Polynomial terminology on linear differential equations?

Question

I've seen people refer to $n$th order linear differential equations as an "$n$th order linear polynomial of the function $y$ and its derivatives."

I can see how it definitely share the same form as a polynomial:

$a_0(x)y+a_1(x)y'+a_2(x)y''+\cdots+a_n(x)y^{(n)}=b(x)$

I also see this in how people characterize homogeneous LDE's as such because they are homogeneous polynomials (i.e all the terms equal $0$). I get the analog being drawn but these aren't truly polynomials right? How can we say these are polynomials if they are not being exponentiated?

Is there a more general notion of polynomial or '$n$th order' that would encompass $n$th order differential equations and $n$th order polynomials?

Answer 1

A polynomial of degree one in the variables $z_0,z_1,\dots,z_n$ has the form $$a_0 z_0 + a_1 z_1 + \dots + a_n z_n + b.$$ Now substitute $z_0=y$, $z_1=y'$ etc. (and allow all coefficients to depend on $x$).

(By the way, in the Wikipedia page that you link to there is no mention of an “$n$th order linear polynomial”, whatever that would be; it just says “linear polynomial”.)