-2 4* y" - 47' + 47 = x sinx
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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![### Problem #2
**Objective: Solve the Differential Equation**
\[ y'' - 4y' + 4y = x \sin x \]
In this problem, we aim to find the function \( y(x) \) that satisfies the given second-order differential equation. In the equation:
- \( y'' \) represents the second derivative of \( y \) with respect to \( x \).
- \( y' \) represents the first derivative of \( y \) with respect to \( x \).
- \( y \) is the unknown function of \( x \) we need to determine.
The right-hand side of the equation includes the term \( x \sin x \), indicating that this is a non-homogeneous differential equation with a sinusoidal forcing function. This will involve finding the complementary solution \( y_c \) and a particular solution \( y_p \) to form the general solution \( y = y_c + y_p \).
#### Steps to Solve:
1. **Find the Complementary Solution \( y_c \)**
- Solve the homogeneous equation \( y'' - 4y' + 4y = 0 \).
2. **Find the Particular Solution \( y_p \)**
- Use an appropriate method (e.g., undetermined coefficients or variation of parameters) to solve for \( y_p \) given the non-homogeneous term \( x \sin x \).
3. **Combine the Solutions**
- Combine the complementary solution and particular solution to get the general solution.
#### Detailed Diagrams:
(If there are any visuals or step-by-step graphical representations, they would be included here with thorough explanations.)
- **Graph 1**: A plot of the complementary solution \( y_c \).
- **Graph 2**: A plot showing the effects of the non-homogeneous term \( x \sin x \).
- **Graph 3**: The final general solution combining \( y_c \) and \( y_p \).
By solving each part systematically, we can better understand the behavior of the solution to this differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e83de08-e5e7-4f69-a8ed-52731b075ebc%2F6a34f1f2-54aa-4f21-9c80-3b1a87b55651%2F5gx3tgp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem #2
**Objective: Solve the Differential Equation**
\[ y'' - 4y' + 4y = x \sin x \]
In this problem, we aim to find the function \( y(x) \) that satisfies the given second-order differential equation. In the equation:
- \( y'' \) represents the second derivative of \( y \) with respect to \( x \).
- \( y' \) represents the first derivative of \( y \) with respect to \( x \).
- \( y \) is the unknown function of \( x \) we need to determine.
The right-hand side of the equation includes the term \( x \sin x \), indicating that this is a non-homogeneous differential equation with a sinusoidal forcing function. This will involve finding the complementary solution \( y_c \) and a particular solution \( y_p \) to form the general solution \( y = y_c + y_p \).
#### Steps to Solve:
1. **Find the Complementary Solution \( y_c \)**
- Solve the homogeneous equation \( y'' - 4y' + 4y = 0 \).
2. **Find the Particular Solution \( y_p \)**
- Use an appropriate method (e.g., undetermined coefficients or variation of parameters) to solve for \( y_p \) given the non-homogeneous term \( x \sin x \).
3. **Combine the Solutions**
- Combine the complementary solution and particular solution to get the general solution.
#### Detailed Diagrams:
(If there are any visuals or step-by-step graphical representations, they would be included here with thorough explanations.)
- **Graph 1**: A plot of the complementary solution \( y_c \).
- **Graph 2**: A plot showing the effects of the non-homogeneous term \( x \sin x \).
- **Graph 3**: The final general solution combining \( y_c \) and \( y_p \).
By solving each part systematically, we can better understand the behavior of the solution to this differential equation.
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