On Application of M.N. Lagutinski Method to Integration of Differential Equations in Symbolic Form. Part 1

The method of M.N. Lagutinski (1871-1915) allows to find rational integrals and Darboux polynomials for given differential ring and thus can be used for integration of ordinary differential equations in symbolic form. A realization of Lagutinski method was made under free opensource mathematics software system Sage and will be presented in this article with application for symbolic integration of 1st order differential equations. In the first part of the article basic concepts of the Lagutinski method is briefly stated for polynomials rings. Then this method is applied to search of algebraic integrated curves for given ordinary differential equations of the form d + d with , Q[, ]. It is shown how the Lagutinski method allows to look for curves of the given order or to prove that there are not such curves. In particular questions about the optimization of computations and integration in micronomials are considered. The theory and its realization in Sage are tested on numerous examples from standard for Russia text-book by A.F. Filippov. Some recommendations for optimization of the Lagutinski method usage are made in the conclusion of the article.

sage, sagemath, Lagutinski method, algebraic integral curves, sage, sagemath