DISCRETE INVARIANTS IN THE CLASS OF ORDINARY DIFFERENTIAL EQUATIONS AND THEIR INTEGRATION

Khakimova Zilya Nailyevna, Shakhova Ekaterina Anatolyevna

The class of ordinary differential equations of the 2nd Order with multiplier right-wing parts, as well as some discrete transformation closed on this class of equations, is viewed. The article discusses the method of discrete invariants. An invariant subclass in the studied class of equations relatively to this discrete transformation is found. With the help of the concomitant, the order of the equations of the invariant subclass is reduced, while the invariant subclass of the equations is led to an easily integrated class of equations. The authors of the article managed to find the transformation, the reverse to the concomitant, due to which they managed to calculate the general decisions of all the invariant subclasses considered in the work. The method of discrete invariants is applied to various multiplier classes of equations, including with power right parts with arbitrary parameters, as well as to a multiplier class with functional arbitrariness.

Keywords: An ordinary differential equation (ODE), a discrete group of transformations, the exact solution of the ODE, the invariant of discrete transformation, the conterior, the class of generalized equations of Emden-Fauler (GEFE).