Some new results on a linear equation of the second order

Work on solving the second order linear oscillation defined by Eq. (1.1), with continuous and positive coefficient Φ(x)Φ(x) that satisfies Lipschitz’s condition on semi-axis [0,+∞)[0,+∞) and the divergence of ∫0+∞(Φ−G′2)dx, had started since the 1830s with Sturm’s theorems. This paper presents generalizations as well as a simplification of classical Sturm’s theorems on the location and the position of zero oscillations, which have not been included in Amrein et al. (2005) [5]. Besides, according to results from Dimitrovski and Mijatović (1997) [1], Dimitrovski et al. (2007) [2] and Dimitrovski et al. (2007) [4], we add some ideas and supplements (Theorems 2.4–2.8, 2.10 and 2.11) to the classical Sturm’s theory of oscillations.