A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential

It is known that the solution to a Cauchy problem of linear differential equations: x′(t) = A(t)x(t), with x(t0) = x0, can be presented by the matrix exponential as exp( ∫ t t0 A(s) ds)x0, if the commutativity condition for the coefficient matrix A(t) holds: [ ∫ 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( ∫ t t0 A(s) ds) = A(t) exp( ∫ 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. AMS subject classifications: 15A99, 34A30, 15A24, 34A12