This paper studies the commutativity and stability for the Heun’s linear
time-varying system (LTVS) with both zero and non-zero initial
conditions(ICs). Given a LTVS A of order 2 , we find it’s commutative
pair, that is a new LTVS B of order m ≤ n . Explicit commutative
theories and conditions for second-order LTVSs are derived and solved to
simplify and guarantee the equivalency between the connected
input-output of systems A B and B A . The explicit results obtained are
juxtaposed by simulation in order to investigate the commutativity of
Heun’s differential system, sensitivity of Heun’s system, effects due to
disturbance on Heun’s system, robustness on Heun’s system and problems
regarding the stability of Heun’s system. This findings will help to
fill the gap on stability problem, system behaviors, commutativity
theory, and general theory for solutions of differential equations,
which has significant contribution to science and unlimited application
in engineering, our results are verify using Heun’s differential system
as well as authenticated by Wolfrom Mathematica 1 1 and Matlab.