Differential equations of second order appear in physical applications such as fluid dynamics, electromagnetism, acoustic vibrations, and quantum mechanics. In this paper, necessary and sufficient conditions are established of the solutions to second-order half-linear delay differential equations of the form , under the assumption . We consider two cases when and , where and are the quotient of two positive odd integers. Two examples are given to show effectiveness and applicability of the result.

中文翻译：

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具有多个时滞的二阶微分方程：振动定理和应用

二阶微分方程出现在物理应用中，例如流体动力学，电磁学，声振动和量子力学。在本文中，建立了以下形式的二阶半线性时滞微分方程解的充要条件，该假设下。我们考虑两种情况 和 ，其中和是两个正整数的商。给出两个例子来说明结果的有效性和适用性。