Volume 1, Issue 4
A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential

Adv. Appl. Math. Mech., 1 (2009), pp. 573-580.

Published online: 2009-01

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• Abstract

It is known that the solution to a Cauchy problem of linear differential equations: $$x'(t)=A(t)x(t), \quad \textup{with}\quad x(t_0)=x_0,$$ 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 $$\D \frac d {dt}\, \exp({\int_{t_0}^t A(s)\, ds})=A(t)\,\exp({\int_{t_0}^t A(s)\, ds}),$$ but do not possess the commutativity condition. The presented matrices consist of finite-times continuously differentiable entries or smooth entries.

• Keywords

Cauchy problem chain rule commutativity condition fundamental matrix solution

• AMS Subject Headings

15A99 34A30 15A24 34A12

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COPYRIGHT: © Global Science Press

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@Article{AAMM-1-573, author = {Wen-Xiu Ma, 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 = {

It is known that the solution to a Cauchy problem of linear differential equations: $$x'(t)=A(t)x(t), \quad \textup{with}\quad x(t_0)=x_0,$$ 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 $$\D \frac d {dt}\, \exp({\int_{t_0}^t A(s)\, ds})=A(t)\,\exp({\int_{t_0}^t A(s)\, ds}),$$ but do not possess the commutativity condition. The presented matrices consist of finite-times continuously differentiable entries or smooth entries.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.09-m0946}, url = {http://global-sci.org/intro/article_detail/aamm/8386.html} }
TY - JOUR T1 - A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential AU - Wen-Xiu Ma, Xiang Gu & Liang Gao 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 KW - chain rule KW - commutativity condition KW - fundamental matrix solution AB -

It is known that the solution to a Cauchy problem of linear differential equations: $$x'(t)=A(t)x(t), \quad \textup{with}\quad x(t_0)=x_0,$$ 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 $$\D \frac d {dt}\, \exp({\int_{t_0}^t A(s)\, ds})=A(t)\,\exp({\int_{t_0}^t A(s)\, ds}),$$ but do not possess the commutativity condition. The presented matrices consist of finite-times continuously differentiable entries or smooth entries.

Wen-Xiu Ma, Xiang Gu & Liang Gao. (1970). A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential. Advances in Applied Mathematics and Mechanics. 1 (4). 573-580. doi:10.4208/aamm.09-m0946
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