5. x' -1 - = × 2 -3,
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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 \( t \rightarrow \infty \).
![Here is the transcription and explanation suitable for an educational website:
---
**Topic: Linear Algebra - Systems of Differential Equations**
**Example Problem**
**Given System:**
\[ \mathbf{x}' = \begin{pmatrix} -1 & -\frac{1}{2} \\ 2 & -3 \end{pmatrix} \mathbf{x} \]
---
**Explanation:**
In this problem, we are given a system of differential equations written in matrix form. Here, \(\mathbf{x}'\) represents the derivative of the vector \(\mathbf{x}\) with respect to time, and \(\mathbf{x}\) is a column vector.
The matrix on the right-hand side is:
\[ \begin{pmatrix} -1 & -\frac{1}{2} \\ 2 & -3 \end{pmatrix} \]
This matrix contains the coefficients that define the interactions between the variables in the vector \(\mathbf{x}\).
### Key Points:
1. **Matrix Elements**:
- The entry in the first row and first column (\(-1\)) affects the rate of change of the first variable in \(\mathbf{x}\).
- The entry in the first row and second column (\(-\frac{1}{2}\)) also impacts the first variable, but through the second variable.
- Similarly, the second row elements (\(2\) and \(-3\)) affect the second variable in \(\mathbf{x}\).
2. **Understanding the System**:
This system of differential equations can be solved to understand how the variables in \(\mathbf{x}\) evolve over time. The coefficients in the matrix determine the nature of this evolution, which could include aspects like growth, decay, oscillations, etc.
### Application:
Such systems are commonly found in various fields including physics, engineering, economics, and biological systems. By solving these, one can predict the behavior of the system under given conditions.
---
This detailed explanation provides a context and fundamental understanding for students learning about systems of differential equations in the context of linear algebra.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3632b9ca-0e22-48c8-943d-c9e08fc0f04c%2Fd35624f7-8d43-489c-8cac-b18ecb2cae4b%2Fsflw3f_processed.png&w=3840&q=75)
Transcribed Image Text:Here is the transcription and explanation suitable for an educational website:
---
**Topic: Linear Algebra - Systems of Differential Equations**
**Example Problem**
**Given System:**
\[ \mathbf{x}' = \begin{pmatrix} -1 & -\frac{1}{2} \\ 2 & -3 \end{pmatrix} \mathbf{x} \]
---
**Explanation:**
In this problem, we are given a system of differential equations written in matrix form. Here, \(\mathbf{x}'\) represents the derivative of the vector \(\mathbf{x}\) with respect to time, and \(\mathbf{x}\) is a column vector.
The matrix on the right-hand side is:
\[ \begin{pmatrix} -1 & -\frac{1}{2} \\ 2 & -3 \end{pmatrix} \]
This matrix contains the coefficients that define the interactions between the variables in the vector \(\mathbf{x}\).
### Key Points:
1. **Matrix Elements**:
- The entry in the first row and first column (\(-1\)) affects the rate of change of the first variable in \(\mathbf{x}\).
- The entry in the first row and second column (\(-\frac{1}{2}\)) also impacts the first variable, but through the second variable.
- Similarly, the second row elements (\(2\) and \(-3\)) affect the second variable in \(\mathbf{x}\).
2. **Understanding the System**:
This system of differential equations can be solved to understand how the variables in \(\mathbf{x}\) evolve over time. The coefficients in the matrix determine the nature of this evolution, which could include aspects like growth, decay, oscillations, etc.
### Application:
Such systems are commonly found in various fields including physics, engineering, economics, and biological systems. By solving these, one can predict the behavior of the system under given conditions.
---
This detailed explanation provides a context and fundamental understanding for students learning about systems of differential equations in the context of linear algebra.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
