We establish some new oscillation criteria for the second order neutral delay differential equation $${\left[r\left(t\right)\right|[x\left(t\right)+p\left(t\right)x\left[\tau \left(t\right)\right]]}^{\text{'}}{{|}^{\alpha -1}{[x\left(t\right)+p\left(t\right)x\left[\tau \left(t\right)\right]]}^{\text{'}}]}^{\text{'}}+q\left(t\right)f\left(x\left[\sigma \left(t\right)\right]\right)=0.$$ The obtained results supplement those of Dzurina and Stavroulakis, Sun and Meng, Xu and Meng, Baculíková and Lacková. We also make a slight improvement of one assumption in the paper of Xu and Meng.

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 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.