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$\frac{d\stackrel{→}{c}}{dt}=kA\stackrel{→}{c}$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 $\stackrel{→}{x}\left(t\right)=c_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+c_n{e}^{{\mathrm{λ}}_{n}t}{\stackrel{→}{v}}_n$and $\stackrel{→}{c}\left(t\right)=c_1{e}^{k{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+c_2{e}^{k{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+c_n{e}^{k{\mathrm{λ}}_{n}t}{\stackrel{→}{v}}_n$where $c_1,c_2,...,c_n$is constant.

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

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

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

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

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

Substitute the value $\stackrel{→}{x}\left(kt\right)$for$c_1{e}^{{\mathrm{λ}}_1\left(kt\right)}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_2\left(kt\right)}{\stackrel{→}{v}}_2+...+c_n{e}^{{\mathrm{λ}}_n\left(kt\right)}{\stackrel{→}{v}}_n$in the equation $\stackrel{→}{c}\left(t\right)=c_1{e}^{{\mathrm{λ}}_1\left(kt\right)}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_2\left(kt\right)}{\stackrel{→}{v}}_2+...+c_n{e}^{{\mathrm{λ}}_n\left(kt\right)}{\stackrel{→}{v}}_n$as follows.

localid="1659535938530" $$\begin{gathered} \stackrel{→}{c}\left(t\right)=c_1{e}^{{\mathrm{λ}}_1\left(kt\right)}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_2\left(kt\right)}{\stackrel{→}{v}}_2+...+c_n{e}^{{\mathrm{λ}}_n\left(kt\right)}{\stackrel{→}{v}}_n \\ \stackrel{→}{c}\left(t\right)=\stackrel{→}{x}\left(kt\right) \end{gathered}$$

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 $\frac{d\stackrel{→}{c}}{dt}=kA\stackrel{→}{c}$respectively then$\stackrel{→}{c}\left(t\right)=\stackrel{→}{x}\left(kt\right)$ is a solution