Notes on the zeros of the solutions of the non-homogeneous Airy's equation

TL;DR

This paper investigates the distribution of zeros in solutions to the non-homogeneous Airy's equation, identifying families of solutions with specific zero properties and introducing a recursion to describe zero distribution.

Contribution

It introduces a recursion for zero distribution and characterizes solutions with simple and double zeros, extending previous results on the homogeneous case.

Findings

Existence of solutions with simple zeros

Existence of solutions with double zeros at specified points

A recursion describing zero distribution and its limitations

Abstract

We present some observations on the distribution of the zeros of solutions of the nonhomogeneous Airy's equation. We show the existence of a principal family of solutions, with simple zeros, and particular solutions, characterized by a double zero in a given position of the complex plane. A recursion, describing the distribution of the zeros, is introduced and the limits of its applicability are discussed. The results can be considered a generalization of previous works on the distribution of the zeros for the solutions of the corresponding homogeneous equation