First verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general solution of the system. 3 4-8-8-8 X'= 12 16 2 x; x₁ = e 2 - 16 - 20 -2 12 - 12 2 -16-20 -2 12 16 2 x₁ = 2 -12-12 -16-20-2 12 16 -12-12 2 3e-4t - 2e-4t 2e-4t |=X₁' C
First verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general solution of the system. 3 4-8-8-8 X'= 12 16 2 x; x₁ = e 2 - 16 - 20 -2 12 - 12 2 -16-20 -2 12 16 2 x₁ = 2 -12-12 -16-20-2 12 16 -12-12 2 3e-4t - 2e-4t 2e-4t |=X₁' C
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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![### System of Differential Equations and Verification of Solutions
#### Problem Statement
First, verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general solution of the system.
#### Differential System
\[
\mathbf{x}' =
\begin{bmatrix}
-16 & -20 & -2 \\
12 & 16 & 2 \\
-12 & -12 & 2
\end{bmatrix}
\mathbf{x}
\]
Given vectors:
\[
\mathbf{x}_1 = e^{-4t}
\begin{bmatrix}
3 \\
-2 \\
2
\end{bmatrix}, \quad
\mathbf{x}_2 = e^{2t}
\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}, \quad
\mathbf{x}_3 = e^{4t}
\begin{bmatrix}
1 \\
-1 \\
0
\end{bmatrix}
\]
### Verification of Solutions
\[
\begin{bmatrix}
-16 & -20 & -2 \\
12 & 16 & 2 \\
-12 & -12 & 2
\end{bmatrix}
\mathbf{x}_1 =
\begin{bmatrix}
3e^{-4t} \\
-2e^{-4t} \\
2e^{-4t}
\end{bmatrix} = \mathbf{x}_1'
\]
### Explanation
The problem involves verification that certain vector functions of time, \( \mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3 \), are solutions to a given system of linear differential equations. The solution involves multiplying the given matrix by each vector and ensuring the result equals the derivative of each respective vector.
### Graphical Explanation
The image includes diagrams of matrices and vectors which describe a linear differential equation system. The provided vectors are potential solutions. The verification process involves ensuring these vector results match when substituted back into the matrix differential equation.
### Task
1. Verify solutions \( \mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3 \).
2. Use the Wronskian to prove linear independence.
3. Write the general solution considering verified solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F671be518-f9e6-4df9-9304-33791fefeccb%2F44f80483-5f4e-481e-8741-19b8a0702a67%2Fm6wlzb_processed.png&w=3840&q=75)
Transcribed Image Text:### System of Differential Equations and Verification of Solutions
#### Problem Statement
First, verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general solution of the system.
#### Differential System
\[
\mathbf{x}' =
\begin{bmatrix}
-16 & -20 & -2 \\
12 & 16 & 2 \\
-12 & -12 & 2
\end{bmatrix}
\mathbf{x}
\]
Given vectors:
\[
\mathbf{x}_1 = e^{-4t}
\begin{bmatrix}
3 \\
-2 \\
2
\end{bmatrix}, \quad
\mathbf{x}_2 = e^{2t}
\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}, \quad
\mathbf{x}_3 = e^{4t}
\begin{bmatrix}
1 \\
-1 \\
0
\end{bmatrix}
\]
### Verification of Solutions
\[
\begin{bmatrix}
-16 & -20 & -2 \\
12 & 16 & 2 \\
-12 & -12 & 2
\end{bmatrix}
\mathbf{x}_1 =
\begin{bmatrix}
3e^{-4t} \\
-2e^{-4t} \\
2e^{-4t}
\end{bmatrix} = \mathbf{x}_1'
\]
### Explanation
The problem involves verification that certain vector functions of time, \( \mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3 \), are solutions to a given system of linear differential equations. The solution involves multiplying the given matrix by each vector and ensuring the result equals the derivative of each respective vector.
### Graphical Explanation
The image includes diagrams of matrices and vectors which describe a linear differential equation system. The provided vectors are potential solutions. The verification process involves ensuring these vector results match when substituted back into the matrix differential equation.
### Task
1. Verify solutions \( \mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3 \).
2. Use the Wronskian to prove linear independence.
3. Write the general solution considering verified solutions.
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