(2 -5 3. x' = -2,
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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differential equation probelm. Please tell me how to draw a direction field and a phase portrait. I know for this question the phase portrait should looks like circles, I am not clearly how to determine the dimention and etc of this phase portrait.
![### Transcription for Educational Website
#### Problem 3:
\[
\mathbf{x'} = \begin{pmatrix}
2 & -5 \\
1 & -2
\end{pmatrix} \mathbf{x}
\]
In this problem, the equation represents a transformation of the vector \(\mathbf{x}\) using a 2x2 matrix. The matrix:
\[
\begin{pmatrix}
2 & -5 \\
1 & -2
\end{pmatrix}
\]
is applied to the vector \(\mathbf{x}\) to produce a new vector \(\mathbf{x'}\). This is an example of a linear transformation in two-dimensional space.
#### Explanation:
- **Matrix Elements**: The matrix consists of two rows and two columns. The elements are as follows:
- First row: 2, -5
- Second row: 1, -2
- **Transformation Process**: The transformation involves multiplying each row of the matrix by the vector \(\mathbf{x}\). This results in a new vector, \(\mathbf{x'}\).
This linear transformation can represent various operations such as scaling, rotation, or shearing depending on the matrix values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf7ac4f3-c9e5-426e-a855-51d402583d0c%2Ff6edfa35-7a66-44ba-923a-667293ca826e%2Fcs0mwm_processed.png&w=3840&q=75)
Transcribed Image Text:### Transcription for Educational Website
#### Problem 3:
\[
\mathbf{x'} = \begin{pmatrix}
2 & -5 \\
1 & -2
\end{pmatrix} \mathbf{x}
\]
In this problem, the equation represents a transformation of the vector \(\mathbf{x}\) using a 2x2 matrix. The matrix:
\[
\begin{pmatrix}
2 & -5 \\
1 & -2
\end{pmatrix}
\]
is applied to the vector \(\mathbf{x}\) to produce a new vector \(\mathbf{x'}\). This is an example of a linear transformation in two-dimensional space.
#### Explanation:
- **Matrix Elements**: The matrix consists of two rows and two columns. The elements are as follows:
- First row: 2, -5
- Second row: 1, -2
- **Transformation Process**: The transformation involves multiplying each row of the matrix by the vector \(\mathbf{x}\). This results in a new vector, \(\mathbf{x'}\).
This linear transformation can represent various operations such as scaling, rotation, or shearing depending on the matrix values.
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