Use the technique of variation of parameters to find a particular solution to the DE y" – 2y + y = ui y1 + uy2 Hint: f (x) S
Use the technique of variation of parameters to find a particular solution to the DE y" – 2y + y = ui y1 + uy2 Hint: f (x) S
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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![**Title: Variation of Parameters for Solving Differential Equations**
**Objective:**
Learn to apply the technique of **variation of parameters** to find a particular solution to the given differential equation (DE).
---
**Problem Statement:**
Given the following second-order linear differential equation:
\[ y'' - 2y' + y = \frac{e^x}{\sqrt{x}} \]
Use the technique of **variation of parameters** to find a particular solution to this DE.
---
**Hint:**
To use variation of parameters, consider the following system of equations:
\[
\left\{
\begin{array}{l}
u_1' y_1 + u_2' y_2 = 0 \\
u_1' y_1' + u_2' y_2' = f(x)
\end{array}
\right.
\]
---
**Explanation:**
1. **Identify the complementary solution \(y_c\):**
Solve the homogeneous part of the equation \(y'' - 2y' + y = 0\) to find the complementary solution \( y_c \).
2. **Form the Ansatz for the particular solution \(y_p\):**
Since we are using variation of parameters, assume that the particular solution can be written as:
\[
y_p = u_1 y_1 + u_2 y_2
\]
where \( u_1 \) and \( u_2 \) are functions to be determined, and \( y_1 \) and \( y_2 \) are solutions to the homogeneous equation.
3. **Differentiate the assumed solution \(y_p\):**
Compute the first and second derivatives of \( y_p \) and substitute them back into the original DE.
4. **Use the system of equations to find \(u_1'\) and \(u_2'\):**
Solve the given system:
\[
\left\{
\begin{array}{l}
u_1' y_1 + u_2' y_2 = 0 \\
u_1' y_1' + u_2' y_2' = f(x)
\end{array}
\right.
\]
to find \( u_1' \) and \( u_2' \).
5. **Integrate to find](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0b9151a0-b51c-4e98-8bd5-faad9be5df3c%2F62c6199c-be14-4d6b-b474-0d7d99ec3058%2F9ptxnx.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Variation of Parameters for Solving Differential Equations**
**Objective:**
Learn to apply the technique of **variation of parameters** to find a particular solution to the given differential equation (DE).
---
**Problem Statement:**
Given the following second-order linear differential equation:
\[ y'' - 2y' + y = \frac{e^x}{\sqrt{x}} \]
Use the technique of **variation of parameters** to find a particular solution to this DE.
---
**Hint:**
To use variation of parameters, consider the following system of equations:
\[
\left\{
\begin{array}{l}
u_1' y_1 + u_2' y_2 = 0 \\
u_1' y_1' + u_2' y_2' = f(x)
\end{array}
\right.
\]
---
**Explanation:**
1. **Identify the complementary solution \(y_c\):**
Solve the homogeneous part of the equation \(y'' - 2y' + y = 0\) to find the complementary solution \( y_c \).
2. **Form the Ansatz for the particular solution \(y_p\):**
Since we are using variation of parameters, assume that the particular solution can be written as:
\[
y_p = u_1 y_1 + u_2 y_2
\]
where \( u_1 \) and \( u_2 \) are functions to be determined, and \( y_1 \) and \( y_2 \) are solutions to the homogeneous equation.
3. **Differentiate the assumed solution \(y_p\):**
Compute the first and second derivatives of \( y_p \) and substitute them back into the original DE.
4. **Use the system of equations to find \(u_1'\) and \(u_2'\):**
Solve the given system:
\[
\left\{
\begin{array}{l}
u_1' y_1 + u_2' y_2 = 0 \\
u_1' y_1' + u_2' y_2' = f(x)
\end{array}
\right.
\]
to find \( u_1' \) and \( u_2' \).
5. **Integrate to find
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