  We consider the Cauchy problem for the Burgers hierarchy with general time
dependent coefficients. The closed form for the Green's function of the
corresponding linear equation of arbitrary order $N$ is shown to be a sum of
generalised hypergeometric functions. For suitably damped initial conditions we
plot the time dependence of the Cauchy problem over a range of $N$ values. For
$N=1$, we introduce a spatial forcing term. Using connections between the
associated second order linear Schr\"{o}dinger and Fokker-Planck equations, we
give closed form expressions for the corresponding Green's functions of the
sinked Bessel process with constant drift. We then apply the Green's function
to give time dependent profiles for the corresponding forced Burgers Cauchy
problem.
