Abstract
It is known that the solution to a Cauchy problem of linear differential equations:
x'(t) = A(t)x(t), with x(t(0)) = x(0),
can be presented by the matrix exponential as exp(integral(t)(t0) A(s) ds)x(0), if the commutativity condition for the coefficient matrix A(t) holds:
[integral(t)(t0) A(s) ds, A(t)] = 0.
A natural question is whether this is true without the commutativity condition. To give a definite answer to this question, we present two classes of illustrative examples of coefficient matrices, which satisfy the chain rule
d/dt exp(integral(t)(t0) A(s) ds) A(t) exp(integral(t)(t0) A(s) ds),
but do not possess the commutativity condition. The presented matrices consist of finite-times continuously differentiable entries or smooth entries.