Matrix Differential Operator Method of Finding a Particular Solution to a Nonhomogeneous Linear Ordinary Differential Equation with Constant Coefficients

TL;DR

This paper introduces a matrix differential operator approach, utilizing inverse and pseudoinverse matrices, including Moore-Penrose pseudoinverses, to efficiently find particular solutions to nonhomogeneous linear ODEs with constant coefficients.

Contribution

It develops a novel matrix differential operator method that employs block matrices and pseudoinverses for solving specific nonhomogeneous linear ODEs with constant coefficients.

Findings

Effective use of block matrices for solution calculation

Application of Moore-Penrose pseudoinverse for singular matrices

Simplified computation of particular solutions

Abstract

The article presents a matrix differential operator and a pseudoinverse matrix differential operator for finding a particular solution to nonhomogeneous linear ordinary differential equations (ODE) with constant coefficients with special types of the right-hand side. Calculation requires the determination of an inverse or pseudoinverse matrix. If the matrix is singular, the Moore-Penrose pseudoinverse matrix is used for the calculation, which is simply calculated as the inverse submatrix of the considered matrix. It is shown that block matrices are effectively used to calculate a particular solution.