On Application of M.N. Lagutinski Method to Integration of Differential Equations in Symbolic Form. Part 2

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. The second part is devoted to integration of given differential equation d + d with , Q[, ] in quadratures. According to the theorem of M. Singer the problem of integration in quadratures is equivalent to the finding of integrating factor of the form = exp d + d where , Q[, ]. The function can be found as a root of Darboux polynomial for some auxiliary differentiation of the ring Q[, , ]. By Lagutinski method we can find all Darboux polynomials for given differentiation of polynomial ring if degrees of required polynomials are less than given boundary and thus we can find integration factor of the form stated above. The theory and its realization in Sage are tested on numerous examples from standard for Russia text-book by A. F. Filippov.

sage, sagemath, Lagutinski method, integration in quadratures, sage, sagemath