求解一类二阶线性齐次微分方程边值问题的相似结构法

摘要

研究二阶线性齐次微分方程边值问题｛y″+p(x)y′+q(x)y=0,[Ey+(1+ EF)y′]x=a=D,[Gy+Hy′]x=b=0其中,D、E、F、G、H、a和b均为已知的实常数,且D≠0,G2+H2≠0,a＜b；已知函数p(x)∈C1[a,b],q(x)∈C2 [a,b]的相似结构法.首先证明方程的相似核函数是方程的特殊形式的解,然后给出求解方程的相似结构法,最后用算例表明所获相似结构法是求解二阶线性齐次微分方程边值问题的有效方法.%This paper investigates the similar structuring method of solving boundary value problem of the second-order linear hom-ogeneous differential equation as follows:rn{y" +p(x)y' + q(x)y = 0, [Ey+(l +EF)y']x=a=D, [Gy + Hy']x=b= 0,rnwhere D,E,F,G,H,a,b are known real numbers, D≠0,G2 + H2≠0,a < b; p(x) ∈ C1[a,b] and q(x) ∈C2[a,b]. First, we prove that its kernel function is the solution of a special kind of the second-order linear homogeneous differential equation. Then we give the similar structuring method of solving boundary value problem of the second-order linear homogeneous differential equation. Finally, we give an example which shows that the similar structuring method is very efficient to solve the boundary value problem of second-order linear homogeneous differential equation.