用计算机求方程Ry″+Py′+Qy=e^(λ·x)Pm(x)的理论特解

WORK OUT THE THE ORETICAL SOLUTION OF THE Ry″+Py′+Qy=e~λxP_m(x) USING COMPUTERS

二阶常系数非齐次线性微分方程Ry″+Py′+Qy=e^(λx)Pm(x)(其中R、P、Q为实常数,λ为复常数,Pm(x)是关于x的m次多项式)通常都用比较系数法求其特解。但是这种方法当m≥2时就显得繁琐,求解速度缓慢,而且这种方法难于在电子计算机上实现。本文通过有限次的使用向量函数的线性变换.给出Ry″+Py′+Qy=e^(λx)·Pm(x)的特解的一种简单、快速的公式算法,利用框图描绘出计算过程,并对这公式算法编制程序,运用电子计算机去求方程Ry″+Py′+Qy=e^(λx)·Pm(x)的特解,将由手算特解变为电算求解。

The method of comparison coefficients is often used to work out the particular solution of the nonhomogeneos second-order linear equation with constant coefficients Ry″+Py′+Qy=e^(λχ)P_m(x), where P, q, and R are real numbers,λ is a constant complex number, P_m(x) is a polynomial of degree m of x. However, when the degree of P_m(x) m≥2, this method seems to be inconvenient, the speed of which is slower and the accuracy of which is less. particularly, it is rather difficult for the method to be used in calculating with electronic computers. In this paper, using the linear transformation of vector function wi- thin finit times, a simple and rapid formulary method is proposed to find out the particular solution of the equation Ry″+Py′+Qy=e^(λχ)P_m(x). At the same time, the computative process is indicated by using the block-diag- ram, and the programmes about the method are made up, and electronic computers are used to work out the particular solution of the equation. The method makes the particular solution be worked out by computers instead of by hand and therefore,a theoretical particular solution can be worked out automatically.