Solving third order linear differential equations in terms of second order equations

ABSTRACT

This paper presents a simplified version of a method by Michael Singer for reducing a third order linear ode to a second order linear ode whenever possible. An implementation is available as well.

REFERENCES

• 1: Manuel Bronstein , Thom Mulders , Jacques-Arthur Weil, On symmetric powers of differential operators, Proceedings of the 1997 international symposium on Symbolic and algebraic computation, p.156-163, July 21-23, 1997, Kihei, Maui, Hawaii, United States [doi>10.1145/258726.258771]
• 2: Manuel Bronstein, An improved algorithm for factoring linear ordinary differential operators, Proceedings of the international symposium on Symbolic and algebraic computation, p.336-340, July 20-22, 1994, Oxford, United Kingdom [doi>10.1145/190347.190436]
• 3: R. Chalkley, Relative invariants for homogeneous linear differential equations, J. Differential Equations, 80 107--153, (1989).
• 4: T. Cluzeau, M. van Hoeij, A Modular Algorithm to Compute the Exponential Solutions of a Linear Differential Operator, J. Symb. Comput., 38 1043--1076, (2004).
• 5: E. Compoint, J-A. Weil, Absolute reducibility of differential operators and Galois groups, J. Algebra, 275 77--105, (2004).
• 6: G. Fano, Ueber lineare homogene Differentialgleichungen mit algebraischen Relationen zwischen den Fundamentallosungen, Math.Ann., 53 493--590, (1900).
• 7: M. van Hoeij, An implementation and examples for algorithm ReduceOrder www.math.fsu.edu/~hoeij/files/ReduceOrder
• 8: Mark Van Hoeij, Factorization of differential operators with rational functions coefficients, Journal of Symbolic Computation, v.24 n.5, p.537-561, Nov. 1997 [doi>10.1006/jsco.1997.0151]
• 9: M. van Hoeij, Decomposing a 4'th order linear differential equation as a symmetric product, Banach Center Publications, 58 89--96, (2002).
• 10: M. vanHoeij, J. Cremona, Solving conics over function fields, accepted for publication in JNTB. www.math.fsu.edu/~hoeij/files/ConicProgram.
• 11: M. van Hoeij, M. van der Put, Descent for differential modules and skew fields. J. of Algebra, 296 18--55 (2006).
• 12: M. van Hoeij, J-A. Weil, An algorithm for computing invariants of differential Galois groups, J. Pure Appl. Algebra, 117-118 353--379 (1997).
• 13: Mark van Hoeij , Jean-François Ragot , Felix Ulmer , Jacques-Arthur Weil, Liouvillian solutions of linear differential equations of order three and higher, Journal of Symbolic Computation, v.28 n.4-5, p.589-609, Oct./Nov. 1999 [doi>10.1006/jsco.1999.0316]
• 14: K.A. Nguyen, M. van der Put, Solving linear differential equations, preprint, (2006).
• 15: W. R. Oudshoorn , M. van der Put, Lie symmetries and differential Galois groups of linear equations, Mathematics of Computation, v.71 n.237, p.349-361, January 2002 [doi>10.1090/S0025-5718-01-01397-7]
• 16: A.C. Person, Solving Homogeneous Linear Differential Equations of Order 4 in Terms of Equations of Smaller Order, PhD thesis, http://www.lib.ncsu.edu/theses/available/etd-08062002-104315/ (2002).
• 17: M. van der Put, M.F. Singer, Galois Theory of linear Differential Equations, Grundlehren der mathematischen Wissenschaften, 328 Springer (2003).
• 18: Josef Schicho, Rational parametrization of real algebraic surfaces, Proceedings of the 1998 international symposium on Symbolic and algebraic computation, p.302-308, August 13-15, 1998, Rostock, Germany [doi>10.1145/281508.281655]
• 19: M.F. Singer, Solving Homogeneous Linear Differential Equations in Terms of Second Order Linear Differential Equations, Am. J. of Math., 107 663--696, (1985).
• 20: M.F. Singer, Algebraic Relations Among Solutions of Linear Differential Equations: Fano's Theorem, Am. J. of Math., 110 115--143, (1988).
• 21: M.F. Singer, F. Ulmer, Linear Differential Equations and Products of Linear Forms, J. of Pure and Appl. Algebra, 117-118 549--563, (1997).
• 22: Michael F. Singer , Felix Ulmer, Galois groups of second and third order linear differential equations, Journal of Symbolic Computation, v.16 n.1, p.9-36, July 1993 [doi>10.1006/jsco.1993.1032]
• 23: Felix Ulmer, Liouvillian solutions of third order differential equations, Journal of Symbolic Computation, v.36 n.6, p.855-889, December 2003 [doi>10.1016/S0747-7171(03)00065-8]
• 24: Christiaan van de Woestijne, Surface parametrisation without diagonalisation, Proceedings of the 2006 international symposium on Symbolic and algebraic computation, July 09-12, 2006, Genoa, Italy [doi>10.1145/1145768.1145823]

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Algorithm design and analysis

General Terms:
Algorithms

Keywords:
linear differential equations, reduction of order