Solving Second Order Linear Differential Equations via Algebraic Invariant Curves (original) (raw)

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

Journal of Nonlinear Mathematical Physics, 2008

In this paper we compute first integrals of nonlinear ordinary differential equations using the extended Prelle-Singer method, as formulated by Chandrasekar et al in J. Math. Phys. 47 (2), 023508, (2006). We find a new first integral for the Painlevé-Gambier XXII equation. We also derive the first integrals of generalized two-dimensional Kepler system and the Liénard type oscillators.

First Integrals and Darboux Polynomials of Homogeneous Linear Differential Systems

1995

This paper studies rational and Liouvillian first integrals of homogeneous linear differential systems Y′=AY over a differential field k. Following [26], our strategy to compute them is to compute the Darboux polynomials associated with the system. We show how to explicitly interpret the coefficients of the Darboux polynomials as functions on the solutions of the system; this provides a correspondence between Darboux polynomials and semi-invariants of the differential Galois groups, which in turn gives indications regarding the possible degrees for Darboux polynomials (particularly in the completely reducible cases). The algorithm is implemented and we give some examples of computations.

How to solve linear differential equations: An outline

Programming and Computer Software, 2000

There are several definitions of closed form solutions to linear differential equations. In this paper, we look for the so-called Liouvillian solutions. Through examples, we give an overview of how the differential Galois theory leads to algorithms to find the Liouvillian solutions. We will outline the general ideas and results, but will give examples instead of proofs.

On Liouvillian integrability of the first–order polynomial ordinary differential equations

Journal of Mathematical Analysis and Applications, 2012

Recently the authors provided an example of an integrable Liouvillian planar polynomial differential system that has no finite invariant algebraic curves, see [8]. In this note we prove that if a complex differential equation of the form y ′ = a 0 (x) + a 1 (x)y + • • • + a n (x)y n with a i (x) polynomials for i = 0, 1,. .. , n, a n (x) ̸ = 0 and n ≥ 2 has a Liouvillian first integral, then it has a finite invariant algebraic curve. So, this result applies to the Riccati and Abel polynomial differential equations. We shall prove that in general this result is not true when n = 1, i.e. for linear polynomial differential equations.

Liouvillian Solutions of Linear Differential Equations of Order Three and Higher

Journal of Symbolic Computation, 1999

gave an algorithm to compute Liouvillian ("closed-form") solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van to compute semiinvariants and a theorem in in such a way that, by computing one semi-invariant that factors into linear forms, one gets all coefficients of the minimal polynomial of an algebraic solution of the Riccati equation, instead of only one coefficient. These coefficients come "for free" as a byproduct of our algorithm for computing semi-invariants. We specifically detail the algorithm in the cases of equations of order three (order two equations are handled by the algorithm of , see also Weil (1996) or Fakler (1997)).

A Kovacic-style algorithm for liouvillian solutions of third order differential equations

2000

The Kovacic algorithm [7] and its improvements (e.g. [18]) giveexplicit formulas for the Liouvillian solutions of second order lineardifferential equations. In this paper we extend this approach to thirdorder equations. The result will either be a formula or a proof thatthe differential Galois group is one of 12 finite groups (it is well knownhow to solve a differential equation with

On first integrals of second-order ordinary differential equations

2013

Here we discuss first integrals of a particular representation associated with second-order ordinary differential equations. The linearization problem is a particular case of the equivalence problem together with a number of related problems such as defining a class of transformations, finding invariants of these transformations, obtaining the equivalence criteria, and constructing the transformation. The relationship between the integral form, the associated equations, equivalence transformations, and some examples are considered as part of the discussion illustrating some important aspects and properties.

On liouvillian solutions of linear differential equations

Applicable Algebra in Engineering, Communication and Computing, 1992

This paper deals with the problem of finding liouvillian solutions of a homogeneous linear differential equation L(y) = 0 of order n with coefficients in a differential field k. For second order linear differential equations with coefficients in ko(x), where ko is a finite algebraic extension of Q, such an algorithm has been given by J. Kovacic and implemented. A general decision procedure for finding liouvillian solutions of a differential equation of order n has been given by M.F. Singer, but the resulting algorithm, although constructive, is not in implementable form even for second order equations. Both algorithms use the fact that, if L(y)= 0 has a liouvillian solution, then, L(y)= 0 has a solution z such that u = z'/z is algebraic over k. Using the action of the differential galois group on u and the theory of projective representation we get sharp bounds ~-(n) for the algebraic degree of u for differential equations of arbitrary order n. For second order differential equations we get the bound ~-(2)=12 used in the algorithm of J. Kovacic and for third order differential equation we improve the bound given by M.F. Singer from 360 to ~(3)< 36. We also show that not all values less than or equal to ~(n) are possible values for the algebraic degree of u. For second order differential equations we rediscover the values 2, 4, 6, and 12 used in the Kovacic Algorithm and for third order differential equations we get the possibilities 3, 4, 6, 7, 9, 12, 21, and 36. We prove that if the differential Galois group of L(y)= 0 is a primitive unimodular linear group, then all liouvillian solutions are algebraic. From this it follows that, if a third order differential equation L(y)= 0 is not of Fuchsian type, then the logarithmic derivative of some liouvillian solution of L(y) = 0 is algebraic of degree 3. We also derive an upper bound for the minimal number Jff(n) of possible degrees m of the minimal polynomial of an algebraic solution of the riccati equation associated with L(y)= O.