SIGMA 2 (2006), 059, 8 pages math-ph/0508065
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
It is known, due to Mordukhai-Boltovski, 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
computationally-intensive 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
right-hand side of such equation. If both the numerator and the
denominator of the right-hand 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
differential equations; exact solution; first integral; integrating factor.
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