Analysing the structure of the integrating factors for first-order ordinary differential equations with Liouvillian functions in the solution

Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time

“…In [23,21,22] the authors give an algorithm to compute such integrating factors. The key point is the computation of exponential factors.…”

A semi-algorithm to find elementary first order invariants of rational second order ordinary differential equations

“…(1) has a first order elementary invariant, there is an algebraic function S that satisfies eq. (13). So, by definition, there is a polynomial P (that contains S) such that P = 0 defines S. From theorem 2, this implies that the polynomial P is either an eigenpolynomial of D S or is an absolute invariant of the Lie transformation group defined by D S .✷ These theoretical results provide us an algorithm to find S. Briefly, in words, what we have to do is the following:…”

“…

A semi-algorithm to find elementary first order invariants of rational second order ordinary differential equations

AbstractHere we present a method to find elementary first integrals of rational second order ordinary differential equations (SOODEs) based on a Darboux type procedure [16,12,13]. Apart from practical computational considerations, the method will be capable of telling us (up to a certain polynomial degree) if the SOODE has an elementary first integral and, in positive case, finds it via quadratures.

…”

Solutions of some second order ODEs by the extended Prelle-Singer method and symmetries

“…In [10,11] Duarte et al have extended the PS method to include such situations. Subsequently in a series of papers [6,8,9] Chandrasekar et al have uncovered rational and even non rational first integrals for a large class of oscillator type equations, by appropriately modifying and also extending the basic idea behind the PS procedure.…”

“…An extension of this method provides the form of an integrating factor when the solution is expressible in terms of Liouvillian functions. Recently Duarte et al [10,11] have extended the technique to second-order ODEs. Essentially, their objective was to look for a wider class of possible integrating factors.…”