State-to-state differential cross sections (DCSs) are computed quantum mechanically in full dimensionality for the title reaction using a reactant-product decoupling scheme. The DCSs are calculated at three collision energies of 0.25, 0.28, and 0.34 eV, corresponding to the existing experimental results. In good agreement with experiment, the calculated DCSs are dominated by backward scattering, thanks to the direct rebound mechanism, and the DOH product has two quanta of OD stretching vibration in the newly formed OD bond. In addition, the vibrational excitation of the OH reactant is found to result in a very different but predictable vibrational distribution of the DOH product. It is further shown at the state-to-state level that the DCSs of the DOH(v, v, v) product state from the OH(v = 1) reactant state resemble the ones of the DOH(v, v, v-1) product state from the OH(v = 0) reactant state, thanks to the spectator nature of the OH moiety.