Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Transcribed Image Text:### General Solutions of Systems
In each of Problems 1 through 6, express the general solution of the given system of equations in terms of real-valued functions. Also, draw a direction field and a phase portrait. Describe the behavior of the solutions as \( t \to \infty \).
![### Linear Transformation Example 3
Consider the system of differential equations represented as:
\[ \mathbf{x'} = \begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix} \mathbf{x} \]
where:
- \( \mathbf{x} \) is a vector with components \( x_1 \) and \( x_2 \).
- \( \mathbf{x'} \) represents the derivative of the vector \( \mathbf{x} \) with respect to time.
In this system, the matrix
\[ \begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix} \]
defines a linear transformation that influences the behavior of the vector \( \mathbf{x} \) over time.
#### Explanation:
- The matrix is a 2x2 matrix containing the coefficients that determine how each component of \( \mathbf{x} \) contributes to each component of \( \mathbf{x'} \).
- In this example, the system shows that the derivative \( x_1' \) (the rate of change of \( x_1 \)) is influenced by both \( x_1 \) and \( x_2 \) via the coefficients 2 and -5 respectively. That is: \( x_1' = 2x_1 - 5x_2 \).
- Similarly, \( x_2' \) (the rate of change of \( x_2 \)) is influenced by both \( x_1 \) and \( x_2 \) via the coefficients 1 and -2 respectively. That is: \( x_2' = x_1 - 2x_2 \).
#### Applications:
Understanding such linear transformations is crucial in fields such as:
- Systems of linear differential equations
- Physics for modeling dynamic systems
- Engineering for control systems analysis
Learning how to solve these systems involves determining the eigenvalues and eigenvectors of the transformation matrix, which gives insights into the stability and nature of the system's solutions.](https://content.bartleby.com/qna-images/question/3632b9ca-0e22-48c8-943d-c9e08fc0f04c/2c9dd3ab-05cc-4429-84f6-e2d7e543d72d/itn40d_processed.png)
Transcribed Image Text:### Linear Transformation Example 3
Consider the system of differential equations represented as:
\[ \mathbf{x'} = \begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix} \mathbf{x} \]
where:
- \( \mathbf{x} \) is a vector with components \( x_1 \) and \( x_2 \).
- \( \mathbf{x'} \) represents the derivative of the vector \( \mathbf{x} \) with respect to time.
In this system, the matrix
\[ \begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix} \]
defines a linear transformation that influences the behavior of the vector \( \mathbf{x} \) over time.
#### Explanation:
- The matrix is a 2x2 matrix containing the coefficients that determine how each component of \( \mathbf{x} \) contributes to each component of \( \mathbf{x'} \).
- In this example, the system shows that the derivative \( x_1' \) (the rate of change of \( x_1 \)) is influenced by both \( x_1 \) and \( x_2 \) via the coefficients 2 and -5 respectively. That is: \( x_1' = 2x_1 - 5x_2 \).
- Similarly, \( x_2' \) (the rate of change of \( x_2 \)) is influenced by both \( x_1 \) and \( x_2 \) via the coefficients 1 and -2 respectively. That is: \( x_2' = x_1 - 2x_2 \).
#### Applications:
Understanding such linear transformations is crucial in fields such as:
- Systems of linear differential equations
- Physics for modeling dynamic systems
- Engineering for control systems analysis
Learning how to solve these systems involves determining the eigenvalues and eigenvectors of the transformation matrix, which gives insights into the stability and nature of the system's solutions.
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