Antiderivatives and linear differential equations using matrices

Abstract

We show how to find the closed-form solutions for antiderivatives of $x^n{e}^{ax}sinbx$ and $x^n{e}^{ax}cosbx$ for all $n\in {\mathbb{N}}_0$ and $a,b\in ℝ$ with $a^2+b^2\ne 0$ 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 $x^n{e}^{ax}sinbx$ or $x^n{e}^{ax}cosbx$.