Differential Polynomials(?) 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.

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 $P(\lambda)$  
