Keyword: «first integrals»

The paper is devoted to teaching theory of normal systems solutions in the course “Differential Equations” and to the problems that arise in the presentation of the material. In studying the discipline, students face difficulties in finding the general solution of the normal systems of ordinary differential equations (ODE). One of the methods for solving systems is the method of allocation of integrable combinations, that is getting out of the system such equations that can be integrated and thus obtain the first integrals. Their aggregate determines the general decision or the general integral of the system. In this regard, the article describes the normal system of ODE and their symmetrical forms of recording. The basic concepts are interpreted: Cauchy problem, theorem of existence and uniqueness of normal ODE systems solutions, first integrals for systems. It illustrates in detail the method of finding integrable combinations, first integrals and general solutions of various systems on a wide range of tasks. The paper will be useful for students of technical universities and lecturers during seminars on the subject “Differential equations”.