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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
34A34 Nonlinear ordinary differential equations and systems, general theory
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
Software:
FiOrDii; ODEtools
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References:
[1] Stephani, H., Differential equations: their solution using symmetries, () · Zbl 0645.53045
[2] Bluman, G.W.; Kumei, S., Symmetries and differential equations, Applied mathematical sciences, vol. 81, (1989), Springer Verlag · Zbl 0718.35004
[3] Olver, P.J., Applications of Lie groups to differential equations, (1986), Springer-Verlag · Zbl 0656.58039
[4] Cheb-Terrab, E.S.; Duarte, L.G.S.; da Mota, L.A.C.P., Computer algebra solving of first order ODEs using symmetry methods, Comput. phys. commun., 101, 254, (1997) · Zbl 0927.65091
[5] Cheb-Terrab, E.S.; Duarte, L.G.S.; da Mota, L.A.C.P., Computer algebra solving of second order ODEs using symmetry methods, Comput. phys. commun., 108, 90, (1998) · Zbl 0930.65079
[6] Prelle, M.; Singer, M., Elementary first integral of differential equations, Trans. am. math. soc., 279, 215, (1983) · Zbl 0527.12016
[7] R. Shtokhamer, Solving first order differential equations using the Prelle-Singer algorithm, Technical report 88-09, Center for Mathematical Computation, University of Delaware, 1988. · Zbl 0673.68026
[8] C.B. Collins, Algebraic invariants curves of polynomial vector fields in the plane, Preprint. Canada: University of Waterloo, 1993.; C.B. Collins, Quadratic vector fields possessing a centre, Preprint. Canada: University of Waterloo, 1993.
[9] Christopher, C., Liouvillian first integrals of second order polynomial differential equations, Electron. J. differen. equat., 49, 7, (1999), electronic · Zbl 0939.34002
[10] Christopher, C.; Llibre, J., Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. differen. equat., 16, 1, 5-19, (2000) · Zbl 0974.34005
[11] Llibre, J., Integrability of polynomial differential systems, (), 437-531, (Chapter 5) · Zbl 1088.34022
[12] Duarte, L.G.S.; Duarte, S.E.S.; da Mota, L.A.C.P., A method to tackle first order ordinary differential equations with Liouvillian functions in the solution, J. phys. A: math. gen., 35, 3899-3910, (2002) · Zbl 1040.34006
[13] Duarte, L.G.S.; Duarte, S.E.S.; da Mota, L.A.C.P., Analyzing the structure of the integrating factors for first order ordinary differential equations with Liouvillian functions in the solution, J. phys. A: math. gen., 35, 1001-1006, (2002) · Zbl 1002.34002
[14] Duarte, L.G.S.; Duarte, S.E.S.; da Mota, L.A.C.P.; Skea, J.F.E., Extension of the prelle – singer method and a MAPLE implementation, Comput. phys. commun., holanda, 144, 1, 46-62, (2002) · Zbl 0994.65082
[15] Davenport, J.H.; Siret, Y.; Tournier, E., Computer algebra: systems and algorithms for algebraic computation, (1993), Academic Press Great Britain · Zbl 0865.68064
[16] Duarte, L.G.S.; Duarte, S.E.S.; da Mota, L.A.C.P.; Skea, J.E.F., Solving second order ordinary differential equations by extending the prelle – singer method, J. phys. A: math. gen., 34, 3015-3024, (2001) · Zbl 1017.34010
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