Solve the given initial value problem. x'(t) = x(t) = 12 - 3 5 4 x(t), x(0)= 1 - 5
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
![The problem requires solving an initial value problem for a system of differential equations. The system is given in matrix form:
\[
\mathbf{x}'(t) =
\begin{bmatrix}
12 & -3 \\
5 & 4
\end{bmatrix}
\mathbf{x}(t),
\quad \mathbf{x}(0) =
\begin{bmatrix}
-1 \\
-5
\end{bmatrix}
\]
**Explanation of Components:**
1. **Differential Equation:**
- \(\mathbf{x}'(t)\) represents the derivative of a vector function \( \mathbf{x}(t) \) with respect to time \( t \).
- The matrix
\[
\begin{bmatrix}
12 & -3 \\
5 & 4
\end{bmatrix}
\]
is the coefficient matrix that determines how the components of \( \mathbf{x}(t) \) interact with each other.
2. **Initial Condition:**
- \(\mathbf{x}(0) =
\begin{bmatrix}
-1 \\
-5
\end{bmatrix}\)
specifies the initial values of the vector function at \( t = 0 \).
3. **Solution Box:**
- \( \mathbf{x}(t) = \) where the solution to the differential equation will be entered. This solution will describe how \( \mathbf{x}(t) \) evolves over time given the initial condition.
The objective is to find \(\mathbf{x}(t)\), which satisfies both the differential equation and the initial condition. This involves finding the eigenvalues and eigenvectors of the coefficient matrix or using methods such as matrix exponentiation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F047a7e6a-f025-4b5b-ab83-4ffe14f69253%2F98a55a8e-9b6f-490a-96f1-f7232b60861b%2Fzt6ktkh_processed.png&w=3840&q=75)
Transcribed Image Text:The problem requires solving an initial value problem for a system of differential equations. The system is given in matrix form:
\[
\mathbf{x}'(t) =
\begin{bmatrix}
12 & -3 \\
5 & 4
\end{bmatrix}
\mathbf{x}(t),
\quad \mathbf{x}(0) =
\begin{bmatrix}
-1 \\
-5
\end{bmatrix}
\]
**Explanation of Components:**
1. **Differential Equation:**
- \(\mathbf{x}'(t)\) represents the derivative of a vector function \( \mathbf{x}(t) \) with respect to time \( t \).
- The matrix
\[
\begin{bmatrix}
12 & -3 \\
5 & 4
\end{bmatrix}
\]
is the coefficient matrix that determines how the components of \( \mathbf{x}(t) \) interact with each other.
2. **Initial Condition:**
- \(\mathbf{x}(0) =
\begin{bmatrix}
-1 \\
-5
\end{bmatrix}\)
specifies the initial values of the vector function at \( t = 0 \).
3. **Solution Box:**
- \( \mathbf{x}(t) = \) where the solution to the differential equation will be entered. This solution will describe how \( \mathbf{x}(t) \) evolves over time given the initial condition.
The objective is to find \(\mathbf{x}(t)\), which satisfies both the differential equation and the initial condition. This involves finding the eigenvalues and eigenvectors of the coefficient matrix or using methods such as matrix exponentiation.
Expert Solution
Step 1: Find eigenvalues
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