Abstract: Similarities and differences in solving dynamic equations between precise time-integration method with augmented matrix (PTI-AM) and extended precise time-integration method (EPTI) were analyzed. The explicit, discrete and recursive expressions for both methods were deduced, respectively, with the evolutionary random excitations in a general combined form of polynomial, exponential, and sinusoid/cosine functions. Both recursive expressions can be transformed into polynomial functions corresponding to the integral steps. With the same number of terms in the Taylor series, the recursive expression for EPTI contains additional high-order terms besides all the terms in PTI-AM. Therefore, EPTI has higher precision than PTI-AM does. If those additional high-order terms are neglected, the two methods have an identical discrete and recursive expression. In this respect, the two methods are essentially the same despite of different programming realization. An engineering example was presented, showing that the computing precision of both methods was as high as 10-significant-figures, and the computing time of EPTI was over 1 order of magnitude less than that of PTI-AM.