91Ó°ÊÓ

Consider ${\stackrel{→}{x}}_1\left(t\right)$and ${\stackrel{→}{x}}_2\left(t\right)$is a solution of the linear system $\frac{d\stackrel{→}{x}}{dt}$where A is $n×n$matrix.

Assume ${\mathrm{λ}}_1,{\mathrm{λ}}_2,{\mathrm{λ}}_3,...,{\mathrm{λ}}_n$be the Eigen values of the A matrix 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 ${\stackrel{→}{x}}_2\left(t\right)=b_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+b_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+b_n{e}^{{\mathrm{λ}}_{n}t}{\stackrel{→}{v}}_n$where $c_1,c_2,.......,c_n$and $b_1,b_2,...,b_n$are constant.

If x(t) is the solution of the linear system $\frac{\mathbf{d}\stackrel{\mathbf{→}}{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}$then $\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}{\mathit{c}}_{\mathbf{1}}{\mathit{e}}^{{\mathbf{λ}}_{\mathbf{1}}\mathbf{,}\mathbf{t}}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{e}}^{{\mathbf{λ}}_{\mathbf{2}}\mathbf{,}\mathbf{t}}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{+}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{}\mathbf{+}{\mathit{c}}_{\mathbf{n}}{\mathit{e}}^{{\mathbf{λ}}_{\mathbf{n}}\mathbf{,}\mathbf{t}}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{n}}$where${\mathit{λ}}_{\mathit{1}}\mathit{,}{\mathit{λ}}_{\mathit{2}}\mathit{,}{\mathit{λ}}_{\mathit{3}}\mathit{,}\mathit{.}\mathit{.}\mathit{.}\mathit{,}{\mathit{λ}}_{\mathit{n}}$be the Eigen values and${\stackrel{\mathit{→}}{\mathit{v}}}_{\mathit{1}}\mathit{,}{\stackrel{\mathit{→}}{\mathit{v}}}_{\mathit{2}}\mathit{,}{\stackrel{\mathit{→}}{\mathit{v}}}_{\mathit{3}}\mathit{,}\mathit{.}\mathit{.}\mathit{.}\mathit{,}{\stackrel{\mathit{→}}{\mathit{v}}}_{\mathit{n}}$of $\mathit{n}\mathbf{×}\mathit{n}$matrix A.

Adding the equations ${\stackrel{→}{x}}_1\left(t\right)=c_1{e}^{{\mathrm{λ}}_1,t}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_2,t}{\stackrel{→}{v}}_2+...+c_n{e}^{{\mathrm{λ}}_n,t}{\stackrel{→}{v}}_n$and ${\stackrel{→}{x}}_2\left(t\right)=b_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+b_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+b_n{e}^{{\mathrm{λ}}_{n}t}{\stackrel{→}{v}}_n$as follows:

${\stackrel{→}{x}}_1\left(t\right)+{\stackrel{→}{x}}_2\left(t\right)=\left\{c_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+c_n{e}^{\mathrm{λ}nt}{\stackrel{→}{v}}_n\right\}+\left\{b_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+b_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+b_n{e}^{\mathrm{λ}nt}{\stackrel{→}{v}}_n\right\}$

Simplify the equation

${\stackrel{→}{x}}_1\left(t\right)+{\stackrel{→}{x}}_2\left(t\right)=\left\{c_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+c_n{e}^{\mathrm{λ}nt}{\stackrel{→}{v}}_n\right\}+\left\{b_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+b_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+b_n{e}^{\mathrm{λ}nt}{\stackrel{→}{v}}_n\right\}$as follows.

$$\begin{gathered} {\stackrel{→}{x}}_1\left(t\right)+{\stackrel{→}{x}}_2\left(t\right)=\left\{c_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+c_n{e}^{\mathrm{λ}nt}{\stackrel{→}{v}}_n\right\}+\left\{b_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+b_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+b_n{e}^{\mathrm{λ}nt}{\stackrel{→}{v}}_n\right\} \\ {\stackrel{→}{x}}_1\left(t\right)+{\stackrel{→}{x}}_2\left(t\right)=c_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+c_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+c_n{e}^{\mathrm{λ}nt}{\stackrel{→}{v}}_n+b_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+b_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+b_n{e}^{\mathrm{λ}nt}{\stackrel{→}{v}}_n \\ {\stackrel{→}{x}}_1\left(t\right)+{\stackrel{→}{x}}_2\left(t\right)=\left\{c_1+b_1\right\}{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+\left\{c_2+b_2\right\}{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+\left\{c_n+b_n\right\}{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_n \end{gathered}$$

Assume $c_n+b_n=d_n$, substitute the values $d_n$for $c_n+b_n$, for all in the equation ${\stackrel{→}{x}}_1\left(t\right)+{\stackrel{→}{x}}_2\left(t\right)=\left\{c_1+b_1\right\}{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+\left\{c_2+b_2\right\}{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+\left\{c_n+b_n\right\}{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_n$as follows:

$$\begin{gathered} {\stackrel{→}{x}}_1\left(t\right)+{\stackrel{→}{x}}_2\left(t\right)=\left\{c_1+b_1\right\}{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+\left\{c_2+b_2\right\}{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+\left\{c_n+b_n\right\}{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_n \\ {\stackrel{→}{x}}_1\left(t\right)+{\stackrel{→}{x}}_2\left(t\right)=d_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+d_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+d_n{e}^{{\mathrm{λ}}_{n}t}{\stackrel{→}{v}}_n \\ \stackrel{→}{x}\left(t\right)=d_1{e}^{{\mathrm{λ}}_{1}t}{\stackrel{→}{v}}_1+d_2{e}^{{\mathrm{λ}}_{2}t}{\stackrel{→}{v}}_2+...+d_n{e}^{{\mathrm{λ}}_{n}t}{\stackrel{→}{v}}_n \end{gathered}$$

By the definition of solution of the linear system, $\stackrel{→}{x}\left(t\right)=\stackrel{→}{x_1}\left(t\right)+\stackrel{→}{x_2}\left(t\right)$is a solution.

Hence, if${\stackrel{→}{x}}_1\left(t\right)$ and ${\stackrel{→}{x}}_2\left(t\right)$is a solution of the system $\frac{d\stackrel{→}{x}}{dt}A\stackrel{→}{x}$ then$\stackrel{→}{x}\left(t\right)=\stackrel{→}{x_1}\left(t\right)+\stackrel{→}{x_2}\left(t\right)$ is a solution.