Entire Solutions Of Certain Type Of Non-linear Differential Equations

In the theory of differential equations in the complex plane C,it's interesting and quite difficult to prove the existence or uniqueness of entire or meromorphic solutions of some differential equation,particularly for the non-linear ones.Many scholars have done some research on this and obtained some results.However,after reviewing it,we can find that the relevant conclusions of the previous research have only two exponential terms on the right side of the differential equation.What are the results if there are three exponential terms?What are the results if there are four exponential terms or more?For this problem,this thesis study the existence problem of the solution of the non-linear differential equation with three exponential terms,and further extended to the existence problem of the solution of the corresponding equation with the sum of multiple exponential terms on the right.In this thesis,using the theory of Nevanlinna's value distribution and differential equa-tions,we study the entire solutions of the following type nonlinear differential equations in the complex planefn(z)+P(z,f,f',...,.f(t)=P1e?1z+P2e?2z+P3e?3z,where Pj and ?i are distinct nonzero constants for j=1,2,3,|?1|>|?2|>|?3| and P(z,f,f',...,f(t)is an algebraic differential polynomial on f(z)with degree not more than n-1.We obtain an entire solutions:f(z)=a1e?1z/n,where a1 is nonzero constant such that a1n=P1.Futhermore,we study the equation which the three exponents is replaced by any finite exponents on the right hand side,that is:fn(z)+P(z,f,f',…,f(t)=P1e?1z+P2e?2z+…+Pme?mz.we get the same entire solutions:f(z)=a1e?1z/n,a1n=P1.