In view of how to solve the general solution and special solution to meet some initial conditions of the second order differential equation within a class of complex plane, the solution and its derivative of the differential equation, and its asymptotic expressions in different areas of the problem, this paper proposed to use the theory of integral equation and differential operator eigenvalue and eigenfunction of gradual theoretical derivation and proved the relevant conclusions. By introducing the integral kernel method which satisfies certain conditions in the integral equation, the boundedness and continuity of the integral equation solution are proved, which provide the theoretical support for subsequent conclusion. By introducing a kind of good nature generalized integral function and by using the method of iterative approximation, the special solution of the differential equation and its asymptotic expression of its derivative in a specific area are given. According to the results, the precision of the special solution of the differential equation is improved, meanwhile, the method for further improving the asymptotic expression precision of the special solution of the differential equation is discussed.