Meromorphic differential equations having all monomials as solutions (Q1203538)

In this paper one considers ordinary differential equations with meromorphic coefficients of the form \(F(y(x),y^{(1)}(x),\dots,y^{(r)}(x))=0\), where \(F(Y_ 0,Y_ 1,\dots,Y_ r)\) is a polynomial in the variables \(Y_ 0,\dots,Y_ r\) with coefficients in the field \(\mathbb{C}\{x\}[x^{-1}]\) of convergent Laurent series in \(x\) over \(\mathbb{C}\). It is shown that essentially the only differential equation of this form with the property that each monomial \(x^ n\) (\(n=0,1,2,\dots\)) is a solution, is the equation \(xy\ddot y- x\dot y^ 2+y\dot y=0\).