Symbolic Solutions of First-Order Algebraic ODEs

Algebraic ordinary differential equations are described by polynomial relations between the unknown function and its derivatives. There are no general solution methods available for such differential equations. However, if the hypersurface determined by the defining polynomial of an algebraic ordinary differential equation admits a parametrization, then solutions can be computed and the solvability in certain function classes may be decided. After an overview of methods developed in the last decade we present a new and rather general method for solving algebraic ordinary differential equations.