The Solution Of The Linear Differential Equation With Mikusinski Operator Coefficient

This article mainly discusses the solution of the linear differential equation with operator coefficient in Mikusinski operator domain.For the general solution of the linear differential equation with constant operator coefficient has been studied which the second part is mainly base on to discuss a special situation namely the solution of the No-isometric linear differential equation with constant operator coefficient. The solution of the linear differential equation with general constant operator coefficient is the analytic function, whose properties are important to the study of this kind of equation. The usual method of the No-isometric is to convert the differential equation to operator equation, then use the forms of power series of the mobile operator to solve this kind of the differential equation. We can prove the power series is convergent.But solution of the differential equation with general operator coefficient has not worked out. Only about the Variable operator which have studied comprehensive. In this paper, make a further discuss on the basis of the research methods. First, from the special operator coefficient which include polynomial and the rational expression of the differential operator. The Main principle is the important properties of the rational expression of the differential operator and the convergence of the mobile operator power series. The decomposition theory of the polynomial of the differential operator and the properties of the linear differential equation with the polynomial of the differential operator have studied, which provides theoretical basis and scientific basis.The fourth chapter discusses the solution of the linear differential equation with the general operator coefficient. Through the special operator coefficient and extend to extended to the general operator coefficient. The solution of the linear differential equation with special operator mainly make use of the properties of the special operator, but the general operator don not have it, so we want to find a kind of operator coefficient which make the polynomial decompose to the convergent power series of the mobile operator. But not all the operator which can satisfy the condition is found. And not all the operator can satisfy the condition. So the linear differential equation with the general operator coefficient will have further research.