On the zeros of the solutions of a class of second order linear differential equations
type: Item
instance of: scholarly article
title: On the zeros of the solutions of a class of second order linear differential equations
zbMATH Open document ID: 0545.34006
author: Anna Maria Bresquar
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1981-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/107799
review text: Il est question de l'équation oscillante (1) \(x''(t)+a(t)x'(t)+b(t)x(t)=0,\) où a(t), b(t) sont fonctions à valeurs réelles et localement sommables sur la demi-droite \([c,+\infty)\). Soit J un ensemble, qui satisfait aux conditions \(J\subset [c,+\infty)\), \(\mu([c,+\infty)\backslash J)=0,\) tel que l'on ait pour \(t\in J\) lim a(t)\(=h\), lim b(t)\(=k\) pour \(t\to +\infty\), avec \(4k-h^ 2>0\). Soient \(t_ n\) les zéros d'une solution x(t) de (1) et \(\tau_ n\) les zéros de x'(t) dans une demi-droite, dans laquelle on a \(t_ n<\tau_ n<t_{n+1}\). Des résultats établis par l'auteur nous rappelons les suivantes: \[ \lim(t_{n+1}-t_ n)=2\pi /\sqrt{4k-h^ 2},\quad \lim(\tau_ n-t_ n)=2/\sqrt{4k-h^ 2}\quad arc\cot g (h/\sqrt{4k-h^ 2}) \] pour \(n\to +\infty\). Le travail ce termine avec deux exemples.
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34A99
zbMATH DE Number: 3867740
zbMATH Keywords: comparison theorems
zbMATH Keywords: upper bounds
zbMATH Keywords: conjugate points
zbMATH Keywords: asymptotic formulae
zbMATH Keywords: lower bounds
reviewed by: Silvio Cinquini
MaRDI profile type: Publication
cites work: Q5614828
cites work: Oscillation and Comparison Theorems for Second Order Linear Differential Equations with Integrable Coefficients
cites work: Disconjugacy
cites work: Q5330830
cites work: Oscillation theory
cites work: On approximating conjugate, focal, and \(\sigma\)-points for linear differential equations of second order
cites work: Some Comparison Theorems for Conjugate And σ-Points
cites work: Q3847596
cites work: Q5336523
cites work: BEHAVIOR OF THE SOLUTIONS OF THE EQUATION $ \ddot x+p(t)\dot x+q(t)x=0$ IN THE NONOSCILLATORY CASE
cites work: The Existence of Conjugate Points for Selfadjoint Differential Equations of Even Order
cites work: Relation between the distribution of the zeros of the solutions of a 2nd order linear differential equation and the boundedness of these solutions
cites work: On the Distance Between Zeroes
cites work: Q5613350
cites work: Oscillation on Finite or Infinite Intervals of Second Order Linear Differential Equations(1)
description: scientific article; zbMATH DE number 3867740