Null sets for algebraic differential equations (Q917755)

Consider an algebraic differential equation \(P(x,y(x),y'(x),...,y^{(n)}(x))=0\) where P is a nontrivial polynomial in all its variables. For a subset N of an interval I, it may happen that if y satisfies the equation for all \(x\in I\setminus N\), then y must satisfy the equation on all of I. In such a case, the set N is called a null set for P on I. The main results of the paper state that if N has inner Lebesgue measure zero, then N is a null set for all P and all I that contain N, and that the converse is true when there exists a suitable n-parameter family of solutions of the differential equation. In particular, the converse is valid whenever the equation is linear, in standard form, with polynomial coefficients.