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 $\gamma \le \frac{{\alpha }^2-4\alpha +5}{8}$.