Solvable Systems of Linear Differential Equations

TL;DR

This paper extends the asymptotic iteration method (AIM) to solve systems of two first-order linear differential equations, establishing new solvable classes and connections to Riccati equations.

Contribution

The work adapts AIM for linear systems, re-examines its termination criteria, and introduces new exactly solvable classes with variable coefficients.

Findings

Established a connection between linear systems and Riccati equations.

Reworked AIM theory for systems of equations.

Constructed new classes of solvable differential systems.

Abstract

The asymptotic iteration method (AIM) is an iterative technique used to find exact and approximate solutions to second-order linear differential equations. In this work, we employed AIM to solve systems of two first-order linear differential equations. The termination criteria of AIM will be re-examined and the whole theory is re-worked in order to fit this new application. As a result of our investigation, an interesting connection between the solution of linear systems and the solution of Riccati equations is established. Further, new classes of exactly solvable systems of linear differential equations with variable coefficients are obtained. The method discussed allow to construct many solvable classes through a simple procedure.