On the 36th order discrete group for a multiplicative class of 2nd order differential Equations

A class of ordinary differential equations of 2nd order with multiplicative right side is examined. The transformations closed in this class of equations are found. Discrete groups of transformations of orders 6, 12 and 36 are constructed for the studied class of equations or for its subclasses. The method of enclosing (extending) a class of equations to construct a discrete group of 36th order is applied. The method of obtaining exact solutions (general and partial) of the equations corresponding to the vertices of the graph, if the solution of at least one of these equations is known, i.e. the method of "multiplication" of solvable cases in the studied classes of equations, is presented. As an example, the equation of free continuous oscillations of a pendulum is examined.