Question

Question:Consider the system

$\frac{\mathbf{d}\overrightarrow{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\left[\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right]\overrightarrow{\mathbf{x}}\mathbf{.}$

For the values of and given in Exercises 16 through 19, sketch the trajectories for all nine initial values shown in the following figure. For each of the points, trace out both the future and the past of the system.

$\mathbf{17}\mathbf{.}\mathbf{}{\mathit{\lambda }}_{\mathbf{1}}\mathbf{=}\mathbf{1}\mathbf{,}{\mathit{\lambda }}_{\mathbf{2}}\mathbf{=}\mathbf{2}$

Short answer

Answer

The trajectories for all nine initial values is

Step-by-step solution

Draw the future and past of the system.

Consider the Eigen values ${\lambda }_1=1and{\lambda }_2=2$.

As , ${\lambda }_2>{\lambda }_1>0$the future and past of the system toward outside.

Draw the graph of $E_{1}and{E}_2$as follows.

Draw the trajectory of the system.

If the Eigen values of the system are ${\lambda }_1=1and{\lambda }_2=2$such that ${\lambda }_2>{\lambda }_1>0$then the phase portraits of the system is

By the definition of trajectory, draw the graph of the trajectory of the system as follows.

Hence, the trajectories for all nine initial values and trace of future and past of the system is sketched.