# About terminology: is there a name for the property that the solution to a differential equation is a function instead of a value

A differential equation is a higher tier equation.

For comparison: the equation to find the roots of a polynomial:
if a solution exists the solution is a value (or multiple values, if there are multiple roots.)

In the case of differential equations:
if for a differential equation a solution exists the solution is a function.

In that sense differential equations are a next level up; a higher tier.

Is there a name convention for that property?

• Look up “functional” (as a noun) to see if it is a good fit for what you have in mind. Apr 9, 2023 at 16:56
• Is it "differential operator" the name you are looking for ? The word "operator" being used in general for a function that maps functions onto functions. Apr 9, 2023 at 17:32
• @ElchananSolomon 'functional' is the closest thing, I suppose. With a differential equation the differential relation must be satisfied for the entire domain concurrently. In that sense a differential equation is global. (The reason for mentioning that: I have seen people claim that a differential equation is local.) Apr 9, 2023 at 17:33
• It is local in the sense that you can check if a differentiable function satisfies the DE in a point just using the value and derivatives of the function in that point. Of course, to be a solution it has to satisfy the DE in every point of its domain interval. // The solution of the DE (or an IVP) is just a point in a function space. Apr 9, 2023 at 20:22