Qualitative comparison of the nonoscillatory behavior of the equations $$L_{n}y\left(t\right)+H(t,y\left(t\right))=0$$ and $$L_{n}y\left(t\right)+H(t,y\left(g\left(t\right)\right))=0$$ is sought by way of finding different nonoscillation criteria for the above equations. $L_n$ is a disconjugate operator of the form $$L_n=\frac{1}{p_n\left(t\right)}\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{p_{n-1}\left(t\right)}\frac{\mathrm{d}}{\mathrm{d}t}...\frac{\mathrm{d}}{\mathrm{d}t}\frac{·}{p_0\left(t\right)}.$$ Both canonical and noncanonical forms of $L_n$ have been studied.

In this paper, oscillation and asymptotic behaviour of solutions of $$y^{\text{'}\text{'}\text{'}}+a\left(t\right)y^{\text{'}\text{'}}+b\left(t\right)y^{\text{'}}+c\left(t\right)y=0$$ have been studied under suitable assumptions on the coefficient functions $a,b,c\in C\left(\right[\sigma ,\infty ),R)$, $\sigma \in R$, such that $a\left(t\right)\ge 0$, $b\left(t\right)\le 0$ and $c\left(t\right)<0$.