The Investigation Of The Growth Of Solutions Of Higher Order Differential Equations With Meromorphic Coefficients

In this paper,we investigated some properties of solutions of the higher order linear differential equations by using the value distribution theory and methods of Nevanlinna. The properties mainly contain the growth of solutions of the differential equations,the growth of solutions of the differential equations in angular domains,and the borel direction. It includes following four chapters.Chapter1:Preface. We introduced the related knowledge and definitions of Nevanlinna’s characteristic in complex plane, the Borel direction and Nevanlinna’s characteristic on domain as the prior knowledge of the full text.Chapter2:We mainly investigated the growth of solutions of the differential equation f(k)+Ak-1f(k1)+…+Af=0(k≥2) by using the fundamental theory and method of Nevanlinna, where Aj(1≤j≤k-1) and A(?)0are meromorphic functions. Assume that A is transcendental and A has a deficient value∞. We proved that every solution f(?)0of the equation is of infinite order under giving some different condition on Ai (1≤j≤k-1).Chapter3:We discussed the growth of solutions of the higher order differential equation f(k)+Ak-1f(k1)+…Aof=0in angular domains, where A0and Aj(1≤j≤k-1) are meromorphic functions. Assume that A0has a finite deficient value a and ρ(Aj)=0(1≤j≤k-1).Under some conditions we proved that every solution f(?)0of the equation is of infinite order in some given angular domains.Chapter4:We researched the Borel direction of solutions of a class second order linear differential equationis.We obtained the exact estimation of quantum of Borel direction of transcendental solution of the linear differential equationis.