ON FINDING CLASSES OF DIFFERENTIAL EQUATIONS INVARIANT WITH RESPECT TO DISCRETE TRANSFORMATIONS

Khakimova Zilya Nailevna

Classes of multiplicative ordinary differential and integro-differential equations containing arbitrary functions are considered. For the classes of equations under consideration, a discrete group (of the 12th order) of transformations is found that preserves the form of the equations. The possibility of extending this group of transformations with the help of the discrete dihedral group of the 6th order is indicated. The method of "replenishment" ("extension") of differential equation classes is applied in the work, with the help of which it is possible to find classes of equations that are invariant with respect to a given discrete transformation. A method is indicated for obtaining new discrete transformations from those found earlier (by means of lowering and raising the order of differential equations) closed in "extended" classes of equations.

Keywords:ordinary differential equation (ODE), integro-differential equation, discrete transformation, discrete transformation group, discrete transformation pseudogroup, dihedral group, exact solution of ODE