A generalized Allwright formula and the vector Riccati equation

TL;DR

This paper extends the classical Allwright formula to systems of differential equations using the Kronecker product, linking it with vector Riccati equations and generalizing the characterization of solutions as fractional linear functions.

Contribution

It introduces a generalized Allwright formula for systems of differential equations and connects it with vector Riccati equations, broadening the classical scalar case.

Findings

Generalized Allwright formula for systems of differential equations.

Connection established between the formula and vector Riccati equations.

Extension of the scalar solution characterization to systems.

Abstract

A classical formula of Allwright on the general solution of a scalar differential equation is generalized to a system of differential equations by means of the Kronecker product.The Allwright formula is connected with the Riccati equation, and in a similar way the generalized formula is connected with a special type of a differential system called a vector Riccati equation. Moreover,the classical result that a scalar differential equation is a Riccati equation if and only if its general solution is a fractional linear function of the starting value, is also generalized to a differential system.