一类三阶非齐次欧拉方程特解的简单求法

A Handy Method for Determining the Particular Solutions to theThird-Order Non-homogeneous Euler Equations

摘要

摘要:对形如x ³( d ³y)/( dx ³)+px ²( d ²y)/( dx ²)+qx( dy)/( dx)+ry=x ^λA cos(ω ln|x|)+B sin(ω ln|x|)的三阶非齐次欧拉方程得到了求其特解的一般公式.首先引入有关线性微分方程的两个基本性质,然后利用变量变换化为常系数线性常微分方程,再利用待定系数法和复数法得到了求解该方程特解的简便公式,并用一些算例验证了该方法的有效性和实用性.

Abstract:The general formula for determining the particular solutions is presented in this paper for a kind of third-order Euler equation likex ³( d ³y)/( dx ³)+px ²( d ²y)/( dx ²)+qx( dy)/( dx)+ry=x ^λA cos(ω ln|x|)+B sin(ω ln|x|). Two basic properties for the linear differential equation are first introduced. Then by use of the variable transformation method, the equations are transformed into the linear ordinary differential equations with constant coefficients. And then by the method of undetermined coefficient and complex number method, the simplified formula for determining the particular solution to the equations is worked out. Through some numerical examples the effectiveness and practicality of the method proposed are proven.