On The Growth Of Solutions Of A Certain Type Of Non-linear Differential Equations

In 1920s,the uniqueness theory of meromorphic functions is established by R.Nevanlinna.The Nevanlinna theory is the basis of modern meormorphic function theory,and it has great effect on the development of mathematical branches.Moreover,the Nevanlinna theory provides a useful method to the research of differential equations and makes this area's development full of vitality.In this paper,we research the growth of solutions of a certain type of non-linear differential equations by using the value distribution theory as an important tool.The whole paper divides into four chapters.In the first chapter,we briefly introduce some fundamental results,usual notations and main concepts in Nevanlinna value distribution theory.In the second chapter,we discuss the meromorphic solutions of a class of differential equations.Research an important conjecture which was proposed by Hayman in 1959.We solve some of the well-known speculation,and promote the conclusions of previous.The third chapter,we study the growth of solutions of a certain type of non-linear differential equations.Research an important conclusion of Wittich and improve the conclusions from Zhang and Liao.Discuss the equation with form ofQ?z,f?=a₁?z?e^b1^?z?+a₂?z?e^b2^?z?+p?z?Relates to homogeneous function of some differential equations,we also do further research to improve the results of some scholars.The fourth chapter,we consider the zeros of some differential polynomial,using the logarithmic derivative method,the method to construct auxiliary function,etc.,so improving the result of E.Mues and M.Ozawa to the form ff'+a?f'?²+b.Specifically,we use a different method form others to prove this conclusion.