In this work we consider a given root of a family of $n$-degree polynomials as a one-variable function that depends only on the independent term. Then we prove that this function satisfies several ordinary differential equations (ODE). More concretely, it satisfies several simple separated variables ODE, a first order generalized Abel ODE of degree $n-1$ and an $(n-1)$-th order linear ODE. Although some of our results are not new, our approach is simple and self-contained. For $n=2,3$ and 4 we recover, from these ODE, the classical formulas for solving these polynomials.

中文翻译：

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用常微分方程求解多项式

在这项工作中，我们考虑一个家庭的给定根 $n$-次多项式作为仅依赖于独立项的单变量函数。然后我们证明这个函数满足几个常微分方程（ODE）。更具体地说，它满足几个简单的分离变量 ODE，一个度数的一阶广义 Abel ODE$n-1$ 和 $(n-1)$-th 阶线性 ODE。尽管我们的一些结果并不新鲜，但我们的方法很简单且自成体系。为了$n=2,3$ 和 4 我们从这些 ODE 中恢复用于求解这些多项式的经典公式。