Use the method of variation of parameters to find a particular solution of the following differential equation. y" + 4y = sin 5x To use the method of variation of parameters, setup the determinant needed to calculate the Wronskian. W= w=0
Use the method of variation of parameters to find a particular solution of the following differential equation. y" + 4y = sin 5x To use the method of variation of parameters, setup the determinant needed to calculate the Wronskian. W= w=0
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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![**Finding a Particular Solution Using the Method of Variation of Parameters**
In this section, we will use the method of variation of parameters to find a particular solution to the following differential equation:
\[ y'' + 4y = \sin(5x) \]
---
To use the method of variation of parameters, we need to set up the determinant required to calculate the Wronskian (W).
\[ W = \boxed{} \]
This involves finding two linearly independent solutions to the corresponding homogeneous equation, constructing the Wronskian determinant, and using this determinant to find the particular solution for the non-homogeneous equation.
---
**Explanation of the Process:**
1. **Solve the Homogeneous Equation:**
First, we solve the corresponding homogeneous equation:
\[ y'' + 4y = 0 \]
2. **Find Linearly Independent Solutions:**
Determine two linearly independent solutions (let's denote them as \( y_1 \) and \( y_2 \)) for the homogeneous equation.
3. **Construct the Wronskian:**
The Wronskian \( W \) is a determinant created from the solutions \( y_1 \), \( y_2 \), and their derivatives:
\[
W(y_1, y_2) = \begin{vmatrix}
y_1 & y_2 \\
y_1' & y_2' \\
\end{vmatrix}
\]
4. **Calculate the Wronskian:**
Plug in the solutions and their derivatives into the determinant to find \( W \).
5. **Find the Particular Solution:**
Use the method of variation of parameters to form the particular solution \( y_p \) using the formula:
\[
y_p = y_1 \int \frac{y_2 g(x)}{W} \, dx + y_2 \int \frac{y_1 g(x)}{W} \, dx
\]
where \( g(x) = \sin(5x) \).
By following these steps, we can determine the particular solution to the given differential equation using the method of variation of parameters.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe52e616-7bd7-4d85-a476-7e5fcec53ee5%2Fb1d3351b-834e-45a5-a865-17b462a8191a%2Fqza2pa4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding a Particular Solution Using the Method of Variation of Parameters**
In this section, we will use the method of variation of parameters to find a particular solution to the following differential equation:
\[ y'' + 4y = \sin(5x) \]
---
To use the method of variation of parameters, we need to set up the determinant required to calculate the Wronskian (W).
\[ W = \boxed{} \]
This involves finding two linearly independent solutions to the corresponding homogeneous equation, constructing the Wronskian determinant, and using this determinant to find the particular solution for the non-homogeneous equation.
---
**Explanation of the Process:**
1. **Solve the Homogeneous Equation:**
First, we solve the corresponding homogeneous equation:
\[ y'' + 4y = 0 \]
2. **Find Linearly Independent Solutions:**
Determine two linearly independent solutions (let's denote them as \( y_1 \) and \( y_2 \)) for the homogeneous equation.
3. **Construct the Wronskian:**
The Wronskian \( W \) is a determinant created from the solutions \( y_1 \), \( y_2 \), and their derivatives:
\[
W(y_1, y_2) = \begin{vmatrix}
y_1 & y_2 \\
y_1' & y_2' \\
\end{vmatrix}
\]
4. **Calculate the Wronskian:**
Plug in the solutions and their derivatives into the determinant to find \( W \).
5. **Find the Particular Solution:**
Use the method of variation of parameters to form the particular solution \( y_p \) using the formula:
\[
y_p = y_1 \int \frac{y_2 g(x)}{W} \, dx + y_2 \int \frac{y_1 g(x)}{W} \, dx
\]
where \( g(x) = \sin(5x) \).
By following these steps, we can determine the particular solution to the given differential equation using the method of variation of parameters.
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