一类二阶变系数线性齐次微分方程的解

The Solution of a Class of Second Order Homogeneous Linear Differential Equation with Variable Coefficients

文章利用待定函数法,把二阶变系数线性齐次微分方程降阶为一阶线性齐次微分方程,由此得出二阶变系数线性齐次微分方程y″+p(x)y′+q(x)y=0具有特解■的充要条件为■,并给出此类微分方程的一个通解形式,以及应用举例.

In this paper,the second order homogeneous linear differential equations with variable coefficients y″+p(x)y′+q(x)y=0 are reduced to the first order homogeneous linear differential equations by using the method of undetermined function.Thus,the sufficient and necessary conditionλ[p2(x)/(1-λ)2+p′(x)/1-λ]+q(x)=0 is given out for second order homogeneous linear differential equation with variable coefficients y″+p(x)y′+q(x)y=0 existing a special solution as the form y=e^λ/1-λ∫p(x)dx(λ≠0,1).Then,the general solutions of the equation are obtained.Examples are also provided for illustrating the applications.