A Graceful Basis in the Solution Space of an ODE with Constant Coefficients

TL;DR

This paper revisits the classical problem of constructing a fundamental system of solutions for linear ODEs with constant coefficients, focusing on solutions that remain analytic and linearly independent across all roots of the characteristic polynomial.

Contribution

It provides a new approach to constructing solutions that are analytic and linearly independent regardless of the roots' values.

Findings

Established a method for constructing a graceful basis of solutions.

Proved solutions remain analytic for all roots of the characteristic polynomial.

Ensured linear independence of solutions across the entire root spectrum.

Abstract

We revisit the classical problem of construction of a fundamental system of solutions to a linear ODE whose elements remain analytic and linearly independent for all values of the roots of the characteristic polynomial.