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Q18E

Expert-verifiedFound in: Page 426

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**For the values of ${{\mathit{\lambda}}}_{{\mathbf{1}}}{\mathbf{=}}{\mathbf{-}}{\mathbf{1}}$** **and ${{\mathit{\lambda}}}_{{\mathbf{1}}}{\mathbf{=}}{\mathbf{-}}{\mathbf{2}}$** **, sketch the trajectories for all nine initial values shown in the following figures. For each of the points, trace out both future and past of the system.**

The trajectories for all nine initial values is

Consider the Eigen values ${\lambda}_{1}=-1$ and ${\lambda}_{2}=-2$.

As 0 > ${\lambda}_{1}>{\lambda}_{2}$ , the future and past of the system toward outside.

Draw the graph of ${E}_{-1}$ and ${E}_{-2}$ as follows:

**If the Eigen values of the system are ${{\mathit{\lambda}}}_{{\mathbf{1}}}$ and ${{\mathit{\lambda}}}_{{\mathbf{2}}}$**** such that ${{\mathit{\lambda}}}_{{\mathbf{1}}}{\mathbf{>}}{\mathbf{0}}{\mathbf{>}}{{\mathit{\lambda}}}_{{\mathbf{2}}}$ **** then the phase portraits of the system is**

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

Hence the trajectories for all nine initial values and traces of the future and past of the system are sketched.

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