$ and $X(t)=x+(e^t-1)v$. Invert the characteristic flow map: the initial velocity for an endpoint $(x,v)$ is $e^{-t}v$, and the initial position is $x-(1-e^{-t})v$. Constancy along each characteristic curve givesThere is no amplitude factor here: this is scalar advection, not the conservative continuity equation for a compressible density.