Question

Let Abe an$\mathit{n}\mathbf{\times }\mathit{n}$matrix andK a scalar. Consider the following two systems:

$$\begin{gathered} \frac{\mathbf{d}\overrightarrow{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{A}\overrightarrow{\mathbf{x}}\left(I\right) \\ \frac{\mathbf{d}\overrightarrow{\mathbf{c}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{k}\mathit{A}\overrightarrow{\mathbf{c}}\left(II\right) \end{gathered}$$

Show that if$\overrightarrow{\mathbf{x}}\left(t\right)$is a solution of the system$\left(I\right)$then$\overrightarrow{\mathbf{c}}\left(t\right)\mathbf{=}\overrightarrow{\mathbf{x}}\left(kt\right)$is a solution of the system$\left(II\right)$. Compare the vector field of the two system.

Short answer

Yes, $\overrightarrow{c}\left(t\right)=\overrightarrow{x}\left(kt\right)$is a solution of the system$\frac{d\overrightarrow{c}}{dt}=kA\overrightarrow{c}$

Step-by-step solution

Determine the equation of the solution

Consider $\overrightarrow{x}\left(t\right)$and$\overrightarrow{c}\left(t\right)$is a solution of the system $\frac{d\overrightarrow{x}}{dt}=A\overrightarrow{x}$and$\frac{d\overrightarrow{c}}{dt}=kA\overrightarrow{c}$respectively.

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

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

Show that c→t=ektx→t is a solution of the system 

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

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

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

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

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

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

Hence, if $\overrightarrow{x}\left(t\right)$and$\overrightarrow{c}\left(t\right)$is a solution of the system$\frac{d\overrightarrow{x}}{dt}=A\overrightarrow{x}$and $\frac{d\overrightarrow{c}}{dt}=kA\overrightarrow{c}$respectively then$\overrightarrow{c}\left(t\right)=\overrightarrow{x}\left(kt\right)$ is a solution