A Generalized Solution Method to Undamped Constant-Coefficient Second-Order ODEs Using Laplace Transforms and Fourier Series

Authors

Laurie A. Florio, DEVCOM Armaments Center United States Army Armament Graduate SchoolFollow
Ryan D. Hanc, DEVCOM Armaments Center United States Army Armament Graduate SchoolFollow

Publication Date

10-10-2023

Keywords

ordinary differential equations, partial fraction expansion, Laplace transform, Fourier series, harmonic

Disciplines

Mathematics | Physical Sciences and Mathematics | Science and Mathematics Education

Abstract

A generalized method for solving an undamped second order, linear ordinary differential equation with constant coefficients is presented where the non-homogeneous term of the differential equation is represented by Fourier series and a solution is found through Laplace transforms. This method makes use of a particular partial fraction expansion form for finding the inverse Laplace transform. If a non-homogeneous function meets certain criteria for a Fourier series representation, then this technique can be used as a more automated means to solve the differential equation as transforms for specific functions need not be determined. The combined use of the Fourier series and Laplace transforms also reinforces the understanding of function representation through a Fourier series and its potential limitations, the mechanics of finding the Laplace transform of a differential equation and inverse transforms, the operation of an undamped system, and through programming insight into the practical application of both tools including information on the influence of the number of terms in the series solution.

Recommended Citation

Florio, Laurie A. and Hanc, Ryan D. (2023) "A Generalized Solution Method to Undamped Constant-Coefficient Second-Order ODEs Using Laplace Transforms and Fourier Series," CODEE Journal: Vol. 16, Article 5.
Available at: https://scholarship.claremont.edu/codee/vol16/iss1/5