(untitled) (Advances in Applied Mathematics and Mechanics, 2018)
It is known that the solution to a Cauchy problem of linear differential equations: x'(t)=A(t)x(t), withx(t_0)=x_0, can be presented by the matrix exponential as (_t_0^tA(s) ds)x_0, if the commutativity condition for the coefficient matrix A(t) holds: [_t_0^tA(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 (_t_0^t A(s) ds)=A(t) (_t_0^t A(s) ds), but do not possess the commutativity condition. The presented matrices consist of finite-times continuously differentiable entries or smooth entries.