A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential (Advances in Applied Mathematics and Mechanics, 2018)
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.