On the Growth of Solutions of 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 have proved 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 \(\frac{1}{2}\). Furthermore, we give a criterion of the normality relating to algebraic differential equations and use it to derive the growth of some solutions of a certain kind of algebraic differential equations.