$\dfrac{dr}{dt} = -4rt$
$\dfrac{dr}{r} = -4t\,dt$
$\displaystyle \int \dfrac{dr}{r} = -4 \int t\,dt$
$\ln r = -2t^2 + \ln c$
$\ln r = \ln e^{-2t^2} + \ln c$
$\ln r = \ln ce^{-2t^2}$
$r = ce^{-2t^2}$
when $t = 0$, $r = r_o$
$r_o = ce^{-2(0^2)}$
$c = r_o$
then,
$r = r_o \, e^{-2t^2}$
$r = r_o \, \exp (-2t^2)$ answer