At Compute the matrix exponential e for the system x' = Ax given below. x₁ = 37x₁ - 36x₂, x 2 = 34x₁ - 33x₂ At IN
At Compute the matrix exponential e for the system x' = Ax given below. x₁ = 37x₁ - 36x₂, x 2 = 34x₁ - 33x₂ At IN
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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![### Matrix Exponential for the System of Differential Equations
To solve the given system of differential equations using the matrix exponential \( e^{At} \), follow the steps outlined below.
#### System of Equations
Consider the system of linear differential equations:
\[
\begin{cases}
x_1' = 37x_1 - 36x_2, \\
x_2' = 34x_1 - 33x_2.
\end{cases}
\]
#### Step-by-Step Solution
1. **Form the Coefficient Matrix \( A \)**:
The system can be rewritten in matrix form \( \mathbf{x}' = A\mathbf{x} \), where:
\[
A = \begin{pmatrix}
37 & -36 \\
34 & -33
\end{pmatrix}
\]
2. **Compute the Matrix Exponential \( e^{At} \)**:
The matrix exponential \( e^{At} \) is found by solving:
\[
e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots
\]
Here, \( I \) is the identity matrix, and \( t \) is the variable (often time).
3. **Substitute the Computed Exponential**:
After computing the series summation, substitute back into the solution format. The detailed specific calculation might require using specific mathematical software to compute effectively.
#### Final Matrix Exponential Form
The matrix exponential \( e^{At} \) is expressed in the placeholder form as follows:
\[
e^{At} = \boxed{\phantom{\text{Placeholder for Expression}}}
\]
In this format, the boxed area indicates the placeholder where the explicit exponential matrix will be placed once computed properly. This involves the exact mathematical operations that provide the solution to the given system.
---
### Understanding the Graphs and Diagrams (if any)
Since no graphs or diagrams are provided in this example, you may visualize the matrix exponential using software tools to plot trajectories or phase diagrams corresponding to the solutions \( x_1(t) \) and \( x_2(t) \) against time \( t \).
This explanation aims to provide clarity on calculating and applying matrix exponentials for solving systems of linear differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F896deee6-4ebc-4afd-8502-502eb7aa6712%2F21643137-447b-4de3-8c67-a64d06543e77%2Fbkmfrj_processed.png&w=3840&q=75)
Transcribed Image Text:### Matrix Exponential for the System of Differential Equations
To solve the given system of differential equations using the matrix exponential \( e^{At} \), follow the steps outlined below.
#### System of Equations
Consider the system of linear differential equations:
\[
\begin{cases}
x_1' = 37x_1 - 36x_2, \\
x_2' = 34x_1 - 33x_2.
\end{cases}
\]
#### Step-by-Step Solution
1. **Form the Coefficient Matrix \( A \)**:
The system can be rewritten in matrix form \( \mathbf{x}' = A\mathbf{x} \), where:
\[
A = \begin{pmatrix}
37 & -36 \\
34 & -33
\end{pmatrix}
\]
2. **Compute the Matrix Exponential \( e^{At} \)**:
The matrix exponential \( e^{At} \) is found by solving:
\[
e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots
\]
Here, \( I \) is the identity matrix, and \( t \) is the variable (often time).
3. **Substitute the Computed Exponential**:
After computing the series summation, substitute back into the solution format. The detailed specific calculation might require using specific mathematical software to compute effectively.
#### Final Matrix Exponential Form
The matrix exponential \( e^{At} \) is expressed in the placeholder form as follows:
\[
e^{At} = \boxed{\phantom{\text{Placeholder for Expression}}}
\]
In this format, the boxed area indicates the placeholder where the explicit exponential matrix will be placed once computed properly. This involves the exact mathematical operations that provide the solution to the given system.
---
### Understanding the Graphs and Diagrams (if any)
Since no graphs or diagrams are provided in this example, you may visualize the matrix exponential using software tools to plot trajectories or phase diagrams corresponding to the solutions \( x_1(t) \) and \( x_2(t) \) against time \( t \).
This explanation aims to provide clarity on calculating and applying matrix exponentials for solving systems of linear differential equations.
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