A short proof of solution formulas for the linear differential equations with constant coefficients

TL;DR

This paper presents a shorter, more comprehensible proof for the solution formulas of linear differential equations with constant coefficients, especially addressing cases with multiple roots of the characteristic polynomial.

Contribution

It introduces a novel, simplified proof technique that improves understanding and reduces complexity compared to traditional methods.

Findings

Proof is shorter and more accessible

Clarifies the role of polynomial factors in solutions

Enhances educational approaches for differential equations

Abstract

The solution of equations from the title is well known since the Euler's time. However, its proof in the case of multiple roots of the characteristic polynomial is rather long and technical and even appearance of the factors $x^m$ looks artificial. A new proof, proposed below, seems not only shorter, but also more comprehensible for the students of any level.