
SIGMA 2 (2006), 059, 8 pages mathph/0508065
https://doi.org/10.3842/SIGMA.2006.059
Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms
Yuri N. Kosovtsov
Lviv Radio Engineering Research Institute, 7 Naukova Str., Lviv, 79060 Ukraine
Received August 31, 2005, in final form May 12, 2006; Published online June 08, 2006
Abstract
It is known, due to MordukhaiBoltovski, Ritt, Prelle,
Singer, Christopher and others, that if a given rational ODE has a
Liouvillian first integral then the corresponding integrating
factor of the ODE must be of a very special form of a product of
powers and exponents of irreducible polynomials. These results
lead to a partial algorithm for finding Liouvillian first
integrals. However, there are two main complications on the way to
obtaining polynomials in the integrating factor form. First of
all, one has to find an upper bound for the degrees of the
polynomials in the product above, an unsolved problem, and then
the set of coefficients for each of the polynomials by the
computationallyintensive method of undetermined parameters. As a
result, this approach was implemented in CAS only for first and
relatively simple second order ODEs. We propose an algebraic
method for finding polynomials of the integrating factors for
rational ODEs of any order, based on examination of the resultants
of the polynomials in the numerator and the denominator of the
righthand side of such equation. If both the numerator and the
denominator of the righthand side of such ODE are not constants,
the method can determine in finite terms an explicit expression
of an integrating factor if the ODE permits integrating factors of
the above mentioned form and then the Liouvillian first
integral. The tests of this procedure based on the proposed
method, implemented in Maple in the case of rational integrating
factors, confirm the consistence and efficiency of the
method.
Key words:
differential equations; exact solution; first integral; integrating factor.
pdf (190 kb)
ps (145 kb)
tex (11 kb)
References
 Olver P.J., Applications of Lie groups to differential equations, New York,
SpringerVerlag, 1993.
 Darboux G.,
Mémoire sur les équations différentielles
algébriques du premier ordre et du premier degré
(Mélanges),
Bull. Sci. Math., 1878, V.2, 6096, 12144, 151200.
 Prelle M., Singer M., Elementary first integral of differential equations,
Trans. Amer. Math. Soc., 1983, V.279, 215229.
 MordukhaiBoltovski D., Researches on the integration
in finite terms of differential equations of the first
order, Communicatios de la Societe Mathematique de Kharkov,
19061909, V.X, 3464, 231269 (in Russian) (English transl.:
pages 3464 by B. Korenblum and M.J. Prelle, SIGSAM
Bulletin, 1981, V.15, N 2, 2032).
 Ritt R.H., Integration
in finite terms. Liouville's theory of elementary functions, New
York, Columbia Univ. Press, 1948.
 Singer M., Liouvillian first integrals
of differential equations, Trans. Amer. Math. Soc., 1992,
V.333, 673688.
 Christopher C., Liouvillian first integrals of second order
polynomial differential equations, Electron. J.
Differential Equations, 1999, N 49, 7 pages.
 Man Y.K., MacCallum M.A.H.,
A rational approach to the PrelleSinger algorithm, J.
Symbolic Comput., 1997, V.24, 3143.
 Man Y.K., First integrals of autonomous systems of differential equations
and PrelleSinger procedure, J. Phys. A: Math. Gen., 1994,
V.27, L329L332.
 Duarte L.G.S., Duarte S.E.S., da Mota L.A.C.P.,
Analyzing the structure of the integrating factor for first
order differential equations with Liouvillian functions in the
solutions, J. Phys. A: Math. Gen., 2002, V.35, 10011006.
 Duarte L.G.S., Duarte S.E.S., da Mota L.A.C.P., A method to
tackle firstorder ordinary differential equations with
Liouvillian functions in the solution, J. Phys. A: Math.
Gen., 2002, V.35, 38993910, mathph/0107007.
 Shtokhamer R., Glinos N., Caviness B.F., Computing elementary first
integrals of differential equations, in Proccedings of the Conference on
Computers and Mathematics (July 30  August 1, Stanford, California), 1986,
V.1, 19.
 Man Y.K., Computing closed form solutions of first
order ODEs using the PrelleSinger procedure, J. Symbolic
Comput., 1993, V.16, 423443.
 Duarte L.G.S., Duarte S.E.S., da Mota L.A.C.P., Skea J.E.,
An extension of the PrelleSinger method and a MAPLE
implementation, Comput. Phys. Comm., 2002, V.144, 4662.
 Duarte L.G.S., da Mota L.A., Skea J.E.F.,
Solving second order equations by extending the PS method,
mathph/0001004.
 Kosovtsov Yu.N.,
The structure of general solutions and integrability conditions
for rational firstorder ODE's, mathph/0207032.
 Kosovtsov Yu.N., The rational generalized integrating
factors for firstorder ODEs, mathph/0211069.
 Garcia I.A., Gine J., Generalized cofactors and nonlinear
superposition principles, Appl. Math. Lett., 2003, V.16,
11371141.
 Garcia I.A., Giacomini H., Gine J., Generalized nonlinear
superposition principles for polynomial planar vector fields,
J. Lie Theory, 2005, V.15, 89104.
 Giacomini H., Gine J., An algorithmic method to determine
integrability for polynomial planar vector fields, European
J. Appl. Math., to appear.
 Kosovtsov Yu.N., 2 Procedures for finding
integrating factors of some ODEs of any order, Maple Application
Center, 2004, see
here.
 Kosovtsov Yu.N., Particular solutions and integrating factors of
some ODEs of any order, Maple Application Center, 2005, see
here.

