Find a particular solution of the indicated linear system that satisfies the initial conditions x₁ (0) = 7, x₂ (0) = -3. *-[2-²]*×*=~[]•~•~•[] X₁ 12 -7 1 6 The particular solution is x₁ (t) = and x2₂ (t) = [
Find a particular solution of the indicated linear system that satisfies the initial conditions x₁ (0) = 7, x₂ (0) = -3. *-[2-²]*×*=~[]•~•~•[] X₁ 12 -7 1 6 The particular solution is x₁ (t) = and x2₂ (t) = [
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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![### Linear Systems and Initial Conditions
#### Problem Statement:
Find a particular solution of the indicated linear system that satisfies the initial conditions \( x_1 (0) = 7 \), \( x_2 (0) = -3 \).
#### Given System of Linear Differential Equations:
\[
x' =
\begin{bmatrix}
7 & -2 \\
12 & -7
\end{bmatrix}
x
\]
with \( x_1 = e^{5t}
\begin{bmatrix}
1 \\
1
\end{bmatrix} \)
and \( x_2 = e^{-5t}
\begin{bmatrix}
1 \\
6
\end{bmatrix} \).
#### Initial Conditions:
\[
x_1 (0) = 7, \quad x_2 (0) = -3.
\]
#### Solution:
Find the particular solution.
#### Conclusion:
The particular solution is:
\[
x_1 (t) = \textcolor{blue}{\Box} \quad \text{and} \quad x_2 (t) = \textcolor{blue}{\Box}
\]
In this problem, indicate the forms of \( x_1 (t) \) and \( x_2 (t) \) by solving the initial value problem for the given differential equations. Fill in the boxes with your answers.
---
**Explanation:**
The image contains a linear system of differential equations, given initial conditions, and two exponential solutions of the system. The goal is to find the particular solution satisfying the initial conditions provided.
1. **Initial Conditions:**
- \( x_1 (0) = 7 \)
- \( x_2 (0) = -3 \)
2. **System of Differential Equations Representation:**
\[
x' =
\begin{bmatrix}
7 & -2 \\
12 & -7
\end{bmatrix} x
\]
3. **Exponential Solutions:**
- \( x_1 = e^{5t}
\begin{bmatrix}
1 \\
1
\end{bmatrix} \)
- \( x_2 = e^{-5t}
\begin{bmatrix}
1 \\
6
\end{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbeeb7c66-0798-43fa-9a0f-eb1dc3206797%2F76dbecd6-9fe8-44ec-926c-930f01bd2f59%2Fn3pic3o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Systems and Initial Conditions
#### Problem Statement:
Find a particular solution of the indicated linear system that satisfies the initial conditions \( x_1 (0) = 7 \), \( x_2 (0) = -3 \).
#### Given System of Linear Differential Equations:
\[
x' =
\begin{bmatrix}
7 & -2 \\
12 & -7
\end{bmatrix}
x
\]
with \( x_1 = e^{5t}
\begin{bmatrix}
1 \\
1
\end{bmatrix} \)
and \( x_2 = e^{-5t}
\begin{bmatrix}
1 \\
6
\end{bmatrix} \).
#### Initial Conditions:
\[
x_1 (0) = 7, \quad x_2 (0) = -3.
\]
#### Solution:
Find the particular solution.
#### Conclusion:
The particular solution is:
\[
x_1 (t) = \textcolor{blue}{\Box} \quad \text{and} \quad x_2 (t) = \textcolor{blue}{\Box}
\]
In this problem, indicate the forms of \( x_1 (t) \) and \( x_2 (t) \) by solving the initial value problem for the given differential equations. Fill in the boxes with your answers.
---
**Explanation:**
The image contains a linear system of differential equations, given initial conditions, and two exponential solutions of the system. The goal is to find the particular solution satisfying the initial conditions provided.
1. **Initial Conditions:**
- \( x_1 (0) = 7 \)
- \( x_2 (0) = -3 \)
2. **System of Differential Equations Representation:**
\[
x' =
\begin{bmatrix}
7 & -2 \\
12 & -7
\end{bmatrix} x
\]
3. **Exponential Solutions:**
- \( x_1 = e^{5t}
\begin{bmatrix}
1 \\
1
\end{bmatrix} \)
- \( x_2 = e^{-5t}
\begin{bmatrix}
1 \\
6
\end{
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