Some Notes on the Solutions of non Homogeneous Differential Equations

TL;DR

This paper presents methods for solving various non-homogeneous differential equations using Eisenstein and Fourier series, including a complete solution for a second-order linear ODE with arbitrary non-homogeneous parts.

Contribution

It introduces elementary solutions via Eisenstein series and Fourier series for non-homogeneous differential equations, and provides a complete solution for a specific second-order case.

Findings

Solutions expressed in closed form using Eisenstein series

Fourier series methods applied to linear differential operators

Complete solution for a second-order non-homogeneous ODE with arbitrary functions

Abstract

We solve some forms of non homogeneous differential equations in one and two dimensions. By expanding the solution into whell-posed closed form-Eisenstein series the solution itself is quite simple and elementary. Also we consider Fourier series solutions of linear differential operator equations. In the third section we study operators which are functions of the Leibnitz derivative. The last result is the complete solution of a non homogenus 2-degree ODE with linear coeficients. The non homogenous part is an arbirtary function of $L_2(\bf R\rm)$