Homogeneous, isobaric, and autonomous algebraic differential equations (Q1178472)

Let \(u\) be a \(C^ \infty\)-function. The author proves that if \(u\) satisfies an algebraic differential equation (ADE), then \(u\) must satisfy an ADE which is autonomous, homogeneous, and isobario. The assumption that \(u\) is a \(\mathbb{Q}^ \infty\)-function is necessary as shows the following example. Let \(u(x)=(e^ x-1)^ 2\) if \(x\geq 0\), and \(u(x)=0\) if \(x<0\). Then \(u\in C^ 1(\mathbb{R})\), \(\mathbb{R}\) the real line, and \(u\) satisfies the first-order ADE \((y')^ 2-4yy'+4y^ 2-4y=0\) but it satisfies no homogeneous or isobaric ADE. For related results see \textit{L. A. Rubel} and \textit{M. F. Singer} [J. Differ. Equations 75, 354-370 (1988; Zbl 0674.34004)].