SIAM Review, Volume 64, Issue 2, Page 485-499, May 2022.
In the study of ordinary differential equations (ODEs) of the form $\hat{L}[y(x)]=f(x)$, where $\hat{L}$ is a linear differential operator, two related phenomena can arise: resonance, where $f(x)\propto u(x)$ and $\hat{L}[u(x)]=0$, and repeated roots, where $f(x)=0$ and $\hat{L}=\hat{D}^n$ for $n\geq 2$. We illustrate a method to generate exact solutions to these problems by taking a known homogeneous solution $u(x)$, introducing a parameter $\epsilon$ such that $u(x)\rightarrow u(x;\epsilon)$, and Taylor expanding $u(x;\epsilon)$ about $\epsilon = 0$. The coefficients of this expansion $\frac{\partial^k u}{\partial\epsilon^k}\big{|}_{\epsilon=0}$ yield the desired resonant or repeated root solutions to the ODE. This approach, whenever it can be applied, is more insightful and less tedious than standard methods such as reduction of order or variation of parameters. We provide examples of many common ODEs, including constant coefficient, equidimensional, Airy, Bessel, Legendre, and Hermite equations. While the ideas can be introduced at the undergraduate level, we could not find any existing elementary or advanced text that illustrates these ideas with appropriate generality.

中文翻译：

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使用微扰方法生成常微分方程的谐振和重复根解

SIAM 评论，第 64 卷，第 2 期，第 485-499 页，2022 年 5 月。
在 $\hat{L}[y(x)]=f(x)$ 形式的常微分方程 (ODE) 的研究中，其中 $\hat{L}$ 是线性微分算子，两个相关的现象可以出现：共振，其中 $f(x)\propto u(x)$ 和 $\hat{L}[u(x)]=0$，以及重复根，其中 $f(x)=0$ 和 $\帽子{L}=\hat{D}^n$ 为 $n\geq 2$。我们通过采用已知齐次解 $u(x)$、引入参数 $\epsilon$ 使得 $u(x)\rightarrow u(x;\epsilon)$ 和泰勒将 $u(x;\epsilon)$ 扩展为关于 $\epsilon = 0$。此展开式 $\frac{\partial^ku}{\partial\epsilon^k}\big{|}_{\epsilon=0}$ 的系数产生 ODE 的所需谐振或重复根解。这种方法，只要可以应用，比标准方法（例如减少阶数或参数变化）更具洞察力且不那么繁琐。我们提供了许多常见 ODE 的示例，包括常系数、等维、艾里、贝塞尔、勒让德和 Hermite 方程。虽然这些想法可以在本科阶段介绍，但我们找不到任何现有的初级或高级文本可以适当概括地说明这些想法。