http://www.mmrc.iss.ac.cn/pub/mm23.pdf/mmpolyrational.pdf

A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs

We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For an autonomous first order ODE, we give an algorithm to compute a rational general solution if it exists. The algorithm is based on the relation between rational solutions of the first order ODE and rational parametrizations of the plane algebraic curve defined by the first order ODE and Pade approximants.

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Rational general solutions of algebraic ordinary differential equations

Given a second order linear differential equations with coefficients in a field k=C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem of Klein states that, in the most interesting cases of the Kovacic algorithm (i.e when the projective differential Galois group is finite), the differential equation must be a pullback (a change of variable) of a standard hypergeometric equation. This provides a way to represent solutions of the differential equation in a more compact way than the format provided by the Kovacic algorithm. Formulas to make Klein's theorem effective were given in [4, 2, 3]. In this paper we will give a simple algorithm based on such formulas. To make the algorithm more easy to implement for various differential fields k, we will give a variation on the earlier formulas, namely we will base the formulas on invariants of the differential Galois group instead of semi-invariants.

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Solving second order linear differential equations with Klein's theorem

We report a new analytical method for finding the exact solution of homogeneous linear ordinary dierential equations with arbitrary order and variable coecients. The method is based on the definition of jump trans- fer matrices and their extension into limiting dierential form. The approach reduces the nth-order dierential equation to a system of n linear dieren- tial equations with unity order. The full analytical solution is then found by the perturbation technique. The important feature of the presented method is that it deals with the evolution of independent solutions, rather than its derivatives. We prove the validity of method by direct substitution of the so- lution in the original dierential equation. We discuss the general properties of dierential transfer matrices and present several analytical examples, showing the applicability of the method.

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Analytical solution of linear ordinary differential equations by differential transfer matrix method

In this paper, we give a necessary and sufficient condition for an algebraic ODE to have an algebraic general solution. For a first order autonomous ODE, we give an optimal bound for the degree of its algebraic general solutions and a polynomial-time algorithm to compute an algebraic general solution if it exists. Here an algebraic ODE means that an ODE given by a differential polynomial.

/pdf/algebraic-general-solutions-of-algebraic-ordinary-2l8o4rnn00.pdf

Algebraic general solutions of algebraic ordinary differential equations

We describe a new algorithm for computing special function solutions of the form y(x) = m(x)F(ξ(x)) of second order linear ordinary differential equations, where m(x) is an arbitrary Liouvillian function, ξ(x) is an arbitrary rational function, and F satisfies a given second order linear ordinary differential equation. Our algorithm, which is based on finding an appropriate point transformation between the equation defining F and the one to solve, is able to find all rational transformations for a large class of functions F, in particular (but not only) the 0F1 and 1F1 special functions of mathematical physics, such as Airy, Bessel, Kummer and Whittaker functions. It is also able to identify the values of the parameters entering those special functions, and can be generalized to equations of higher order.

https://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf

Sébastien Lafaille

Papers

Solutions of linear ordinary differential equations in terms of special functions