@Article{AAMM-1-4,
author = {Wen-Xiu, Ma and Xiang, Gu and Liang, Gao},
title = {A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential},
journal = {Advances in Applied Mathematics and Mechanics},
year = {2009},
volume = {1},
number = {4},
pages = {573--580},
abstract = {<p style="text-align: justify;">It is known that the solution to a Cauchy problem of linear
differential equations: $$x&#39;(t)=A(t)x(t), \quad {with}\quad
x(t_0)=x_0,$$ can be presented by the matrix exponential as
$\exp({\int_{t_0}^tA(s)\,ds})x_0,$ if the commutativity condition
for the coefficient matrix $A(t)$ holds:
$$\Big[\int_{t_0}^tA(s)\,ds,A(t)\Big]=0.$$ 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 $$ \frac d {dt}\, \exp({\int_{t_0}^t A(s)\,
ds})=A(t)\,\exp({\int_{t_0}^t A(s)\, ds}),$$ but do not possess the
commutativity condition. The presented matrices consist of
finite-times continuously differentiable entries or smooth entries.</p>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.09-m0946},
url = {https://global-sci.com/article/73703/a-note-on-exact-solutions-to-linear-differential-equations-by-the-matrix-exponential}
}