二元一阶常系数线性微分方程组的新解法

A New Solution for the Bivariate First Order Linear Differential Equations with Constant Coefficients

对于二元一阶常系数线性微分方程组:x′=Ax+f(t),引入特征根方程|A-λE|=0的特征行向量K=(k_1,k_2)(其中K满足:K^T(A-λE)=0)概念,将二元一阶常系数线性微分方程组,化为二元一次代数线性方程:k_1x_1+k_2x_2=C_1e^(λt)+e^(λt)∫(k_1f_1+k_2f_2)e^(-λt)dt,并结合代数线性方程和一阶线性微分方程的理论,给出原微分方程组的解.

By employing the eigenvector K =（k1,k2） which satisfies the equations KT（A-λE）=0 of the characteristic equations ｜A-λE｜ = 0,the bivariate first order linear differential equations with constant coefficients can be transformed into the bivariate linear algebraic equations k1x1＋ k2x2= C1eλt＋ eλt∫（k1f1＋ k2f2）e-λtdt. Then combining the theories of the linear algebraic equations and the first order linear differential equations,the solutions of the original differential equations are given.