Use the method of variation of parameters to solve the initial value problem x' = Ax + f(t), (a)=x₂ using the following values. ^ [:)]~-~-~-[*] A = x(0)= e At 2 4 1 16 4 f(t): --8-8-8 =(t) = + 2 5 2 1 + 4t -t 16t 1-4t
Use the method of variation of parameters to solve the initial value problem x' = Ax + f(t), (a)=x₂ using the following values. ^ [:)]~-~-~-[*] A = x(0)= e At 2 4 1 16 4 f(t): --8-8-8 =(t) = + 2 5 2 1 + 4t -t 16t 1-4t
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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![**Solving an Initial Value Problem with Variation of Parameters**
To solve the initial value problem \( \mathbf{x}' = A \mathbf{x} + \mathbf{f}(t) \), with the condition \(\mathbf{x}(a) = \mathbf{x}_a\), we will use the method of variation of parameters and the following values:
**Matrix \( A \):**
\[
A = \begin{bmatrix}
4 & -1 \\
16 & -4
\end{bmatrix}
\]
**Function \( \mathbf{f}(t) \):**
\[
\mathbf{f}(t) = \begin{bmatrix}
5 \\
2
\end{bmatrix}
\]
**Initial Condition \( \mathbf{x}(0) \):**
\[
\mathbf{x}(0) = \begin{bmatrix}
8 \\
2
\end{bmatrix}
\]
**Exponential Matrix \( e^{At} \):**
\[
e^{At} = \begin{bmatrix}
1 + 4t & -t \\
16t & 1 - 4t
\end{bmatrix}
\]
The solution for \( \mathbf{x}(t) \) is represented as:
**General Solution for \( \mathbf{x}(t) \):**
\[
\mathbf{x}(t) = \begin{bmatrix}
8 \\
2
\end{bmatrix}
+
\begin{bmatrix}
\blacksquare \\
\blacksquare
\end{bmatrix}
t
+
\begin{bmatrix}
\blacksquare \\
\blacksquare
\end{bmatrix}
t^2
\]
**Explanation of the Components:**
- **Matrix \( A \):** The system matrix that defines the linear part of the differential equation.
- **Function \( \mathbf{f}(t) \):** The non-homogeneous part, providing external input.
- **Initial Condition \( \mathbf{x}(0) \):** The state of the system at \( t = 0 \).
- **Exponential Matrix \( e^{At} \):** Describes the system's homogeneous solution over time.
- **General Solution \( \mathbf{x}(t) \):** Combines the homogeneous and particular solutions to meet the initial condition.
The solution involves](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3200e892-7b83-4670-8aa5-c4c84f2a6adb%2Fdfbaaeb7-149b-4a97-a066-bd6b35cf36dc%2Feibgyal_processed.png&w=3840&q=75)
Transcribed Image Text:**Solving an Initial Value Problem with Variation of Parameters**
To solve the initial value problem \( \mathbf{x}' = A \mathbf{x} + \mathbf{f}(t) \), with the condition \(\mathbf{x}(a) = \mathbf{x}_a\), we will use the method of variation of parameters and the following values:
**Matrix \( A \):**
\[
A = \begin{bmatrix}
4 & -1 \\
16 & -4
\end{bmatrix}
\]
**Function \( \mathbf{f}(t) \):**
\[
\mathbf{f}(t) = \begin{bmatrix}
5 \\
2
\end{bmatrix}
\]
**Initial Condition \( \mathbf{x}(0) \):**
\[
\mathbf{x}(0) = \begin{bmatrix}
8 \\
2
\end{bmatrix}
\]
**Exponential Matrix \( e^{At} \):**
\[
e^{At} = \begin{bmatrix}
1 + 4t & -t \\
16t & 1 - 4t
\end{bmatrix}
\]
The solution for \( \mathbf{x}(t) \) is represented as:
**General Solution for \( \mathbf{x}(t) \):**
\[
\mathbf{x}(t) = \begin{bmatrix}
8 \\
2
\end{bmatrix}
+
\begin{bmatrix}
\blacksquare \\
\blacksquare
\end{bmatrix}
t
+
\begin{bmatrix}
\blacksquare \\
\blacksquare
\end{bmatrix}
t^2
\]
**Explanation of the Components:**
- **Matrix \( A \):** The system matrix that defines the linear part of the differential equation.
- **Function \( \mathbf{f}(t) \):** The non-homogeneous part, providing external input.
- **Initial Condition \( \mathbf{x}(0) \):** The state of the system at \( t = 0 \).
- **Exponential Matrix \( e^{At} \):** Describes the system's homogeneous solution over time.
- **General Solution \( \mathbf{x}(t) \):** Combines the homogeneous and particular solutions to meet the initial condition.
The solution involves
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