Exponential expansions of solutions to an ODE

We consider an ordinary differential equation of a very general form. Let we have found a power expansion of its solution with exponential addendums. In Section 1 we show how the addendums can be prolongated in the exponential expansions of solutions to the initial equation.We explain a method of calculation of critical numbers. Their absence is sufficient for the existence of the expansions. In section 2 we show a method for calculation of exponential expansions of solutions corresponding to a horisontal edge of the polygon of the equation. In Section 3 we proof statements of Sections 1 and 2. Examples from the Painlevé equations are given.