Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients

Authors:
Michael F. Singer
Affiliations:
Department of Mathematics, Box 8205, North Carolina State University, Raleigh, NC 27695, USA
Venue:
Journal of Symbolic Computation
Year:
1991

Citing 4
Cited 22

Elementary and Liouvillian Solutions of Linear Differential Equations

Journal of Symbolic Computation
Integration of elementary functions

Journal of Symbolic Computation
Liouvillian solutions of linear differential equations with Liouvillian coefficients

Proceedings of the third conference on Computers and mathematics
Algebraic Extensions and Algebraic Closure in Scratchpad II

ISAAC '88 Proceedings of the International Symposium ISSAC'88 on Symbolic and Algebraic Computation

Integration of elementary functions

Journal of Symbolic Computation
The Risch differential equation on an algebraic curve

ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
On solutions of linear ordinary differential equations in their coefficient field

Journal of Symbolic Computation
Linear ordinary differential equations: breaking through the order 2 barrier

ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
D'Alembertian solutions of linear differential and difference equations

ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
An improved algorithm for factoring linear ordinary differential operators

ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
On polynomial solutions of linear operator equations

ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
An algorithm for computing exponential solutions of first order linear differential systems

ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Solving linear ordinary differential equations over C (x, e∫ f(x)dx)

ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Linear differential equations, iterative logarithms and orderings on monomial differential extensions

ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Factorization of linear differential operators in exponential extensions

ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Cited References

Computer algebra handbook
Factoring systems of linear PDEs with finite-dimensional solution spaces

Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Polynomial solutions for first order ODEs with constant coefficients

ACM SIGSAM Bulletin
Closed form solutions of linear odes having elliptic function coefficients

ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Solving second order linear differential equations with Klein's theorem

Proceedings of the 2005 international symposium on Symbolic and algebraic computation
A recursive method for determining the one-dimensional submodules of Laurent-Ore modules

Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Solving difference equations whose coefficients are not transcendental

Theoretical Computer Science
What Might "Understand a Function" Mean?

Calculemus '07 / MKM '07 Proceedings of the 14th symposium on Towards Mechanized Mathematical Assistants: 6th International Conference
Integration in finite terms of non-liouvillian functions

ACM Communications in Computer Algebra
A reduced form for linear differential systems and its application to integrability of Hamiltonian systems

Journal of Symbolic Computation
Computer algebra in nanosciences: modeling electronic states in quantum dots

CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing

Quantified Score

Hi-index	0.00

Visualization

Abstract

Let L(y) = b be a linear differential equation with coefficients in a differential field K. We discuss the problem of deciding if such an equation has a non-zero solution in K and give a decision procedure in case K is an elementary extension of the field of rational functions or is an algebraic extension of a transcendental liouvillian extension of the field of rational functions. We show how one can use this result to give a procedure to find a basis for the space of solutions, liouvillian over K, of L(y) = 0 where K is such a field and L(y) has coefficients in K.