Ilea, Veronica; Otrocol, Diana; Rus, Ioan A. Some properties of solutions of the homogeneous nonlinear second order differential equations. (English) Zbl 1389.34044 Mathematica 57(80), No. 1-2, 38-43 (2015). Summary: In this paper we consider the following nonlinear homogeneous second order differential equations \[ F (x, y, y', y'') = 0. \] We present for the solutions, \(y \in C^2 [a, b]\), of this equation, extremal principle, Sturm-type, Nicolescu-type and Butlewski-type separation theorems. Some applications and examples are given. Open problems are also presented. MSC: 34A09 Implicit ordinary differential equations, differential-algebraic equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 58E50 Applications of variational problems in infinite-dimensional spaces to the sciences Keywords:homogeneous nonlinear second-order differential equation; zeros of solutions; Sturm-type theorem; Nicolescu-type theorem; Butlewski-type theorem; bilocal problem; Cauchy problem; open problem; extremal principle PDFBibTeX XMLCite \textit{V. Ilea} et al., Mathematica 57(80), No. 1--2, 38--43 (2015; Zbl 1389.34044)