Rational general solutions of algebraic ordinary differential equations

Abstract

We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For an autonomous first order ODE, we give an algorithm to compute a rational general solution if it exists. The algorithm is based on the relation between rational solutions of the first order ODE and rational parametrizations of the plane algebraic curve defined by the first order ODE and Padé approximants.

Algebraic general solutions of algebraic ordinary differential equations

“…Let D (α 0 ;α) be the determinant of A (α 0 ;α) (y). Note that if n = 1, D (α 0 ,α) is just equal to Dn,m in [9]. Lemma 2.5.…”

“…Let T = tdeg(F ), the total degree of F . Theorem 9 given in [9] shows that the number of the points on F (y, y1) = 0 which make S(y, y1) or y1 vanish is at most T 2 .…”

“…In [12], Hubert gave a method to compute a basis of the general solutions of first order ODEs and applied it to study the local behavior of the solutions. In [9,10], Feng and Gao gave a necessary and sufficient condition for an algebraic ODE to have a rational type general solution and a polynomial-time algorithm to compute a rational general solution if it exists.…”

“…In this paper, the idea proposed in [9] is generalized to compute algebraic function solutions. In Section 2, we give a sufficient and necessary condition for an algebraic ODE to have an algebraic general solution, by constructing a class of differential equations whose solutions are all algebraic functions.…”

On Factorization and Solution of Multidimensional Linear Partial Differential Equations

“…For polynomial b i (x, y) this is one of the famous fields of research: study of polynomial vector fields in the plane. Recently an essential advance was made in [3,6,7]; one may hope that a complete algorithm may be found. Still the problem of finding complete solutions of (19) in a suitable "constructive differential field algorithmically is a challenging problem, as well as the problem of finding solutions for the equation (18).…”