An Alternative Method for the Undetermined Coefficients and the Annihilator Methods

TL;DR

This paper introduces a straightforward formula for finding particular solutions of linear ODEs with constant coefficients when the nonhomogeneous term is a combination of polynomials, trigonometric, and exponential functions, using linear systems.

Contribution

It presents a new, simplified method for solving certain linear differential equations, improving on traditional undetermined coefficients and annihilator methods.

Findings

Simplified formula for particular solutions

Efficient solution via lower triangular linear systems

Validated with two example problems

Abstract

This paper exhibits a very simple formula for a particular solution of a linear ordinary differential equation with constant real coefficients, P(d/dt)x = f, f a function given by a linear combination of polynomials, trigonometrical and exponential real functions products. This is done by solving lower triangular linear systems with constant coefficients. Two examples are given.