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Linearization criteria for a system of two second-order ordinary differential equations. (English) Zbl 1213.34056

For a system of second-order ordinary differential equations
\[ x''=f(t,x,x'),\quad x\in\mathbb R^n, \]
conditions of its linearizability to the form \(z''=0\), \(z\in\mathbb R^n\), are well known. However, an arbitrary linear system needs not to be equivalent via an invertible point transformation to this simplest form. The paper provides criteria for a system of two second-order equations to be mapped to a linear system of general form. Necessary and sufficient conditions for linearization by means of a point transformation are given in terms of the system coefficients. The results obtained are illustrated by examples.

MSC:

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