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

作　　者：Wen-Xiu Ma Xiang Gu Liang Gao 

机构地区：[1]Department of Mathematics and Statistics,University of South Florida,Tampa,FL 33620-5700,USA [2]Department of Physics,University of South Florida,Tampa,FL 33620-5700,USA [3]Department of Applied Mathematics,Northwestern Polytechnical University,Xi’an,Shaanxi 710072,P.R.China

出　　处：《Advances in Applied Mathematics and Mechanics》2009年第4期573-580,共8页应用数学与力学进展（英文）

基　　金：supported in part by the Established Researcher Grant and the CAS Faculty Development Grant of the University of South Florida,Chunhui Plan of the Ministry of Education of China,Wang Kuancheng Foundation,the National Natural Science Foundation of China(Grant Nos.10332030,10472091 and 10502042);the Doctorate Foundation of Northwestern Polytechnical University(Grant No.CX200616).

摘　　要：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(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition for the coefficient matrix A(t)holds:[∫_(t0)^(t)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(∫_(t0)^(t)A(s)ds)=A(t)exp(∫_(t0)^(t)A(s)ds),but do not possess the commutativity condition.The presented matrices consist of finite-times continuously differentiable entries or smooth entries.

关 键 词：Cauchy problem chain rule commutativity condition fundamental matrix solution 

分 类 号：O17[理学—数学]