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6. x' = 2호 1 2 X

6. x' = 2호 1 2 X

Advanced Engineering Mathematics
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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### General Solutions and Phase Portraits In each of Problems 1 through 6, find the general solution of the given system of equations. Also, draw a direction field and a phase portrait. Describe how the solutions behave as \( t \rightarrow \infty \).
Transcribed Image Text:### General Solutions and Phase Portraits In each of Problems 1 through 6, find the general solution of the given system of equations. Also, draw a direction field and a phase portrait. Describe how the solutions behave as \( t \rightarrow \infty \).
The image contains a mathematical expression that appears to be a system of linear equations written in matrix form. Here is the transcription of the image: --- **6.** \(\mathbf{x}' = \begin{pmatrix} 2 & \frac{1}{2} \\ -\frac{1}{2} & 1 \end{pmatrix} \mathbf{x}\) --- Explanation: - \(\mathbf{x}'\) and \(\mathbf{x}\) are vectors, indicating that this is a transformation of the vector \(\mathbf{x}\) to the vector \(\mathbf{x}'\). - The matrix \(\begin{pmatrix} 2 & \frac{1}{2} \\ -\frac{1}{2} & 1 \end{pmatrix}\) is a 2x2 matrix that represents the linear transformation applied to the vector \(\mathbf{x}\). In this transformation: - The first element of \(\mathbf{x}'\) is calculated as \(2x_1 + \frac{1}{2}x_2\), where \(x_1\) and \(x_2\) are the components of the vector \(\mathbf{x}\). - The second element of \(\mathbf{x}'\) is calculated as \(-\frac{1}{2}x_1 + x_2\). This type of matrix equation is common in linear algebra and is used to describe various linear transformations such as rotations, scaling, and shearing in vector space.
Transcribed Image Text:The image contains a mathematical expression that appears to be a system of linear equations written in matrix form. Here is the transcription of the image: --- **6.** \(\mathbf{x}' = \begin{pmatrix} 2 & \frac{1}{2} \\ -\frac{1}{2} & 1 \end{pmatrix} \mathbf{x}\) --- Explanation: - \(\mathbf{x}'\) and \(\mathbf{x}\) are vectors, indicating that this is a transformation of the vector \(\mathbf{x}\) to the vector \(\mathbf{x}'\). - The matrix \(\begin{pmatrix} 2 & \frac{1}{2} \\ -\frac{1}{2} & 1 \end{pmatrix}\) is a 2x2 matrix that represents the linear transformation applied to the vector \(\mathbf{x}\). In this transformation: - The first element of \(\mathbf{x}'\) is calculated as \(2x_1 + \frac{1}{2}x_2\), where \(x_1\) and \(x_2\) are the components of the vector \(\mathbf{x}\). - The second element of \(\mathbf{x}'\) is calculated as \(-\frac{1}{2}x_1 + x_2\). This type of matrix equation is common in linear algebra and is used to describe various linear transformations such as rotations, scaling, and shearing in vector space.
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