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A semi-algorithm to find elementary first order invariants of rational second order ordinary differential equations. (English) Zbl 1115.65077

This paper is concerned with the explicit calculation (up to a certain polynomial degree) of first order invariants in rational second order differential equations \[ y''= M(x,y,y')/N(x,y,y'), \tag{1} \] where \(M\) and \(N\) are polynomials in \( x, y , y', \) provided that there exist such an integral. The authors propose a semi-algorithmic approach closely related to the one used by M. J. Prelle and M. F. Singer [Trans. Am. Math. Soc. 279, 215–229 (1983; Zbl 0527.12016)] for a first order equation \( y' = R(x,y)\) where \(R\) is a rational function of its arguments that allows to obtain a solution in terms of elementary functions that can be computed after a sufficient number of steps. Several theoretical results on the form of first order invariants of (1), provided that they exist, are given leading to a calculation of them. Finally, some examples as well as remarks about the explicit computation are also given.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations

Citations:

Zbl 0527.12016

Software:

ODEtools; FiOrDii
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Full Text: DOI arXiv

References:

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