Simple conditions for parametrically excited oscillations of generalized Mathieu equations

The following equation is considered in this paper:x″+(−α+βcos⁡(γt))x=0, where α, β and γ   are real parameters and γ>0γ>0. This equation is referred to as Mathieu's equation when γ=2γ=2. The parameters determine whether all solutions of this equation are oscillatory or nonoscillatory. Our results provide parametric conditions for oscillation and nonoscillation; there is a feature in which it is very easy to check whether these conditions are satisfied or not. Parametric oscillation and nonoscillation regions are drawn to help understand the obtained results.