The Euler half-linear differential equation \[ \left[t^{\alpha-1}\,\Phi(x')\right]'+\gamma\,t^{\alpha-p-1}\,\Phi(x)=0\,,\quad \alpha\neq p \] is a typical example of the so-called \textit{conditionally oscillatory} half-linear differential equation. It is oscillatory if \(\gamma\,p^p> |p-\alpha|^p\) and non-oscillatory in the opposite case, that is, \(\lambda_0=|p-\alpha|^p/p^p\) is the \textit{oscillation constant}. In this paper, a modified Prüfer transformation is introduced in order to study the half linear second order differential equation \[ \left[t^{\alpha-1}\,r(t)\,\Phi(x')\right]'+t^{\alpha-p-1}\,s(t)\,\Phi(x)=0\,,\quad \Phi(x)=|x|^{p-1}\,\text{sgn}\, x\,, \] where \(p>1\), \(\alpha\neq p\) and \(r\), \(s\) are continuous functions such that \(r(t)>0\) for large \(t\). The authors obtain conditions on the functions \(r\), \(s\) which guarantee that this equation is conditionally oscillatory.