Ordinary differential equations of order zero?

Question

Is $x+y+2=0$ a differential equation without derivatives of order $n$, $n>0$? Could it be called a differential equation (for unknown $y(x)$) of order $0$?

If not, can we define differential equations of order zero?

Answer 1

You could call such equations "zero-order differential equations" but since no derivative is actually involved, the name is more misleading than helpful. I would call it an algebraic equation for $y$, to emphasize the difference between it and differential equations. Shorter and less confusing.

But if one generalizes from differential operators (like the $k$th derivative) to pseudo-differential operators, things get more interesting. There are important pseudo-differential operators of order $0$, such as the Hilbert transform, and one could consider equations involving them: say, solving $$ \int_{\mathbb R} \frac{u(x)}{x-t}\,dt = f(t) $$ for $u$. The nature of pseudo-differential operators of order $0$ is such that they are usually thought of as (singular) integral operators, and so the equations would be usually called integral equations. But occasionally one sees "pseudodifferential equations of order $0$", for example here:

In the present paper we prove the stability of a nodal spline collocation method for (locally) strongly elliptic zero order pseudodifferential equations