91Ó°ÊÓ

Consider$\stackrel{→}{x}\left(t\right)$and$\stackrel{→}{c}\left(t\right)$is a solution of the system$\frac{d\stackrel{→}{x}}{dt}=A\stackrel{→}{x}$and$\frac{\mathrm{d}\stackrel{→}{\mathrm{c}}}{\mathrm{dt}}=\left(\mathrm{A}+{\mathrm{kl}}_{\mathrm{n}}\right)\stackrel{→}{\mathrm{c}}$respectively.

Assume${\mathrm{λ}}_1,{\mathrm{λ}}_2,{\mathrm{λ}}_3,…,{\mathrm{λ}}_n$be the Eigen values of the matrix A then there exist Eigen vectors${\stackrel{→}{v}}_1,{\stackrel{→}{v}}_2.{\stackrel{→}{v}}_3,…,{\stackrel{→}{v}}_n$such that ${\stackrel{→}{x}}_1\left(t\right)=c_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+c_n{e}^{{\mathrm{λ}}_{n}t}{\stackrel{→}{v}}_n$and$\stackrel{→}{c}\left(t\right)=c_1{e}^{\left({\mathrm{λ}}_1+k\right)}{\stackrel{→}{v}}_1+c_2{e}^{\left({\mathrm{λ}}_2+k\right)}{\stackrel{→}{v}}_2+...+c_n{e}^{\left({\mathrm{λ}}_n+k\right)}{\stackrel{→}{v}}_n$where$c_1,c_2,…,c_n$is constant.

If x(t) is the solution of the linear system localid="1659702528208" $\frac{d\stackrel{→}{x}}{dt}=A\stackrel{→}{x}$then${\stackrel{→}{x}}_1\left(t\right)=c_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+c_n{e}^{{\mathrm{λ}}_{n}t}{\stackrel{→}{v}}_n$where${\mathrm{λ}}_1,{\mathrm{λ}}_2,{\mathrm{λ}}_3,…,{\mathrm{λ}}_n$be the Eigen values and of$n×n$matrix A.

Simplify the equation${\stackrel{→}{x}}_1\left(t\right)=c_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+c_n{e}^{{\mathrm{λ}}_{n}t}{\stackrel{→}{v}}_n$as follows.

$$\begin{gathered} \stackrel{→}{c}\left(t\right)=c_1{e}^{\left({\mathrm{λ}}_1+k\right)}{\stackrel{→}{v}}_1+c_2{e}^{\left({\mathrm{λ}}_2+k\right)}{\stackrel{→}{v}}_2+...+c_n{e}^{\left({\mathrm{λ}}_n+k\right)}{\stackrel{→}{v}}_n \\ \stackrel{→}{c}\left(t\right)=c_1{e}^{{\mathrm{λ}}_{1}t}{e}^{kt}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_{2}t}{e}^{kt}{\stackrel{→}{v}}_2+...+c_n{e}^{{\mathrm{λ}}_{n}t}{e}^{kt}{\stackrel{→}{v}}_n \\ \stackrel{→}{c}\left(t\right)=e^{kt}\left\{c_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+c_n{e}^{{\mathrm{λ}}_{n}t}{\stackrel{→}{v}}_n\right\} \\ \stackrel{→}{c}\left(t\right)=e^{kt}\stackrel{→}{x}\left(t\right) \end{gathered}$$

By the definition of solution of the linear system,$\stackrel{→}{c}\left(t\right)=e^{kt}\stackrel{→}{x}\left(t\right)$is a solution.

Hence, if $\stackrel{→}{x}\left(t\right)$and$\stackrel{→}{c}\left(t\right)$is a solution of the system$\frac{d\stackrel{→}{x}}{dt}=A\stackrel{→}{x}$and$\frac{\mathrm{d}\stackrel{→}{\mathrm{c}}}{\mathrm{dt}}=\left(\mathrm{A}+{\mathrm{kl}}_{\mathrm{n}}\right)\stackrel{→}{\mathrm{c}}$respectively then is a solution.