On the meromorphic solutions of some algebraic differential equations (Q1076206)

The authors obtain various kinds of results concerning meromorphic solutions of (*) \(u'=\sum^{n}_{k=0}A_ k(z)u^ k,\) \(A_ n(z)\not\equiv 0\) where each \(A_ k\) is meromorphic in the whole complex plane in order to show that the Riccati differential equation occupies a special position in the equations of the form (*). They also study the growth of meromorphic solutions of the riccati differential equation \(u'=A(z)+u^ 2\) where A is meromorphic. The authors investigate equation (*) under several conditions on \(A_ k's\). For example, the number of distinct meromorphic or entire solutions is estimated from above. In the simple case where each \(A_ k\) is a polynomial, Theorem 3 says that if \(n\geq 3\) then (*) can possess at most a finite number of meromorphic solutions and compare this with the well known theorem that every solution of (*) is meromorphic if \(n\leq 2\). Theorem 2 is related to a question of the late professor E. Hille. One can find several interesting examples in this paper.