Solving polynomial differential equations by transforming them to linear functional-differential equations

TL;DR

This paper introduces a novel method for solving polynomial ordinary differential equations by transforming them into linear functional equations, enabling more straightforward solutions especially for first-order Abel equations.

Contribution

It proposes a new transformation technique that generalizes to higher-order, systems, and partial differential equations, expanding the toolkit for solving complex differential equations.

Findings

Successfully applied to first-order Abel equations

Potential for generalization to higher-order and PDEs

Provides a systematic approach for transforming polynomial ODEs

Abstract

We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order Abel differential equation with two nonlinear terms in order to demonstrate in as much detail as possible the computations necessary for a complete solution. We mention in our section on further developments that the basic transformation idea can be generalized to apply to differential equations of any order, to a system of ordinary differential equations without first differentially eliminating the multiple dependent variables, and even to partial differential equations.