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 . Later, the
divisibility and elimination theory of algebraic differential equations was developed by
C. H. Riquier , M. Janet , J. F. Ritt [21, 22], E. R. Kolchin  et al. The notion
of resultant of two nonlinear differential polynomials was introduced by Ritt  in 1932
under some hypothesis on the differential polynomials by using of pseudo-division. In
1997 G. Carra-Ferro  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 . Other
generalizations of differential resultant can be found in , .
In this paper, we present our joint work on computing differential resultant via
Dixon's resultant  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 .
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resultant, Mathematics and Computers in Simulation 49 8(1999), 49(3):205-220.
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biochemistry and pharmacology, Mathematics and Computers in Simulation 8(2003),
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systems, Lecture Notes in Computer Science 4869, Springer 8(2007), 68-79.
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London Math. Soc. 21 (1921), 14-21.
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