Consider the system of equations. Edit 1 1 3. 3 3. 1 1 1 X. 1 Find a fundamental matrix for the given system of equations. -2t -e 2t 5t V (t) = %3D -2 e2t est -2t e e2t est Find the fundamental matrix $(t) satisfying Þ(0) = I. %3D 1 (t) = 0. 0.
Consider the system of equations. Edit 1 1 3. 3 3. 1 1 1 X. 1 Find a fundamental matrix for the given system of equations. -2t -e 2t 5t V (t) = %3D -2 e2t est -2t e e2t est Find the fundamental matrix $(t) satisfying Þ(0) = I. %3D 1 (t) = 0. 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
![**Consider the system of equations:**
\[
\mathbf{x}' = \begin{pmatrix} 1 & 1 & 3 \\ 1 & 3 & 1 \\ 3 & 1 & 1 \end{pmatrix} \mathbf{x}.
\]
**Find a fundamental matrix for the given system of equations:**
\[
\Psi(t) = \begin{pmatrix}
-e^{-2t} & e^{2t} & e^{5t} \\
0 & -2e^{2t} & e^{5t} \\
e^{-2t} & e^{2t} & e^{5t}
\end{pmatrix}
\]
*Note: The matrix \(\Psi(t)\) has a green check mark indicating it as correct.*
**Find the fundamental matrix \(\Phi(t)\) satisfying \(\Phi(0) = I\):**
\[
\Phi(t) = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\]
*Note: The matrix \(\Phi(t)\) has a red cross mark indicating it is incorrect.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39f57797-3474-4d17-b3c6-523e014d17d6%2Fbc13ca63-1700-41ee-844f-3c9af4fd5298%2Fi34brms_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Consider the system of equations:**
\[
\mathbf{x}' = \begin{pmatrix} 1 & 1 & 3 \\ 1 & 3 & 1 \\ 3 & 1 & 1 \end{pmatrix} \mathbf{x}.
\]
**Find a fundamental matrix for the given system of equations:**
\[
\Psi(t) = \begin{pmatrix}
-e^{-2t} & e^{2t} & e^{5t} \\
0 & -2e^{2t} & e^{5t} \\
e^{-2t} & e^{2t} & e^{5t}
\end{pmatrix}
\]
*Note: The matrix \(\Psi(t)\) has a green check mark indicating it as correct.*
**Find the fundamental matrix \(\Phi(t)\) satisfying \(\Phi(0) = I\):**
\[
\Phi(t) = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\]
*Note: The matrix \(\Phi(t)\) has a red cross mark indicating it is incorrect.*
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