Expansion of Solutions to an Ordinary Differential Equation into Transseries

Abstract

A polynomial ordinary differential equation (ODE) of order $n$ in a neighborhood of zero or infinity of the independent variable is considered. In 2004, a method was proposed for computing its solutions in the form of power series and an exponential addition that involves another power series. The addition contains an arbitrary constant, exists only in a set ${{E}_{1}}$ consisting of sectors of the complex plane, and is found by solving an ODE of order $n - 1$. A hierarchical sequence of exponential additions is possible such that each addition is determined by an ODE of progressively lower order $n - i$ and each exists in its own set ${{E}_{i}}$. In this case, one has to check that the intersection of the existence sets ${{E}_{1}} \cap {{E}_{2}} \cap \ldots \cap {{E}_{i}}$ is nonempty. Each exponential addition extends to its own exponential expansion involving a countable set of power series. Finally, the solution is expanded into a transseries involving a countable set of power series, all of which are summable. The transseries describes families of solutions to the original equation in certain sectors of the complex plane.