As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of -

$\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall k=0,\cdots ,n, \, a_k\in \mathbb{R}, \, a_n\neq 0\,. $

There are interesting motivations for it in here, but I'm mainly interested in the historical aspects.