Simple conditions for parametrically excited oscillations of generalized Mathieu equations

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• 作者：Kazuki Ishibashi ^{ishibashi_kazuaoi@yahoo.co.jp" class="auth_mail" title="E-mail the corresponding author} ; Jitsuro Sugie ; ^{jsugie@riko.shimane-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Mathieu's equation ; Oscillation problem ; Damped linear differential equations ; Variable transformation ; Parametric excitation
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：233-247
• 全文大小：421 K

文摘

The following equation is considered in this paper: where α, β and γ   are real parameters and γ>0$\gamma >0$. This equation is referred to as Mathieu's equation when γ=2$\gamma =2$. 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.