On the growth of meromorphic and entire solutions of second-order algebraic differential equations

Abstract

In this paper, the growth of the meromorphic solutions of the equation $$f'' = L(z,f)(f')^2 + M(z,f)f' + N(z,f)$$ where L, M, N are birational functions, is studied. We prove that if L(z, f) satisfies a quite general condition, then f must be of finite order. Furthermore, if (L(z, f)≡0, and M(z, f), N(z, f) are polynomials in f, then the order of any entire solution of the equation is a positive integral multiple of 1/2.