常系数线性方程组的一种新解法

对于常系数线性微分方程组:dx/dt=Ax(A是n阶实常数矩阵)通过特征根λ和对应的特征行向量K:K~T(A-λE)=0将微分方程组化为线性方程组:1°当有n个互异的特征根λ_1,λ_2,…,λ_n,对应的线性无关的特征行向量为K_1,K_2,…,K_n,若记K_i=(k_1,k_2,…,k_n)(i=1,2,…,n),则有方程组:(n∑i=1 k_ix_i)′=λ_j(n∑i=1 k_ix_I)(j=1,2,…,n);2°当有不同的特征根λ_1,λ_2,…,λ_m其重数分别为n_1,n_2,…,n_m,n_1+n_2+…+n_m=n,对应的线性无关的特征行向量为K_i=(k_1,K_2,…,k_n)(i=1,2,…,m),则有方程组:(n∑i=1 k_rx_r)′=λ_k(n∑i=1 k_rx_r)((A-λ_jE)x_(n_i)=0;i=1),(n∑i=1 k_rx_r)′=λ_j(n∑i=1k_rx_r)+c_(n_i)e~(λ_jt)((A-λ_kE)x_(i-1)=Ex_i,i=2,…,n_i).

A New Method for Solving Linear Equations with Constant Coefficients

For linear differential equations with constant coefficients:(dx)/(dt)=Ax(A is the real constant matrix) The eigenvalue of λ and corresponding characteristic vector K:K~T(AXE)=0,the differential equations into linear equations:1° when there are n different characteristic rootsλ_1,λ_2,…,λ_n,feature vector corresponding to the linear independence is K_1,K_2,…,K_n,if K_i=(k_1,k_2,…,k_n)(i=1,2,…,n),there are equations:(n∑i=1 k_ix_i)' =λ_j(n∑i=1 k_ix_i)(j =1,2,…,n);2° degrees when the characteristic roots of λ_1,λ_2,…,λ_mdifferent,the number of n_1,n_2,…,n_m,n_1+n_2+…+n_m=n respectively,and the feature vector corresponding to the linear independence of K_i=(k_1,k_2,…,k_n)(i=1,2,…,m),there are equations:(n∑i=1 k_rx_r)' =λ_j(n∑i=1 k_rx_r)((A-λ_jE)x_(n_i)=0;i=1),(n∑i=1 k_rx_r)'=λ_j(n∑i=1 K_rx_r)+C_(n_ie~(λ_j~t)((A-λ_jE)x_i-1)=Ex_i,i=2,…,n_i).