DISCRETE SYMMETRIES AND SOLUTIONS IN QUADRATURES FOR SOME HYDRODYNAMIC MODEL

Khakimova Zilya Nailevna

A non-linear second-order ordinary differential equation with a polynomial right-hand side is considered, which arises when solving the system of hydrodynamic equations for an ideal self-gravitating non-relativistic fluid with zero pressure. For the differential equation under consideration, a discrete transformation pseudogroup of the 24th order is found and the graph of this pseudogroup is constructed. By the «reproduction» method, 23 equations of polynomial and fractional-polynomial form were obtained that are integrable in quadratures, just like the original equation.

Keywords:ordinary differential equation (ODE) of the 2nd order, polynomial ODE, fractional-polynomial ODE, discrete transformation group, transformation pseudogroup, dihedral group, exact solution of a differential equation