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Vo, Thieu N.; Grasegger, Georg; Winkler, Franz

Computation of all rational solutions of first-order algebraic ODEs. (English) Zbl 1393.34008

Adv. Appl. Math. 98, 1-24 (2018).
Summary: In this paper, we discuss three different approaches to attack the problem of determining all rational solutions for a first-order algebraic ordinary differential equation (AODE). We first give a sufficient condition for first-order AODEs to have the property that poles of rational solutions can only occur at the zeros of the leading coefficient. A combinatorial approach is presented to determine all rational solutions, if there are any, of the family of first-order AODEs satisfying this condition. Algebraic considerations based on algebraic function theory yield another algorithm for quasi-linear first-order AODEs. And finally ideas from algebraic geometry combine these results to an algorithm for finding all rational solutions of parametrizable first-order AODEs.

Cited in 8 Documents

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
14H45 Special algebraic curves and curves of low genus
68W30 Symbolic computation and algebraic computation
34A09 Implicit ordinary differential equations, differential-algebraic equations

Keywords:

ordinary differential equation; rational solution; algebraic function field; rational curve

Software:

CASA
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Full Text: DOI

References:

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