Oscillation and nonoscillation criteria for the self-adjoint linear differential equation $${\left(t^{\alpha }{y}^{\text{'}\text{'}}\right)}^{\text{'}\text{'}}-\frac{{\gamma }_{2,\alpha }}{t^{4-\alpha }}y=q\left(t\right)y,\alpha \notin \{1,3\},$$ where $${\gamma }_{2,\alpha }=\frac{{(\alpha -1)}^2{(\alpha -3)}^2}{16}$$ and $q$ is a real and continuous function, are established. It is proved, using these criteria, that the equation $${\left(t^{\alpha }{y}^{\text{'}\text{'}}\right)}^{\text{'}\text{'}}-\left(\frac{{\gamma }_{2,\alpha }}{t^{4-\alpha }}+\frac{\gamma }{t^{4-\alpha }{ln}^{2}t}\right)y=0$$ is nonoscillatory if and only if $\gamma \le \frac{{\alpha }^2-4\alpha +5}{8}$.

In this paper two sequences of oscillation criteria for the self-adjoint second order differential equation ${\left(r\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}+p\left(t\right)u\left(t\right)=0$ are derived. One of them deals with the case ${\int }^{\infty }\frac{\mathrm{d}t}{r\left(t\right)}=\infty$, and the other with the case ${\int }^{\infty }\frac{\mathrm{d}t}{r\left(t\right)}<\infty$.

Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation $${(-1)}^n{\left(t^{\alpha }{y}^{\left(n\right)}\right)}^{\left(n\right)}+q\left(t\right)y=0(*)$$ are established. In these criteria, equation $(*)$ is viewed as a perturbation of the conditionally oscillatory equation $${(-1)}^n{\left(t^{\alpha }{y}^{\left(n\right)}\right)}^{\left(n\right)}-\frac{{\mu }_{n,\alpha }}{t^{2n-\alpha }}y=0,$$ where ${\mu }_{n,\alpha }$ is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.