First order rational differential equations depending transcendentally on arbitrary constants (Q916721)

Let \({\mathcal K}\) be a differential field of characteristic 0 with a single differentiation '. In the paper the question is discussed when does the general solution of the first order differential equation (1) \(y'=f(y)\), where \(f\in {\mathcal K}(y)\), depend algebraically on arbitrary constants. It is shown how the notion introduced by Painlevé may be formulated in terms of so called extensions depending algebraically on arbitrary constants in the differential algebra frame. A lemma which describes the properties of tower of such extension is proved. It is also proved that the above property of general solution leads to essential restrictions on \(f\) in (1). In conclusion it is obtained a broad class of equations whose general solutions depend transcendentally on arbitrary constants. [For references see \textit{M. Matsuda}, First order algebraic differential equations. A differential algebraic approach. (Lect. Notes Math. 804) (1980; Zbl 0447.12014) and \textit{K. Nishioka}, Nagoya Math. J. 113, 1-6 (1989; Zbl 0702.12008)].