The differential equation \(y'''=0\) revisited (Q810225)

An N-solution of an algebraic differential equation is a kind of generalized solution, based on the differentially algebraic Nullstellensatz. The author shows that there exist two N-solutions \(y_ 1\) and \(y_ 2\) of \(y'''=0\). The reason in this case is that \(y_ 1+y_ 2\) is not the actual solution (and hence not the N-solution) of any algebraic differential equation. The question remains open whether, if \(y_ 1\) and \(y_ 2\) are N-solutions of \(y'''=0\) and if \(y_ 1+y_ 2\) does satisfy some algebraic differential equation, then \(y_ 1+y_ 2\) must be an N-solution of \(y'''=0\).