In this work, we prove some new oscillation theorems for second-order neutral delay differential equations of the form ${\left(a\left(\xi \right)\left({(v\left(\xi \right)+b\left(\xi \right)v\left(\vartheta \left(\xi \right)\right))}^{\prime }\right)\right)}^{\prime }+c\left(\xi \right)G_1\left(v\left(\kappa \left(\xi \right)\right)\right)+d\left(\xi \right)G_2\left(v\left(\varsigma \left(\xi \right)\right)\right)=0$ under canonical and non-canonical operators, that is, ${\int }_{{\xi }_0}^{\infty }\frac{d\xi }{a\left(\xi \right)}=\infty$ and ${\int }_{{\xi }_0}^{\infty }\frac{d\xi }{a\left(\xi \right)}<\infty$. We use the Riccati transformation to prove our main results. Furthermore, some examples are provided to show the effectiveness and feasibility of the main results.

By using the Riccati technique, which reduces the higher order dynamic equations to a Riccati dynamic inequality, we will establish some new sufficient conditions for the oscillation of half-linear/Emden-Fowler neutral dynamic equation of the form \[ (r(\varrho)((\mathbf{\mathbf{x}}(\varrho)+p(\varrho)\mathbf{x}(\tau (\varrho)))^{\Delta })^{\gamma })^{\Delta }+q(\varrho)\mathbf{x}^{a }(\delta (\varrho))+v(\varrho)\mathbf{x}^{\beta }(\eta (\varrho))=0, \] on a time scale \(\mathcal{T}\), where \(\gamma \), \(a \), and \(\beta \) are quotients of odd positive integers. An example with particular equation is constructed in consistent to the above equation and oscillation criteria are established for its solution.