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

A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential
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Year:    2009

Author:    Wen-Xiu Ma, Xiang Gu, Liang Gao

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 4 : pp. 573–580

Abstract

It is known that the solution to a Cauchy problem of linear differential equations: $$x'(t)=A(t)x(t), \quad {with}\quad x(t_0)=x_0,$$ 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.09-m0946

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 4 : pp. 573–580

Published online:    2009-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    8

Keywords:    Cauchy problem chain rule commutativity condition fundamental matrix solution.

Author Details

Wen-Xiu Ma

Xiang Gu

Liang Gao

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