Transient dynamics of the conversion of reacting species in a Continuous stirred tank reactor (CSTR) or Plug flow reactor (PFR) can be studied using numerical solution to differential equations, linear or non-linear. A fourth order Runge-Kutta method can be used for numerical integration. Conversion as a function of time for different values of Damkohler numbers can be obtained as output response. Transient dynamics of initiated polymerization and thermal polymerization in a CSTR was studied using 4^th order Runge-Kutta method. Multiplicity was found in the transient conversion for the initiated case. The case of 2 CSTRs in series was studied. The multiplicity was seen to shift to the second reactor. Transient dynamic analysis of a PFR for a simple irreversible first order reaction with Dankwert’s boundary conditions at the entrance and exit of the reactor was studied. The wave patterns can be seen in the transient conversion after assuming that the conversion is analytic in space and time.