Delay Differential Equations with a Damping Term: Enhanced Criteria for Testing the Oscillatory Performance of Solutions

The oscillatory characteristics of solutions to damped differential equations are examined in this study. Enhanced properties are obtained for positive solutions of the equation under examination. These properties are then utilized to derive criteria ensuring the oscillation of all solutions of the equation, based on the symmetry principle between positive and negative solutions. Several techniques are employed to establish oscillation criteria that encompass a broader range of cases of the equation under investigation. Through the new approach adopted in this study, criteria are obtained that refine and complement the related previous results. The findings are applied to Euler-type equations and are juxtaposed with prior results to illustrate their significance.