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**On solving systems of autonomous ordinary differential equations by reduction to a variable of an algebra.**
*(English)*
Zbl 1254.34053

Summary: A new technique for solving a certain class of systems of autonomous ordinary differential equations over \(\mathbb K^n\) is introduced (\(\mathbb K\) being the real or complex field). The technique is based on two observations: (1) if \(\mathbb K^n\) has the structure of certain normed, associative, commutative algebras with a unit \(\mathbb A\) over \(\mathbb K\), then there is a scheme for reducing the system of differential equations to an autonomous ordinary differential equation on one variable of the algebra; (2) a technique, previously introduced for solving differential equations over \(\mathbb C\) is shown to work on the class mentioned in the previous paragraph. In particular, it is shown that the algebras in question include algebras linearly equivalent to the tensor product of matrix algebras of certain normal forms.

### MSC:

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

16-XX | Associative rings and algebras |

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\textit{A. Alvarez-Parrilla} et al., Int. J. Math. Math. Sci. 2012, Article ID 753916, 21 p. (2012; Zbl 1254.34053)

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