5. Solve the system by using the matrix exponential 0 1 0 *-[2] + [2]· --[] x' = x x(1) = -3t 3
5. Solve the system by using the matrix exponential 0 1 0 *-[2] + [2]· --[] x' = x x(1) = -3t 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
![**Problem 5: Solving the System Using the Matrix Exponential**
Given the system of differential equations:
\[
x' = \begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} x + \begin{bmatrix} 0 \\ e^{-3t} \end{bmatrix},
\]
with the initial condition:
\[ x(1) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \]
we are to solve this system using the matrix exponential method.
### Steps and Explanation:
1. **Matrix Representation:**
- The system is represented in matrix form where \( x' \) is the derivative of vector \( x \).
- Matrix \( \begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} \) represents the coefficients of \( x \).
- Vector \( \begin{bmatrix} 0 \\ e^{-3t} \end{bmatrix} \) represents the non-homogeneous part of the system.
2. **Initial Condition:**
- The system starts from \( x(1) = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \), which is the initial state at \( t = 1 \).
### Approach to Solution:
To solve using the matrix exponential method, we will:
- Compute the matrix exponential of \( \begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} \).
- Use the initial condition to determine the constants or specific solution.
- Incorporate the non-homogeneous term \( \begin{bmatrix} 0 \\ e^{-3t} \end{bmatrix} \).
This detailed approach involves linear algebra and differential equation techniques which are typically covered in higher-level mathematics courses.
By understanding and computing these components, you will be able to solve the system and understand the behavior of the solution over time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f24dc5d-a43a-4abd-9b6e-c5e020f186b5%2F951fb060-93b6-4be0-9a43-7116b097a90f%2Fie06j09_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 5: Solving the System Using the Matrix Exponential**
Given the system of differential equations:
\[
x' = \begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} x + \begin{bmatrix} 0 \\ e^{-3t} \end{bmatrix},
\]
with the initial condition:
\[ x(1) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \]
we are to solve this system using the matrix exponential method.
### Steps and Explanation:
1. **Matrix Representation:**
- The system is represented in matrix form where \( x' \) is the derivative of vector \( x \).
- Matrix \( \begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} \) represents the coefficients of \( x \).
- Vector \( \begin{bmatrix} 0 \\ e^{-3t} \end{bmatrix} \) represents the non-homogeneous part of the system.
2. **Initial Condition:**
- The system starts from \( x(1) = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \), which is the initial state at \( t = 1 \).
### Approach to Solution:
To solve using the matrix exponential method, we will:
- Compute the matrix exponential of \( \begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} \).
- Use the initial condition to determine the constants or specific solution.
- Incorporate the non-homogeneous term \( \begin{bmatrix} 0 \\ e^{-3t} \end{bmatrix} \).
This detailed approach involves linear algebra and differential equation techniques which are typically covered in higher-level mathematics courses.
By understanding and computing these components, you will be able to solve the system and understand the behavior of the solution over time.
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