Consider an equation of the form: cy"+cy'+cy Or something of the form. Essentially, it's a polynomial but instead of powers, there are derivatives. Do these kind of things have a name? Or are they completely useless?
Note: I KNOW what Taylor Polynomials and the like are, but I mean something in the form of what I have shown.
On
Yes the polynomial associated to $$ ay'' + by' +cy =0$$ is $$P(\lambda )= a \lambda ^2 + b \lambda +c$$ which is called the charateristic polynomial.
This polynomial plays a very important role in finding the solutions to your differential equation.
The genera solution to the differential equation is $$ y=C_1 e^{\lambda _1} +C_2 e^{\lambda _2} $$ where $\lambda _1$ and $\lambda _2$ are solutions to $P(\lambda)$
On
$$L= a_0(x) + a_1(x)\frac{d}{dx}+ \ldots + a_n(x) \frac{d^n}{dx^n} $$
is known as a linear differential operator.
We have $$Ly= a_0(x)y + a_1(x)\frac{dy}{dx}+ \ldots + a_n(x) \frac{d^ny}{dx^n} $$
$$a_0(x)y+a_1(x)y'+\ldots a_n(x)y^{(n)}+b(x)=0$$ is a linear differential equation.
when the $a_i(x)$ is independent of $x$, we describe them as constant coefficients.
There is a rich study of so-called "differential algebra."
https://en.wikipedia.org/wiki/Differential_algebra
However, what you're realizing is the connection between linear algebra and differential equations.
Namely, if $D: C^\infty(X) \longrightarrow C^\infty(X)$ is the derivative operator, and $1$ represents the identity map, then the vectors $y$ which satisfy the following polynomial equation
$$(a_n D^n + \cdots+ a_1D + a_01)y=0$$ are said to be solutions to the differential equation
$$a_n y^{(n)} + \cdots+ a_2y''+ a_1y' + a_0y=0.$$
The missing connection would be the Cayley Hamilton Theorem. Which would say that if $T:V\longrightarrow V$ is a linear operator, with characteristic equation $$a_n \lambda^n + \cdots+ a_1\lambda + a_0=0$$
Then $T$ satisfies this characteristic equation $$a_n T^nv + \cdots+ a_1Tv + a_0v=0$$ for all $v\in V.$