In the paper we offer criteria for oscillation of the third order Euler differential equation with delay $$y^{\text{'}\text{'}\text{'}}\left(t\right)+\frac{k^2}{t^3}y\left(ct\right)=0.$$ We provide detail analysis of the properties of this equation, we fill the gap in the oscillation theory and provide necessary and sufficient conditions for oscillation of equation considered.

We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term $$x^{\text{'}\text{'}\text{'}}\left(t\right)+q\left(t\right)x^{\text{'}}\left(t\right)+r\left(t\right){\left|x\right|}^{\lambda }\left(t\right)\mathrm{sgn}x\left(t\right)=0,t\ge 0.$$ We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case $\lambda \le 1$ and if the corresponding second order differential equation $h^{\text{'}\text{'}}+q\left(t\right)h=0$ is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.

In this paper, the authors present some new results for the oscillation of the second order nonlinear neutral differential equations of the form $${\left(r\left(t\right)\psi \left(x\left(t\right)\right){\left[x\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)\right]}^{\text{'}}\right)}^{\text{'}}+q\left(t\right)f\left(x\left[\sigma \left(t\right)\right]\right)=0$$ . Easily verifiable criteria are obtained that are also new for differential equations without neutral term i.e. for p(t)≡0.

In this paper we present some new oscillatory criteria for the $n$-th order neutral differential equations of the form $${(x\left(t\right)\pm p\left(t\right)x\left[\tau \left(t\right)\right])}^{\left(n\right)}+q\left(t\right)x\left[\sigma \left(t\right)\right]=0.$$ The results obtained extend and improve a number of existing criteria.

The authors study the n-th order nonlinear neutral differential equations with the quasi – derivatives $L_n[x\left(t\right)+{(-1)}^{r}P\left(t\right)x\left(g\left(t\right)\right)]+\delta Q\left(t\right)f\left(x\left(h\left(t\right)\right)\right)=0,$ where $n\ge 2,r\in \{1,2\},$ and $\delta =\pm 1.$ There are given sufficient conditions for solutions to be either oscillatory or they converge to zero.

Some oscillation criteria for solutions of a general perturbed second order ordinary differential equation with damping (r(t)x′ (t))′ + h(t)f (x)x′ (t) + ψ(t, x) = H(t, x(t), x′ (t)) with alternating coefficients are given. The results obtained improve and extend some existing results in the literature.