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

Keywords

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

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