(a) Let N be the number of nuclei of X at time t.
Rate of formation of X = α
Rate of disintegration = λN
Number of nuclei of X will increase until both the rates will become equal.
Therefore,
(b) Net rate of formation of X at time t is,
Integrating with proper limits, we have
This expression shows that number of nuclei of X are increasing exponentially from 0 to α/λ.