On Meromorphic Solutions Of Non-Linear Differential Equations Of Tumura-Clunie Type

Meromorphic solutions of non-linear differential equations of the form fn+P(z,f)=h are investigated, where n >= 2 is an integer, h is a meromorphic function, and P(z,f) is differential polynomial in f and its derivatives with small functions as its coefficients. In the existing literature this equation has been studied in the case when h has the particular form h(z)=p1(z)e alpha 1(z)+p2(z)e alpha 2(z), where p1,p2 are small functions of f and alpha 1,alpha 2 are entire functions. In such a case the order of h is either a positive integer or equal to infinity. In this article it is assumed that h is a meromorphic solution of the linear differential equation h ''+r1(z)h '+r0(z)h=r2(z) with rational coefficients r0,r1,r2, and hence the order of h is a rational number. Recent results by Liao-Yang-Zhang (2013) and Liao (2015) follow as special cases of the main results.