Rational general solutions of planar rational systems of autonomous ODEs

Area of Science:

• Differential Equations
• Algebraic Geometry
• Computational Mathematics

Background:

• Autonomous ordinary differential equations (ODEs) are fundamental in modeling dynamic systems.
• Finding explicit rational solutions for ODEs remains a significant challenge.
• Rational invariant algebraic curves offer a potential pathway to solution discovery.

Purpose of the Study:

• To develop an algorithm for computing explicit rational solutions of rational autonomous ODE systems.
• To utilize rational invariant algebraic curves as the basis for this computation.
• To determine the existence and compute rational general solutions when a rational first integral is present.

Main Methods:

• Derivation of an algorithm based on rational invariant algebraic curves.
• Application of proper rational parametrization to these curves.
• Utilizing linear reparametrizations to obtain rational solutions.
• Developing criteria to identify and compute rational general solutions via first integrals.

Main Results:

• An explicit algorithm for computing rational solutions of rational autonomous ODE systems.
• Successful application of rational parametrization and linear reparametrizations.
• A method to decide the existence of a rational general solution.
• The computation of rational general solutions in applicable cases.

Conclusions:

• Rational invariant algebraic curves provide a powerful tool for solving rational autonomous ODE systems.
• The proposed method offers an effective approach to finding explicit rational solutions.
• The algorithm successfully addresses the computation of rational general solutions when possible.