Preliminary Exploration and Prospects of the Similar Structure of Solutions of Differential Equations

Aiming at the boundary-value problem for second-order homogeneous linear ordinary differential equations (systems) and the mixed problem for second-order homogeneous linear partial differential equations (systems), this paper reviews some results of preliminary explorations with regard to the similar structure theory of their solutions or their Laplace space solutions; analyzes the formation idea of the similar solution and its significance of research; takes apart the relation between the similar structure of it's solution and definite equation and the boundary conditions; finds a way to construct the solution of definite problems by making use of the formula of similar structure together with the similar core function; introduces the existing achievements which have been applied to permeation fluid mechanics. Finally, the Research and application prospect of the Similar Structure theory of differential equation solution is presented. It can be predicted that quite the same as the study of other properties of differential equation solution, the exploration of similar structure theory of differential equation solution will inevitably arouse great interest among Mathematics and Applied Mathematics science and Engineering technology, and this will inspire people to explore more of its deeper theoretical results and applications.