Lviv Radio Engineering Research Institute,

7 Naukova Str.,

Lviv 79060,

UKRAINE

E-mail: kosovtsov@escort.lviv.net

**Finding Liouvillian first integrals of rational ODEs
of any order in finite terms**

**Abstract:**

It is known (due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher
and others) that if given rational ODE has a Liouvillian first integral
then the corresponding integrating factor of the ODE must be of a very
special form $\mu=\prod_i P_i^{a_i} \exp(\prod_j Q_j^{b_j})$, where
$P_i$'s and $Q_j$'s are irreducible polynomials, $a_i$'s and $b_j$'s are
constants. 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. One have to find
first of all upper bound for the degrees of polynomials $P_i$ and $Q_j$
(this problem is unsolved yet) and then the set of coefficients for each
of the polynomials by computation-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 forms of rational ODEs of any order $\frac{d^n
y}{dx^n}= A(x,y,y^{(1)},...,y^{(n-1)}) / B(x,y,y^{(1)},...,y^{(n-1)})$,
where $A$ and $B$ are polynomials, based on examination of the resultants
of $A$ and $B$ (or $A$ and $P_i$ and the like). If one of the polynomials
$P_i$ or $Q_j$ is known, we are able to obtain by elementary computations
all of the possible polynomials as candidates to the integrating factor
form. We consider ways for obtaining such initial polynomial. If both of
$A$ and $B$ of given ODE's are not constant polynomials, the method can
determine in finite terms the explicit expression of an integrating factor
if the ODE permits integrating factors of above mentioned form and then
the Liouvillian first integral. The tests of procedures, based on proposed
method, implemented in Maple in the case of rational integrating
factors (see http://www.maplesoft.com/applications/app_center_view.aspx?AID=1309

and http://www.maplesoft.com/applications/app_center_view.aspx?AID=1652)
confirm the consistence and efficiency of the method.