The Growth And Singular Direct Ions Of Solutions Of Some Types Of Linear Differential Equations

In this thesis,we investigate the complex oscillation properties of the solutions of some types of linear differential equation by applying the theories and methods of the complex analysis.It contains the following four chapters.First chapter,we mainly reviews the status of research on the theory of differential equation of complex oscillation,as well as the research background of this article.we also describes relevant notation definitions and the relevant prior knowledge.Second chapter,we investigate the growth and Borel directions of solutions in angular domains of the differential equation f(k)+Ak1f(k-1)+…+A1f’+A0f=0and corresponding non-homogeneous differential equation with entire coefficients.Under some conditions,we proved that every solution f(?)0of the equation is of infinite order in any angular domain which has σ order Borel direction of A0,and the infinite order Borel direction of the solution is unanimous with the σ order Borel direction of A0,the condition is the same with non-homogeneous except for an extra solution.Third chapter,we prove that the radial oscillation of infinite order solutions of a class of second order differential equation f+Af=0with meromorphic coefficient.When the coefficient is a transcendental meromorphic function which has finite number poles andσ2(A)<∞or λ2(1/A)<σ2(A)=σ (<∞) and infinite deficient value,we can be obtained an equivalent relation between the zero point of convergence index and the Borel direction.Fourth chapter,we study the growth of solutions of a class of second order linear differential equation f+Qf’+hepf=0with meromorphic coefficients.When Q(z) has a finite deficient value or Borel exceptional value and finitely many Borel directions, P(z) is a non-constant polynomial and h(z) is meromorphic function with σ(h)<n,we prove that every solution/f(?)0of the equation has infinite order.