Antiderivatives and linear differential equations using matrices

Meemark, Yotsanan; Sriwongsa, Songpon

doi:10.2140/involve.2019.12.151

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Antiderivatives and linear differential equations using matrices

Yotsanan Meemark and Songpon Sriwongsa

Vol. 12 (2019), No. 1, 151–156

DOI: 10.2140/involve.2019.12.151

Abstract

We show how to find the closed-form solutions for antiderivatives of $x^{n} e^{a x} sin b x$ and $x^{n} e^{a x} cos b x$ for all $n \in ℕ_{0}$ and $a, b \in ℝ$ with $a^{2} + b^{2} \neq 0$ by using an idea of Rogers, who suggested using the inverse of the matrix for the differential operator. Additionally, we use the matrix to illustrate the method to find the particular solution for a nonhomogeneous linear differential equation with constant coefficients and forcing terms involving $x^{n} e^{a x} sin b x$ or $x^{n} e^{a x} cos b x$ .

Keywords

differential operator, inverse of matrix, rectangular form

Mathematical Subject Classification 2010

Primary: 15A09

Secondary: 34A30

Milestones

Received: 3 September 2017

Revised: 26 October 2017

Accepted: 14 December 2017

Published: 31 May 2018

Communicated by Kenneth S. Berenhaut

Authors

Yotsanan Meemark
Department of Mathematics and Computer Science Faculty of Science Chulalongkorn University Bangkok Thailand

Songpon Sriwongsa
Department of Mathematical Sciences University of Wisconsin-Milwaukee Milwaukee, WI United States

Vol. 12, No. 1, 2019