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.