Q52 E Find all the solution of the sys... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 9: Q52 E (page 428)

Find all the solution of the system $\frac{d \overset{鈫�}{x}}{dt} = [\begin{matrix} 位 & 1 \\ 0 & 位 \end{matrix}] \overset{鈫�}{x}$ where is arbitrary constant.

Short Answer

Expert verified

The solution is $\overset{鈫�}{x} (t) = c_{1} e^{位 t} [\begin{matrix} 1 \\ 0 \end{matrix}] + c_{2} e^{位 t} (t [\begin{matrix} 1 \\ 0 \end{matrix}] + [\begin{matrix} 1 \\ 1 \end{matrix}])$ .

Step by step solution

Explanation of the solution

Consider the system as follows.

$\frac{d \overset{鈫�}{x}}{dt} = [\begin{matrix} 位 & 1 \\ 0 & 位 \end{matrix}] \overset{鈫�}{x} \frac{d \overset{鈫�}{x}}{dt} = A \overset{鈫�}{x}$

To find the eigenvalues of A as follows.

$d e t (A - 渭 l) = 0 |\begin{matrix} 位 - 渭 & 1 \\ 0 & 位 - 渭 \end{matrix}| = 0 (位 - 渭) (位 - 渭) - 0 = 0 {(位 - 渭)}^{2} = 0$

Simplify further as follows.

$位 - 渭 = 0 位 = 渭$

Therefore, the eigenvalues are $渭_{1} = 位$ and $渭_{2} = 位$ .

Now, to find the corresponding eigenvector as follows.

$A \overset{鈫�}{v} = 位 \overset{鈫�}{v} [\begin{matrix} 位 & 1 \\ 0 & 位 \end{matrix}] [\begin{matrix} v_{1} \\ v_{2} \end{matrix}] = 位 [\begin{matrix} v_{1} \\ v_{2} \end{matrix}] [\begin{matrix} {位惫}_{1} + v_{2} \\ {位惫}_{2} \end{matrix}] = [\begin{matrix} {位惫}_{1} \\ {位惫}_{2} \end{matrix}] v_{2} = 0, v_{1} = arbitrary$

Simplify further as follows.

$\overset{鈫�}{v} = [\begin{matrix} 1 \\ 0 \end{matrix}]$

Since, the system has only on eigenvector, and find generalized vector which is a solution of a system $(A - 位濒) \overset{鈫�}{u} = \overset{鈫�}{v}$ as follows.

$\overset{鈫�}{x} (t) = c_{1} e^{位 t} [\begin{matrix} 1 \\ 0 \end{matrix}] + c_{2} e^{位 t} (t [\begin{matrix} 1 \\ 0 \end{matrix}] + [\begin{matrix} 1 \\ 1 \end{matrix}])$

Phase portrait of the solution

The phase portrait for $位 < 0$ is given in figure 1 as follows.

The phase portrait for $位 < 0$ is given in figure 2 as follows.

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91影视

Find all the solution of the system $\frac{d \overset{鈫�}{x}}{dt} = [\begin{matrix} 位 & 1 \\ 0 & 位 \end{matrix}] \overset{鈫�}{x}$ where is arbitrary constant.

Short Answer

Step by step solution

Explanation of the solution

Phase portrait of the solution

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Find all the solution of the system $\frac{d \overset{鈫�}{x}}{dt} = [\begin{matrix} 位 & 1 \\ 0 & 位 \end{matrix}] \overset{鈫�}{x}$ where is arbitrary constant.