On the solution calculation of nonlinear ordinary differential equations via exact quadratization

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-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, and whose coefficients are analytic time functions. The solution method consists in reducing the ODE to a certain canonical homogeneous quadratic ODE, named driver-type Riccati equation 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 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 existence of a piecewise-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. (C) 2020 Elsevier Inc. All rights reserved.