Applications of Prüfer transformations in the theory of ordinary differential Equations

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Abstract

his article is a review article on the use of Prüfer Transformations techniques in proving classical theorems from the theory of Ordinary Differential Equations. We consider self-adjoint second order linear differential equations of the form Lx = (p(t)x (t)) + g(t)x(t) = 0, t ∈ (a, b). () We use Prüfer transformation techniques (which are a gener-alization of Poincaré phase-plane analysis) to obtain some of the main theorems of the classical theory of linear differen-tial equations. First we prove theorems from the Oscillation Theory (Sturm Comparison theorem and Disconjugacy theo-rems). Furthermore we study the asymptotic behavior of the equation () when t → ∞ and we obtain necessary and suf-ficient conditions in order to have bounded solutions for (). Finally, we consider a certain type of regular Sturm–Liouville eigenvalue problems with boundary conditions and we study their spectrum via Prüfer transformations.
Original languageEnglish
Pages (from-to)11-31
Number of pages21
JournalIrish Math. Soc. Bulletin
Volume63
Publication statusPublished - 2009

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Ordinary differential equation
Theorem
Sturm's theorem
Disconjugacy
Phase Plane Analysis
Oscillation Theory
Sturm-Liouville Problem
Bounded Solutions
Comparison Theorem
Linear differential equation
Second order differential equation
Eigenvalue Problem
Asymptotic Behavior
Boundary conditions
Necessary
Review
Form

Keywords

  • Ordinary Differential Equations,.
  • Prufer Transformations
  • Sturm Liouville Problems

Cite this

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title = "Applications of Pr{\"u}fer transformations in the theory of ordinary differential Equations",
abstract = "his article is a review article on the use of Pr{\"u}fer Transformations techniques in proving classical theorems from the theory of Ordinary Differential Equations. We consider self-adjoint second order linear differential equations of the form Lx = (p(t)x (t)) + g(t)x(t) = 0, t ∈ (a, b). () We use Pr{\"u}fer transformation techniques (which are a gener-alization of Poincar{\'e} phase-plane analysis) to obtain some of the main theorems of the classical theory of linear differen-tial equations. First we prove theorems from the Oscillation Theory (Sturm Comparison theorem and Disconjugacy theo-rems). Furthermore we study the asymptotic behavior of the equation () when t → ∞ and we obtain necessary and suf-ficient conditions in order to have bounded solutions for (). Finally, we consider a certain type of regular Sturm–Liouville eigenvalue problems with boundary conditions and we study their spectrum via Pr{\"u}fer transformations.",
keywords = "Ordinary Differential Equations,., Prufer Transformations, Sturm Liouville Problems",
author = "George Chailos",
year = "2009",
language = "English",
volume = "63",
pages = "11--31",
journal = "Irish Math. Soc. Bulletin",
issn = "0791-5578",

}

Applications of Prüfer transformations in the theory of ordinary differential Equations. / Chailos, George.

In: Irish Math. Soc. Bulletin , Vol. 63, 2009, p. 11-31.

Research output: Contribution to journalArticle

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AU - Chailos, George

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AB - his article is a review article on the use of Prüfer Transformations techniques in proving classical theorems from the theory of Ordinary Differential Equations. We consider self-adjoint second order linear differential equations of the form Lx = (p(t)x (t)) + g(t)x(t) = 0, t ∈ (a, b). () We use Prüfer transformation techniques (which are a gener-alization of Poincaré phase-plane analysis) to obtain some of the main theorems of the classical theory of linear differen-tial equations. First we prove theorems from the Oscillation Theory (Sturm Comparison theorem and Disconjugacy theo-rems). Furthermore we study the asymptotic behavior of the equation () when t → ∞ and we obtain necessary and suf-ficient conditions in order to have bounded solutions for (). Finally, we consider a certain type of regular Sturm–Liouville eigenvalue problems with boundary conditions and we study their spectrum via Prüfer transformations.

KW - Ordinary Differential Equations,.

KW - Prufer Transformations

KW - Sturm Liouville Problems

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SP - 11

EP - 31

JO - Irish Math. Soc. Bulletin

JF - Irish Math. Soc. Bulletin

SN - 0791-5578

ER -