A special kind of nonoscillatory second order linear differential equations (Q1121424)

The author shows that the differential equation \(y''+p(x)y=0\), with \(p(x)=1\) for \(0\leq x\leq x_ 0\) and \(p(x)=-\eta\) \((\eta >0)\) for \(x_ 0<x\leq 2\pi\), is nonoscillatory if \(x_ 0\geq 0\) is such that: \((i)\quad tg x_ 0<\sqrt{\eta},\) \((ii)\quad x_ 0<(2\pi -x_ 0)\eta,\) \((iii)\quad x_ 0<\pi /2.\) The result is discussed under some other assumptions on \(x_ 0\) and p(x) and compared with similar results obtained on the basis of Adamov's theorem.