The idea of using resultant for solving differential equations can be traced to 1930's. The first work on "differential resultant" was done by O. Ore [19]. Later, the divisibility and elimination theory of algebraic differential equations was developed by C. H. Riquier [20], M. Janet [8], J. F. Ritt [21, 22], E. R. Kolchin [12] et al. The notion of resultant of two nonlinear differential polynomials was introduced by Ritt [21] in 1932 under some hypothesis on the differential polynomials by using of pseudo-division. In 1997 G. Carra-Ferro [1] introduced a notion of differential resultant of two ordinary differential polynomials as quotient of two determinants and proved that a necessary condition for the existence of a common solution of two algebraic differential equations is that the differential resultant is equal to zero based on Macaulay's resultant (see [7, 17, 18]). Methods for using characteristic set theory to solve first order autonomous ODE and partial difference polynomial systems was propsed by X. S. Gao et al in [6]. Other generalizations of differential resultant can be found in [3], [11].

In this paper, we present our joint work on computing differential resultant via Dixon's resultant [10] for first and higher ODEs. Comparing with the classical resultant of Sylvester, the advantage of Dixon's resultant is that it can do one-step elimination of a block of unknowns from a system of polynomial equations like Macauley's. Meanwhile, there are works showing that Dixon's resultant is much faster than Macauley's for certain specific problems (cf. [13, 14, 15, 16]). So the motivation for using Dixon's resultant to solve differential resultants comes very natural. Our results in this paper show that this attempt is also worth to develop for many non-trivial problems listed in E. Kamke's book [9].

[1] G. Carra-Ferro, A resultant theory for the systems of two ordinary algebraic differential equations, Appl. Algebra Engin. Comm. Comput. 8(1997), 539-560.
[3] G. Carra-Ferro, Generalized differential resultant systems of algebraic ODEs and differential elimination theory, Trends in Mathematics: Differential Equations with Symbolic Computation, Birkhauser Verlag Basel, 2006, 327-334.
[5] Mao-Ching Foo and Eng-Wee Chionh, Corner edge cutting and Dixon-resultant quotients, J. Symbolic Computation 37(2004), no. 1, 101-119.
[6] X.S. Gao and M.B. Zhang, Decomposition of differential polynomials with constant coefficients, Proc. ISSAC (2004), 175-182.
[7] I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Discriminants Resultants and Multidimensional Determinants, Birkhauser, Berlin, 1994.
[8] M. Janet, Sur les sist'emes d''equations aux d'eriv'ees partielles, J. Math. 3(1920), 65-151.
[9] E. Kamke, Differentialgleichungen, osungsmethoden und osungen, Akad. Verlag. Geest & Portig K.-G., Leipzig, 1956.
[10] D. Kapur, T. Saxena and Lu Yang, Algebraic and geometric reasoning using Dixon resultants, Proc. ISSAC (1994), 99-107.
[11] A. Kasman and E. Previato, Commutative partial differential operators, Physica D 152- 153 (2001), 66-77.
[12] E. R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, London-New York, 1973.
[13] R.H. Lewis and P.F. Stiller, Solving the recognition problem for six lines using the Dixon resultant, Mathematics and Computers in Simulation 49 8(1999), 49(3):205-220.
[14] R.H. Lewis and S. Bridgett, Conic tangency equations and Apollonius problems in biochemistry and pharmacology, Mathematics and Computers in Simulation 8(2003), 61(2):101-114.
[15] R.H. Lewis and E.A. Coutsias, Algorithmic search for flexibility using resultants of polynomial systems, Lecture Notes in Computer Science 4869, Springer 8(2007), 68-79.
[16] R.H. Lewis, Comparing acceleration techniques for the Dixon and Macaulay resultants (abstract only), ACM Communications in Computer Algebra 8(2008), 42(1-2): 79-81.
[17] F.S. Macaulay, Some formulae in elimination, Proc. London Math. Soc. 35(1902), 3-27.
[18] F.S. Macaulay, Note on the resultant of a number of polynomials of the same degree, Proc. London Math. Soc. 21 (1921), 14-21.
[19] O. Ore, Formale theorie der linearen differentialgleichungen, I, J. Reine Angew. Math. 167 (1932), 221-234.
[20] C. H. Riquier, Les Systemes d' Equations aux Derivees Partielles, Gauthier-Villars, Paris, 1910.
[21] J. F. Ritt, Differential Equations from the Algebraic Standpoint, Coll. Publ. 14, Amer. Math. Soc., New York, 1932.
[22] J. F. Ritt, Differential Algebra, Coll. Publ. 33, Amer. Math. Soc., New York, 1950.