The solution of the differential equation $\frac{dr}{dt}=-rt, r(0) = r_o$ is

$r = r_oe^{-t^2/2}$

The correct answer is Option (3) → $r = r_oe^{-t^2/2}$

Given: $\frac{dr}{dt} = -r\,t,\; r(0)=r_{0}$.

Separate variables: $\frac{dr}{r} = -t\,dt$.

Integrate: $\ln|r| = -\frac{t^{2}}{2} + C$.

Use $r(0)=r_{0}$: $\ln|r_{0}| = C$.

Hence, $ \ln|r| = -\frac{t^{2}}{2} + \ln|r_{0}| \Rightarrow r = r_{0}e^{-t^{2}/2}$.

Solution: $r(t)=r_{0}e^{-t^{2}/2}$.