Solution
= Solution
For this force, $\dot V=V$, hence $V(t)=e^tv$ 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> gives
$$
\boxed{f(t,x,v)=f_0\bigl(x-(1-e^{-t})v,e^{-t}v\bigr)}.
$$
There is no amplitude factor here: this is scalar advection, not the conservative <continuity equation> for a compressible density.