高阶常系数非齐次线性微分方程的新解法

提出了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)的一种新解法.首先将该方程降为n个一阶非齐次线性微分方程组:y1′-w1y1=f(x),y2′-w2y2=y1,…………………yn′-wnyn=yn-1,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.然后求出它的通解y=yn,最后得出了求原方程一个特解的迭代公式.

New Solution for High Order Nonhomogeous Linear Differential Equation with Constant Coefficients

This paper puts forward a new solution for high order nonhomogeous linear differential equation with constant coefficients y~((n))+P_1y~((n-1))+…+P_ny=f(x)(P_1,P_2,…,P_n∈R). First reduces it into n one, order nonhomogeous linear differential equations y_1′-w_1y_1=f(x), y_2′-w_2y_2=y_1, ………………… y_n′-w_ny_n=y_(n-1), w_1,w_2,…,w_n are roots of the characteristic equation t~n+P_1t~(n-1)+…+P_n=0. Second solves the general answer of the equation y=y_n. Last gets a formula for one special answer of this equation.