As far as i remember, the three conditions are underdamped, overdamped & critically damped and they don't have any derivation but a graph b/w distance and time depicting them collectively. You would need a bit explanation of the case in the light of equation for each case i-e b/2m < Wo in case of underdamped, b/2m > Wo for overdamped and equal for critically damped (i may be wrong so proof check these equations) just a 8-10 line explanation for each case with a collective graph will suffice.

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