TY - JOUR
T1 - A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential
AU - Ma , Wen-Xiu
AU - Gu , Xiang
AU - Gao , Liang
JO - Advances in Applied Mathematics and Mechanics
VL - 4
SP - 573
EP - 580
PY - 2009
DA - 2009/01
SN - 1
DO - http://doi.org/10.4208/aamm.09-m0946
UR - https://global-sci.org/intro/article_detail/aamm/8386.html
KW - Cauchy problem, chain rule, commutativity condition, fundamental matrix solution.
AB - <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>