The Darboux property and solutions of algebraic differential equations (Q1065238)

In discussing solutions u of an algebraic differential equation (ADE) or of a system of such equations, care must be taken to enunciate how smooth u is required to be - certain qualitative assertions are true for one degree of smoothness and false for another [\textit{L. Rubel}, J. Differ. Equations 49, 441-452 (1983; Zbl 0475.12029)]. In the present paper, a solution is simply a function u that makes the ADE a true statement when substituted into it. No assumptions of continuity of the highest order derivative are made. Several results are obtained, three of which are stated below. Theorem 3. A solution of an ADE on an interval I must be analytic on a dense open subset of I. Theorem 4. Let P and Q be differential polynomials and let u be a function on I such that: (i) u satisfies \(P(x,y,y',..,y^{(m)})=0\) on I. (ii) u is n-times differentiable on I. (iii) u satisfies \(Q(x,y,y',...,y^{(n)})=0\) on a dense subset of I. Then (iv) u satisfies \(Q(x,y,y',...,y^{(n)})=0\) everywhere on I. Theorem 5. Let \(\Sigma\) be a system of any cardinality, of ADE's in one dependent variable y. Then there exists a countable subsystem \(\Sigma_{\omega}\) of \(\Sigma\) such that \(\Sigma_{\omega}\) and \(\Sigma\) have exactly the same solutions.