There is this abstract notion of a derivation, which really only cares about the property
$D(ab)=aD(b)+D(a)b,$
where $a,b$ are elements of some algebra. This only tangents the ideas, which lead to $\frac{\text d}{\text d x}$ for functions on the real line.
I wonder if there is such a thing as as differential equations in abstract algebra? I guess I can just write down an equation $D(aD(ab))=abaa$ or $D(a)=-ca$ for some algebra and some $D$ and try to figure out if there are actually elements, which satisfy this relation, but I wonder if peolpe are actually doing such things and what their insights turn out to be. Are there investigations of e.g. initially physically motivated equations in terms of this abstract concepts?
The only related variant I can think of are equations of Lie-Groups, however, it seems these can be expressed in terms of the usual derivative as well (since they also carry the manifold structure), so it's not really something new.