Mathieu’s equation originally emerged while studying vibrations on an elliptical drumhead, so naturally, being a linear second-order ordinary differential equation with a Cosine periodic potential, it has many useful applications in theoretical and experimental physics. Unfortunately, there exists no closed-form analytic solution of Mathieu’s equation, so that future studies and applications of this equation, as evidenced in the literature, are inevitably fraught by numerical approximation schemes and nonlinear analysis of so-called stability charts. The present research work, therefore, avoids such analyses by making exceptional use of Laurent series expansions and four-term recurrence relations. Unexpectedly, this approach has uncovered two linearly independent solutions to Mathie’s equation, each of which is in closed form. An exact and general analytic solution to Mathieu’s equation, then, follows in the usual way of an appropriate linear combination of the two linearly independent solutions.

中文翻译：

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Mathieu方程的精确解

Mathieu方程最初是在研究椭圆鼓头上的振动时出现的，因此自然而然地成为具有余弦周期电势的线性二阶常微分方程，在理论和实验物理学中都有许多有用的应用。不幸的是，没有Mathieu方程的闭式解析解，因此，如文献所证明的，该方程的未来研究和应用不可避免地受到数值逼近方案和所谓的稳定性图的非线性分析的困扰。因此，本研究工作通过大量使用Laurent级数展开和四项递归关系来避免此类分析。出乎意料的是，这种方法发现了Mathie方程的两个线性独立解，每个解都是封闭形式。