The author considers the equation (1) \((p(t)y'(t))'+q(t)y(t)=0,\) \(t\geq T\), where p and q are real-valued and \(p(t)>0\) for all \(t\geq T\). This paper contains three parts: The main result of the first part is Theorem 1: Suppose q is piecewise continuous and there exists a function r with \(pr\in C^ 1[T,\infty)\) such that \(\lim_{t\to \infty}\int^{t}_{T}p^{-1}(s)\exp [2\int^{s}_{r}r(v)dv]ds=\infty,\) \[ \lim_{t\to \infty}\int^{t}_{T}\exp [-2\int^{s}_{T}r(v)dv](q-(pr)'+pr^ 2)(s)ds=\infty, \] then equation (1) is oscillatory. In the second part the author gives comparison results for equations of the form (1). In the last part he gives new results in the estimation of the number of zeros of solutions to (1) in an interval [\(\alpha\),\(\beta\) ]\(\subset [T,\infty)\).