Expansion of solutions to an ordinary differential equation into transseries
We consider a polynomial ODE of the order n in a neighbourhood of zero or of infinity of the independent variable. A method of calculation of its solutions in the form of power series and an exponential addition, which contains one more power series, was described. The exponential addition has an arbitrary constant, exists in some set E1 of sectors of the complex plane and can be found from a solution to an ODE of the order n - 1. An hierarchic sequence of such exponential additions is possible, that each of these exponential additions is defined from an ODE of a lower order n - i and exists in its own set Ei. Here we must check the non-emptiness of intersection of the sets E1 Ç ... Ç Ei. Each exponential addition continues into its own exponential expansion, containing countable set of power series. As a result we obtain an expansion of a solution into a transseries, containing countable set of power series, all of which are summable. The transseries describes families of solutions to the initial ODE in some set of sectors of the complex plane.