The integration of ordinary differential equations: factorization and transformations

This paper is based on a uniform theory of factorization and transformation of nth (n≥2) order ordinary differential equations (ODEs) that are used to constructively solve problems of integrability. This method of factorization of differential operators is developed not only in a base differential field, but also in its algebraic and transcendental extensions. For the first time, the method is extended to nonlinear equations. A new method of exact linearization is proposed that includes transformations used earlier. This method allows us to constructively study nonlinear and nonstationary problems in mathematical simulations with the help of a computer algebra system called REDUCE (as well as other systems).