A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs

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Abstract

We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For a first order autonomous ODE F=0, we give an exact degree bound for its rational solutions, based on the connection between rational solutions of F=0 and rational parametrizations of the plane algebraic curve defined by F=0.

For a first order autonomous ODE, we further give a polynomial time algorithm for computing a rational general solution if it exists based on the computation of Laurent series solutions and Padé approximants. Experimental results show that the algorithm is quite efficient.

Keywords

Rational general solution
First order autonomous ODE
Rational parametrizations
Laurent series
Padé approximants
Polynomial time algorithm

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Partially supported by a National Key Basic Research Project of China 2004CB318000.