-1 Find a fundamental matrix of the following system, and then apply x(t) = (t)(0) xo to find a solution satisfying the initial conditions. 5 -6 0 x= 4 -1 40 - 4 x, x(0)= 8 -2-8 2
-1 Find a fundamental matrix of the following system, and then apply x(t) = (t)(0) xo to find a solution satisfying the initial conditions. 5 -6 0 x= 4 -1 40 - 4 x, x(0)= 8 -2-8 2
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
![**Solving Systems of Differential Equations**
To find a fundamental matrix of the given system and apply \( x(t) = \Phi(t)\Phi(0)^{-1} x_0 \) to find a solution that satisfies the initial conditions, follow the steps below:
Given the system of differential equations:
\[ x' = A x, \quad A = \begin{bmatrix}
5 & -6 & 0 \\
4 & -1 & -4 \\
8 & -2 & -8
\end{bmatrix}, \quad x(0) = \begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix} \]
### Step 1: Compute the Fundamental Matrix, Φ(t)
You are provided with two options for the fundamental matrix:
**Option C:**
\[ \Phi(t) = \begin{bmatrix}
20 e^{3t} & 2e^{-7t} & 4 e^{-7t} \\
19 e^{3t} & 2 e^{3t} & 4 e^{-7t}
\end{bmatrix} \]
**Option D:**
\[ \Phi(t) = \begin{bmatrix}
2t e^{3t} & 24 e^{3t} & 20 e^{-7t} \\
19t e^{3t} & 4 e^{3t} & \ 2e^{-7t}
\end{bmatrix} \]
### Step 2: Select the Correct Fundamental Matrix
The correct fundamental matrix needs to be determined by checking against the initial conditions.
### Step 3: Apply Solution Formula
Once the correct fundamental matrix is chosen, apply \( x(t) = \Phi(t) \Phi(0)^{-1} x_0 \).
### Step 4: Find Solution Satisfying the Initial Condition
Solve for \( x(t) \) given \( \Phi(t) \), \( x_0 \), and \( \Phi(0)^{-1} \).
**Example Calculation:**
First, find \( \Phi(0) \) and then compute its inverse \( \Phi(0)^{-1} \).
Let's assume \( \Phi(0) \) is derived from \( \Phi(t) \) when \( t=0 \):
- If \( \Phi(t) \) is chosen from **Option C](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F896deee6-4ebc-4afd-8502-502eb7aa6712%2F9badcea7-5117-4961-80a1-deb9d4800ee2%2Frz1fell_processed.png&w=3840&q=75)
Transcribed Image Text:**Solving Systems of Differential Equations**
To find a fundamental matrix of the given system and apply \( x(t) = \Phi(t)\Phi(0)^{-1} x_0 \) to find a solution that satisfies the initial conditions, follow the steps below:
Given the system of differential equations:
\[ x' = A x, \quad A = \begin{bmatrix}
5 & -6 & 0 \\
4 & -1 & -4 \\
8 & -2 & -8
\end{bmatrix}, \quad x(0) = \begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix} \]
### Step 1: Compute the Fundamental Matrix, Φ(t)
You are provided with two options for the fundamental matrix:
**Option C:**
\[ \Phi(t) = \begin{bmatrix}
20 e^{3t} & 2e^{-7t} & 4 e^{-7t} \\
19 e^{3t} & 2 e^{3t} & 4 e^{-7t}
\end{bmatrix} \]
**Option D:**
\[ \Phi(t) = \begin{bmatrix}
2t e^{3t} & 24 e^{3t} & 20 e^{-7t} \\
19t e^{3t} & 4 e^{3t} & \ 2e^{-7t}
\end{bmatrix} \]
### Step 2: Select the Correct Fundamental Matrix
The correct fundamental matrix needs to be determined by checking against the initial conditions.
### Step 3: Apply Solution Formula
Once the correct fundamental matrix is chosen, apply \( x(t) = \Phi(t) \Phi(0)^{-1} x_0 \).
### Step 4: Find Solution Satisfying the Initial Condition
Solve for \( x(t) \) given \( \Phi(t) \), \( x_0 \), and \( \Phi(0)^{-1} \).
**Example Calculation:**
First, find \( \Phi(0) \) and then compute its inverse \( \Phi(0)^{-1} \).
Let's assume \( \Phi(0) \) is derived from \( \Phi(t) \) when \( t=0 \):
- If \( \Phi(t) \) is chosen from **Option C
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