| Abstract: | It is known that the solution to a Cauchy problem of lineardifferential equations: $$x'(t)=A(t)x(t), quad {with}quadx(t_0)=x_0,$$ can be presented by the matrix exponential as$exp({int_{t_0}^tA(s),ds})x_0,$ if the commutativity conditionfor the coefficient matrix $A(t)$ holds:$$Big[int_{t_0}^tA(s),ds,A(t)Big]=0.$$ A natural question is whetherthis is true without the commutativity condition. To give a definiteanswer to this question, we present two classes of illustrativeexamples of coefficient matrices, which satisfy the chain rule $$ frac d {dt}, exp({int_{t_0}^t A(s),ds})=A(t),exp({int_{t_0}^t A(s), ds}),$$ but do not possess thecommutativity condition. The presented matrices consist offinite-times continuously differentiable entries or smooth entries. |
