Use the method of variation of parameters to solve the initial value problem x'=Ax+ f(t), x(a)=x₂ using the following values. -[3] - e A = x(t) = 2 -1 4 -2 f(t) 8 x(0) = 1 + 2t -t 4t 1-2t
Use the method of variation of parameters to solve the initial value problem x'=Ax+ f(t), x(a)=x₂ using the following values. -[3] - e A = x(t) = 2 -1 4 -2 f(t) 8 x(0) = 1 + 2t -t 4t 1-2t
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 the Initial Value Problem Using the Method of Variation of Parameters**
To solve the initial value problem \( x' = Ax + f(t),\ x(a) = x_a \) using the given values, follow the steps below:
1. **Matrix \( A \):**
\[
A = \begin{bmatrix}
2 & -1 \\
4 & -2
\end{bmatrix}
\]
2. **Function \( f(t) \):**
\[
f(t) = \begin{bmatrix}
8 \\
2
\end{bmatrix}
\]
3. **Initial Condition \( x(0) \):**
\[
x(0) = \begin{bmatrix}
8 \\
3
\end{bmatrix}
\]
4. **Matrix Exponential \( e^{At} \):**
\[
e^{At} = \begin{bmatrix}
1 + 2t & -t \\
4t & 1 - 2t
\end{bmatrix}
\]
To find the solution \( x(t) \), we use the variation of parameters method. The general solution is given by:
\[ x(t) = e^{At} x(0) + e^{At} \int_0^t e^{-A\tau} f(\tau)\,d\tau \]
**Explanation of the Graph/Diagram (if present):**
There is no graph or diagram accompanying the given problem. The focus is on the matrices and the methods required to solve the differential equation.
To complete the solution, perform the following steps:
- Compute \( e^{At} \int_0^t e^{-A\tau} f(\tau)\,d\tau \).
- Add this result to \( e^{At} x(0) \) to get the final solution \( x(t) \).
Finally, the solution \( x(t) \) is:
\[ x(t) = \boxed{\ \ } \]
Insert the detailed calculation steps and final expression for \( x(t) \) within the box provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F896deee6-4ebc-4afd-8502-502eb7aa6712%2F0a2d9cab-742e-471c-95ec-fc292bf5bcdd%2Faqmyjz5_processed.png&w=3840&q=75)
Transcribed Image Text:**Solving the Initial Value Problem Using the Method of Variation of Parameters**
To solve the initial value problem \( x' = Ax + f(t),\ x(a) = x_a \) using the given values, follow the steps below:
1. **Matrix \( A \):**
\[
A = \begin{bmatrix}
2 & -1 \\
4 & -2
\end{bmatrix}
\]
2. **Function \( f(t) \):**
\[
f(t) = \begin{bmatrix}
8 \\
2
\end{bmatrix}
\]
3. **Initial Condition \( x(0) \):**
\[
x(0) = \begin{bmatrix}
8 \\
3
\end{bmatrix}
\]
4. **Matrix Exponential \( e^{At} \):**
\[
e^{At} = \begin{bmatrix}
1 + 2t & -t \\
4t & 1 - 2t
\end{bmatrix}
\]
To find the solution \( x(t) \), we use the variation of parameters method. The general solution is given by:
\[ x(t) = e^{At} x(0) + e^{At} \int_0^t e^{-A\tau} f(\tau)\,d\tau \]
**Explanation of the Graph/Diagram (if present):**
There is no graph or diagram accompanying the given problem. The focus is on the matrices and the methods required to solve the differential equation.
To complete the solution, perform the following steps:
- Compute \( e^{At} \int_0^t e^{-A\tau} f(\tau)\,d\tau \).
- Add this result to \( e^{At} x(0) \) to get the final solution \( x(t) \).
Finally, the solution \( x(t) \) is:
\[ x(t) = \boxed{\ \ } \]
Insert the detailed calculation steps and final expression for \( x(t) \) within the box provided.
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