Exactly solvable quadratic differential equation systems through generalized inversion

TL;DR

This paper introduces a subclass of quadratic differential systems that can be solved exactly using a generalized inversion technique, providing an algorithm to identify and solve such systems analytically.

Contribution

The paper presents a novel method using multi-dimensional inversion to solve a specific subclass of quadratic differential systems analytically.

Findings

Identified a subclass of quadratic systems solvable via generalized inversion.

Developed an algorithm to determine solvability and find solutions.

Applicable to systems with any finite number of variables.

Abstract

We study the autonomous systems of quadratic differential equations of the form $\dot{x}_i(t)=\mathbf{x}(t)^T \mathbf{A}_i \mathbf{x}(t) + \mathbf{v}_i^T \mathbf{x}(t)$ with $\mathbf{x}(t) = (x_1(t),x_2(t),\dots,x_i(t),\dots)$ which, in general, cannot be solved exactly. In the present paper, we present a subclass of analytically solvable quadratic systems, whose solution is realized through a multi-dimensional generalization of the inversion which transforms a quadratic system into a linear system. We provide a constructive algorithm which, on one hand, decides whether the system of differential equations is analytically solvable with the inversion transformation and, on the other hand, provides the solution. The presented results apply for arbitrary, finite number of variables.