An extension of "A new method of computing the state transition matrix of linear time-varying systems"

The class of linear time-varying systems
\dot{x}(t) = A (t)x(t)
where the state transition matrix can be computed by
\Phi(t,0) = exp (A_{1}t) exp (A_{2}t)
is extended to the cases where eigenvalues of
A(t)
need not be independent of
t
.