Solving Differential Equations by Differentiating

TL;DR

This paper explores using Taylor series differentiation to find solutions to differential equations, deriving recursions for series coefficients and analyzing convergence without relying on integration.

Contribution

It introduces a novel approach of solving differential equations by differentiating to find series coefficients, expanding the traditional Taylor series method.

Findings

Recursions for series coefficients can be derived via differentiation.

Series solutions can be analyzed for convergence using Cauchy-Hadamard theorem.

The method provides solutions even when closed-form expressions are unavailable.

Abstract

In this work, we illustrate and explore the use of Taylor series as solutions of differential equations. For a large a number of classes of differential equations in the literature, there are plenty of sources where the well known Taylor Series Method is used to approximate the solution, but here we are focused in seeing the Taylor series as a solution, which in turn prompt us to find the recursions defining the coefficients in the series. Because these recursions are found by differentiating, instead of integrating the differential equation, it is not difficult to prove that the resulting series is a solution. In the case where the series does not have a closed analytic form or it is not a known function, Cauchy-Hadamard theorems can be used to find the radius of convergence and then the series is a solution for the differential equation, in the domain where it converges.