Rational General Solutions of High Order Non-autonomous ODEs

Y. Huang; Lam Xuan Chau Ngo

Rational General Solutions of High Order Non-autonomous ODEs

Y. Huang, Lam Xuan Chau Ngo

Research Institute for Symbolic Computation

Research output: Working paper and reports › Preprint

Abstract

In this paper, we generalize the results of Ngo and Winkler [18, 20, 21] to the case of high order non-autonomous algebraic ODE with a birational parametrization of the corre- sponding algebraic hypersurface. First, we reduce the problem for finding rational general solutions of non-autonomous n -1 (n > 2) order ODE to finding rational general solutions of an associated first order rational system of autonomous ODEs in n indeterminates based on the parametrization of hypersurface. Next, the correspondence of the rational general solutions between the original non-autonomous algebraic ODE and the associated system of autonomous ODEs is proved. Finally, a criterion is presented for existence of rational general solutions of the associated system of autonomous ODEs if the degree bound of its rational general solutions is given. Moreover, we give some nice properties of polynomial system of autonomous ODEs.

Original language	English
Place of Publication	Altenberger Straße 69, 4040 Linz
Publisher	JKU Linz
Number of pages	15
Publication status	Published - Jun 2010

Publication series

Name	RISC Report Series
No.	10-13

Fields of science

101001 Algebra
101002 Analysis
101 Mathematics
102 Computer Sciences
102011 Formal languages
101013 Mathematical logic
101020 Technical mathematics
101025 Number theory
101012 Combinatorics
101005 Computer algebra
101003 Applied geometry
102025 Distributed systems

JKU Focus areas

Computation in Informatics and Mathematics
Engineering and Natural Sciences (in general)

Cite this

@techreport{bbbb72981b3a4a748179c5550b66b81e,

title = "Rational General Solutions of High Order Non-autonomous ODEs",

abstract = "In this paper, we generalize the results of Ngo and Winkler [18, 20, 21] to the case of high order non-autonomous algebraic ODE with a birational parametrization of the corre- sponding algebraic hypersurface. First, we reduce the problem for finding rational general solutions of non-autonomous n -1 (n > 2) order ODE to finding rational general solutions of an associated first order rational system of autonomous ODEs in n indeterminates based on the parametrization of hypersurface. Next, the correspondence of the rational general solutions between the original non-autonomous algebraic ODE and the associated system of autonomous ODEs is proved. Finally, a criterion is presented for existence of rational general solutions of the associated system of autonomous ODEs if the degree bound of its rational general solutions is given. Moreover, we give some nice properties of polynomial system of autonomous ODEs.",

author = "Y. Huang and Ngo, {Lam Xuan Chau}",

year = "2010",

month = jun,

language = "English",

series = "RISC Report Series",

publisher = "JKU Linz",

number = "10-13",

type = "WorkingPaper",

institution = "JKU Linz",

}

TY - UNPB

T1 - Rational General Solutions of High Order Non-autonomous ODEs

AU - Huang, Y.

AU - Ngo, Lam Xuan Chau

PY - 2010/6

Y1 - 2010/6

N2 - In this paper, we generalize the results of Ngo and Winkler [18, 20, 21] to the case of high order non-autonomous algebraic ODE with a birational parametrization of the corre- sponding algebraic hypersurface. First, we reduce the problem for finding rational general solutions of non-autonomous n -1 (n > 2) order ODE to finding rational general solutions of an associated first order rational system of autonomous ODEs in n indeterminates based on the parametrization of hypersurface. Next, the correspondence of the rational general solutions between the original non-autonomous algebraic ODE and the associated system of autonomous ODEs is proved. Finally, a criterion is presented for existence of rational general solutions of the associated system of autonomous ODEs if the degree bound of its rational general solutions is given. Moreover, we give some nice properties of polynomial system of autonomous ODEs.

AB - In this paper, we generalize the results of Ngo and Winkler [18, 20, 21] to the case of high order non-autonomous algebraic ODE with a birational parametrization of the corre- sponding algebraic hypersurface. First, we reduce the problem for finding rational general solutions of non-autonomous n -1 (n > 2) order ODE to finding rational general solutions of an associated first order rational system of autonomous ODEs in n indeterminates based on the parametrization of hypersurface. Next, the correspondence of the rational general solutions between the original non-autonomous algebraic ODE and the associated system of autonomous ODEs is proved. Finally, a criterion is presented for existence of rational general solutions of the associated system of autonomous ODEs if the degree bound of its rational general solutions is given. Moreover, we give some nice properties of polynomial system of autonomous ODEs.

M3 - Preprint

T3 - RISC Report Series

BT - Rational General Solutions of High Order Non-autonomous ODEs

PB - JKU Linz

CY - Altenberger Straße 69, 4040 Linz

ER -