Your final equation should be

dr/r = -k3dt/3 and after integrating,

ln(r) = -k3t/3 + C

C = ln(r0) and all you have to do is substitute and operate:

ln(r) = -k3t/3 + ln(r0)
ln(r) - ln(r0) = -k3t/3
ln(r/r0) = -k3t/3
r/r0 = e-k3t/3
r = r0e-k3t/3