Consider the Initial Value Problem X₁ =0,0₁ dx dt = x(t) = -2 2 -2 X, x(0) = (a) Find the eigenvalues and eigenvectors for the coefficient matrix. [8] 8 2 and X₂ = ₁₂ 0 [18] (b) Find the solution to the initial value problem. Give your solution in real form. [81
Consider the Initial Value Problem X₁ =0,0₁ dx dt = x(t) = -2 2 -2 X, x(0) = (a) Find the eigenvalues and eigenvectors for the coefficient matrix. [8] 8 2 and X₂ = ₁₂ 0 [18] (b) Find the solution to the initial value problem. Give your solution in real form. [81
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Consider the Initial Value Problem**
\[
\frac{dx}{dt} = \begin{bmatrix} -2 & -2 \\ 2 & -2 \end{bmatrix} x, \quad x(0) = \begin{bmatrix} 8 \\ 2 \end{bmatrix}
\]
**(a)** Find the eigenvalues and eigenvectors for the coefficient matrix.
\[
\lambda_1 = \boxed{}, \ \vec{v}_1 = \begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix}, \quad \text{and} \quad \lambda_2 = \boxed{}, \ \vec{v}_2 = \begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix}
\]
**(b)** Find the solution to the initial value problem. Give your solution in real form.
\[
x(t) = \begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix}.
\]
**Use the phase plotter pplane9.m in MATLAB to help you describe the trajectory:**
An ellipse with clockwise orientation
**1.** Describe the trajectory.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6890296c-2fc0-4a83-b66a-ee1a85d807a1%2F1a7c68cf-e9ae-45f8-9f9b-1ea159524d59%2Flkw7gqs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Consider the Initial Value Problem**
\[
\frac{dx}{dt} = \begin{bmatrix} -2 & -2 \\ 2 & -2 \end{bmatrix} x, \quad x(0) = \begin{bmatrix} 8 \\ 2 \end{bmatrix}
\]
**(a)** Find the eigenvalues and eigenvectors for the coefficient matrix.
\[
\lambda_1 = \boxed{}, \ \vec{v}_1 = \begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix}, \quad \text{and} \quad \lambda_2 = \boxed{}, \ \vec{v}_2 = \begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix}
\]
**(b)** Find the solution to the initial value problem. Give your solution in real form.
\[
x(t) = \begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix}.
\]
**Use the phase plotter pplane9.m in MATLAB to help you describe the trajectory:**
An ellipse with clockwise orientation
**1.** Describe the trajectory.
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