Decomposition of a second-order linear time-varying differential system as the series connection of two first order commutative pairs

Necessary and sufficiently conditions are derived for the decomposition of a second order linear time-varying system into two cascade connected commutative first order linear time-varying subsystems. The explicit formulas describing these subsystems are presented. It is shown that a very small class of systems satisfies the stated conditions. The results are well verified by simulations. It is also shown that its cascade synthesis is less sensitive to numerical errors than the direct simulation of the system itself.