Applications of Prüfer Transformations in the Theory of Ordinary Differential Equations

Abstract. This 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 Poincare ́ 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. 1