X'= -13 -17 -2 10 14 - 10 - 10 - 13 - 17 -2 10 14 2x₁ - 3t H 2 x; x₁ = e 2 Flicid 11 3 -2, X₂ = e²¹ 2 system. -1 1 4t X3 = e 13 -17 -2 3e-3t 10 14 2 -2e-3t ==×₁' 1 0
X'= -13 -17 -2 10 14 - 10 - 10 - 13 - 17 -2 10 14 2x₁ - 3t H 2 x; x₁ = e 2 Flicid 11 3 -2, X₂ = e²¹ 2 system. -1 1 4t X3 = e 13 -17 -2 3e-3t 10 14 2 -2e-3t ==×₁' 1 0
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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Question
![**Verifying Linear Independence and General Solution of a System of Differential Equations**
**Objective:**
To verify that the given vectors are solutions of a given system of differential equations. Use the Wronskian to demonstrate their linear independence and finally, write the general solution of the system.
**Given System:**
\[ x' = \begin{bmatrix}
-13 & -17 & -2 \\
10 & 14 & 2 \\
-10 & -10 & 2
\end{bmatrix} \cdot x \]
**Provided Solutions:**
\[ x_1 = e^{3t} \begin{bmatrix}
3 \\
-2 \\
2
\end{bmatrix}, \quad x_2 = e^{2t} \begin{bmatrix}
1 \\
-1 \\
2
\end{bmatrix}, \quad x_3 = e^{4t} \begin{bmatrix}
1 \\
-1 \\
0
\end{bmatrix} \]
**Verification of Solutions:**
We will first confirm that these provided vectors are indeed solutions to the given differential system.
1. For \( x_1 \):
\[
\begin{bmatrix}
-13 & -17 & -2 \\
10 & 14 & 2 \\
-10 & -10 & 2
\end{bmatrix}
\begin{bmatrix}
3e^{3t} \\
-2e^{3t} \\
2e^{3t}
\end{bmatrix}
=
\begin{bmatrix}
3e^{3t} \\
-2e^{3t} \\
2e^{3t}
\end{bmatrix}'
\]
**Explanation of Matrices:**
- The first matrix represents the coefficients in the system of differential equations.
- The vectors \( x_1, x_2, x_3 \) represent specific solutions multiplied by exponential terms \( e^{3t}, e^{2t}, e^{4t} \) respectively.
**Illustration with matrix multiplication for \( x_1 \):**
Verify \( x_1 \):
\[
\begin{bmatrix}
-13 & -17 & -2 \\
10 & 14 & 2 \\
-10 & -10 & 2
\end{bmatrix}
\begin{bmatrix}
3e](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe52e616-7bd7-4d85-a476-7e5fcec53ee5%2F04059dbc-a7be-4bf7-868a-24a13041f93f%2Fr2jkz1e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Verifying Linear Independence and General Solution of a System of Differential Equations**
**Objective:**
To verify that the given vectors are solutions of a given system of differential equations. Use the Wronskian to demonstrate their linear independence and finally, write the general solution of the system.
**Given System:**
\[ x' = \begin{bmatrix}
-13 & -17 & -2 \\
10 & 14 & 2 \\
-10 & -10 & 2
\end{bmatrix} \cdot x \]
**Provided Solutions:**
\[ x_1 = e^{3t} \begin{bmatrix}
3 \\
-2 \\
2
\end{bmatrix}, \quad x_2 = e^{2t} \begin{bmatrix}
1 \\
-1 \\
2
\end{bmatrix}, \quad x_3 = e^{4t} \begin{bmatrix}
1 \\
-1 \\
0
\end{bmatrix} \]
**Verification of Solutions:**
We will first confirm that these provided vectors are indeed solutions to the given differential system.
1. For \( x_1 \):
\[
\begin{bmatrix}
-13 & -17 & -2 \\
10 & 14 & 2 \\
-10 & -10 & 2
\end{bmatrix}
\begin{bmatrix}
3e^{3t} \\
-2e^{3t} \\
2e^{3t}
\end{bmatrix}
=
\begin{bmatrix}
3e^{3t} \\
-2e^{3t} \\
2e^{3t}
\end{bmatrix}'
\]
**Explanation of Matrices:**
- The first matrix represents the coefficients in the system of differential equations.
- The vectors \( x_1, x_2, x_3 \) represent specific solutions multiplied by exponential terms \( e^{3t}, e^{2t}, e^{4t} \) respectively.
**Illustration with matrix multiplication for \( x_1 \):**
Verify \( x_1 \):
\[
\begin{bmatrix}
-13 & -17 & -2 \\
10 & 14 & 2 \\
-10 & -10 & 2
\end{bmatrix}
\begin{bmatrix}
3e
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