Asymptotic Formula of the Second Order Differential Equation Solution in the Complex Plane

DOI：

 作者 单位 崔庆岳， 赵国瑞 广州城建职业学院 人文学院，广州 510925

针对如何求解一类复平面内满足一定初始条件下的二阶微分方程的通解和特解，以及微分方程特解及其导数在不同区域内渐近表达式的问题，提出了利用积分方程理论和微分算子中特征值和特征函数渐近理论推导并证明了相关结论；通过在积分方程中引入满足特定条件的积分核的方法证明了积分方程解的有界性和连续性，从而为后续结论的推导证明提供了理论支撑，另外通过引入一类性质很好的广义积分函数并通过迭代逼近的方法给出了微分方程特解及其导数在特定区域内的渐近表达式；根据所得结果可知，微分方程特解的渐近式的精度得以提高，同时探讨了进一步提高微分方程特解的渐近式精度的方法.

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.