Method of variation of parameters revisited

TL;DR

This paper revisits the method of variation of parameters for solving linear ODEs, providing a systematic construction of solutions, analyzing invariance properties, and discussing applications including Green's functions.

Contribution

It offers a systematic explanation of the variation of parameters method, extending it to systems of ODEs and exploring its invariance and application to Green's functions.

Findings

Systematic construction of particular solutions using linearly independent functions.

Invariance of solutions even when time variation of constants is significant.

Application of VOP method to derive Green's functions for linear systems.

Abstract

The method of variation of parameter (VOP) for solving linear ordinary differential equation is revisited in this article. Historically, Lagrange and Euler explained the method of variation of parameter in the context of perturbation method. In this article, we explain the construction of particular solutions of a linear ordinary differential equation in the light of linearly independent functions in a more systematic way. In addition, we have shown that if the time variation of the constants contribute substantially to the velocity then also the solution remains invariant. VOP method for system of n linear ODE is discussed. Duhamels principle has also been studied in reference to a system of n linear ODE for completeness of this review. Finally, applications of VOP method for constructing Green's function is reported.