求高阶常系数非齐次线性微分方程特解的新方法

New Method for the Special Answer of High order Nonhomogeous Linear Differential Equation with Constant Coefficients

求高阶常系数非齐次线性微分方程:y(n)+P1y(n-1)+…+Pny=f(x)(P1,P2,…,Pn是实数)的特解的一种新方法.首先将该方程降为n个一阶非齐次线性微分方程组:其中w1,w2,…,wn是对应的齐次方程的特征方程:tn+P1tn-1+…+Pn=0的n个根.然后得出了求原方程一个特解的迭代公式.

This paper puts forward a new method for the special answer of high order nonhomogeous linear differential equation with constant coefficients： y（n）＋P1y（n-1）＋…＋Pny=f（x）（P1,P2,…,Pn∈R. First reduces it into n one order nonhomogeous linear differential equations ：{y1-w1y1=f（x） y2-w2y2=y1 ………yn-wnyn=yn-1 } w1 ,w2, …… ,wn are roots of the charcteristic equation ： tn＋P1t^n-1＋…＋Pn=0= 0; then gets a formula for one special answer of this equation.