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Modified Euler type second order half-linear differential equations are considered and a non-oscillation criterion is derived for them. This criterion is the counterpart of a previously obtained oscillation theorem. Thus, from the main result of this paper, it follows that the studied equations are conditionally oscillatory in a very general case. To prove the non-oscillation criterion, a combination of the Riccati technique and the generalized Prüfer angle is used. Since the criterion is new in many cases (especially, in the linear case), several corollaries are formulated and the novelty is illustrated by an example as well.