Rational Solutions of a Rational System of Autonomous ODEs: Generalization to Trivariate Case and Problems

This paper presents a method for computing the explicitly rational solutions of a trivariate rational system of autonomous ordinary differential equations (ODEs) based on the proper parametrization of its invariant algebraic space curve. It is the generalization of the method in [11]. First, an undetermined coefficients method is given for computing the invariant algebraic space curves of the trivariate polynomial system of autonomous ODEs. Then we extend some nice properties of the proper parametrization for the plane curves to the case of space curves. Moreover, an algorithm is provided for computing the rational solutions of the trivariate polynomial system of autonomous ODEs based on previous preparation. Finally, we generalize this algorithm to compute the rational solutions of the trivariate rational system of autonomous ODEs and give some relations and properties of the rational solutions.