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Zbl 0961.34023
Wafo Soh, C.; Mahomed, F.M.
Symmetry breaking for a system of two linear second-order ordinary differential equations. (English)
[J] Nonlinear Dyn. 22, No.1, 121-133 (2000). ISSN 0924-090X; ISSN 1573-269X/e

For a system of two linear second-order ordinary differential equations the following canonical form is obtained: $$x''=a(t)x+b(t)y, \qquad y''=c(t)x-a(t)y,$$ and the Lie algebra of infinitesimal point symmetries depended on coefficients is calculated. The number of symmetries (dimension of symmetries Lie algebra) can be 5, 6, 7, 8 or 15. In particular it is shown that this system of equations is reducible to free particle equations $x''=y''=0$ iff the number of symmetries is equal to 15 and in the case if the number of symmetries is 5 or 6 it cannot be reduced to a system with constant coefficients. For the oscillator-like equations $$x''=-w(t)x,\quad y''=-w(t)y,$$ that has 15 symmetries it is presented a transformation to the free particle equation.
[Oganes M.Khudaverdian (Manchester, UK)]

MSC 2000:: *34C14 Symmetries, invariants
34C20 Transformation of ODE and systems
70G65 Symmetries, Lie-group and Lie-algebra methods

Keywords: linear system; canonical form; group classification of differential equations; point symmetry Lie algebra

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