# On the Solution Calculation of Nonlinear Ordinary Differential Equations via Exact Quadratization

## TL;DR

This paper introduces a general method called exact quadratization for solving a broad class of nonlinear analytic ODEs by transforming them into quadratic form, enabling power series solutions and extending previous work to implicit equations.

## Contribution

The paper develops a comprehensive theory of exact quadratization for real analytic -ODEs, including implicit cases, providing a recursive solution formula and addressing globality issues.

## Key findings

- Provides a recursive formula for power series coefficients.
- Extends quadratization to implicit ODEs.
- Addresses globality of the transformation.

## Abstract

We show a general method allowing the solution calculation, in the form of a power series, for a very large class of nonlinear Ordinary Differential Equations (ODEs), namely the real analytic $\sigma\pi$-ODEs (and, more in general, the real analytic $\sigma\pi$-{\it reducible} ODEs) in many indeterminates, characterized by an ODE-function given by generalized polynomials of the indeterminates and their derivatives, i.e. functions formally polynomial with exponents, though the exponent can be any real number. The solution method consists in reducing the ODE to a certain canonical quadratic ODE, with a larger number of indeterminates, whose solutions include, as sub-solutions, the original solutions, and for which a recursive formula is shown, giving all coefficients of the solution power series directly from the ODE parameters. The reduction method is named {\it exact quadratization} and was formerly introduced in another our article, where we considered explicit ODEs only. In the present paper, which is self-contained to a large extent, we review and complete the theory of exact quadratization by solving issues, such as for instance the {\it globality} of the quadratization, that had remained open, and also extend it to the more general case of an implicitly defined ODE. Finally, we argue that the result can be seen as a partial solution of a differential version of the 22nd Hilbert's problem.