Meromorphic Solutions Of Certain Type Of Algebraic Differential Equations

This thesis deals with certain type of algebraic differential equations in the complex plane C. Value distribution theory offer an important method to the study of meromorphic solution to this kind of equations. The whole thesis consists of three chapters.In the first chapter, we introduce the definition of value sharing and some former results of value distribution theory in complex differential equations, and describe the main results of this thesis.In chapter 2, we introduce some definitions and notations of value distribution theory.In chapter 3, we amend the process of the proof of the following theorem [31].Theorem Suppose that f, g are two nonconstant meromorphic functions and n≥6 is an integer. If fⁿf'gⁿg' = 1, then g(z) = c₁e^cz and f(z) = C₂e^-cz, where c,c₁ and c₂ are constants satisfying (c₁C₂)ⁿ⁺¹c² = -1.It is nature to ask what will be happened when fⁿ and gⁿ in the above theorem are replaced by general polynomials in f and g, respectively. By using Nevanlinna'svalue distribution theory, we study the existence or solvability of meromorphicsolutions of functional equations of the two equations P(f)f'P(g)g' = 1 and P(f)P(g) = 1, where P is a polynomial with three or two distinct zeros at least. Finally, we study when the polynomial Q(z) = zⁿ(z-a)(z-b) is a uniqueness polynomialof meromorphic function, furthermore, we obtain a corollary to uniquely identify meromorphic functions.