Find a particular solution using the Undetermined Coefficients Method of y" -y' + y = 2 sin3x.
Find a particular solution using the Undetermined Coefficients Method of y" -y' + y = 2 sin3x.
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 Undetermined Coefficients**
**Problem Statement:**
Find a particular solution using the Undetermined Coefficients Method for the differential equation:
\[ y'' - y' + y = 2 \sin 3x. \]
**Explanation:**
This problem involves finding a particular solution to the non-homogeneous linear differential equation with constant coefficients. The right-hand side of the equation, \( 2 \sin 3x \), suggests that we use the method of undetermined coefficients.
**Steps to Solve:**
1. **Solve the Corresponding Homogeneous Equation:**
The homogeneous form of the given differential equation is:
\[ y'' - y' + y = 0. \]
2. **Find the General Solution of the Homogeneous Equation:**
Solve the characteristic equation obtained from the homogeneous differential equation.
3. **Guess the Particular Solution:**
For the non-homogeneous equation, make an educated guess (ansatz) for the particular solution based on the method of undetermined coefficients.
4. **Substitute and Determine Coefficients:**
Substitute the guessed particular solution into the original differential equation and adjust the coefficients to satisfy the equation.
5. **Combine Solutions:**
Combine the general solution of the homogeneous equation with the particular solution to get the general solution of the non-homogeneous equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1d700aa-072a-4ff7-be1d-cdca0f10aa9d%2F768640c3-5725-4b35-9514-81a7b02c6ef0%2Fjow3vuo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding a Particular Solution Using the Method of Undetermined Coefficients**
**Problem Statement:**
Find a particular solution using the Undetermined Coefficients Method for the differential equation:
\[ y'' - y' + y = 2 \sin 3x. \]
**Explanation:**
This problem involves finding a particular solution to the non-homogeneous linear differential equation with constant coefficients. The right-hand side of the equation, \( 2 \sin 3x \), suggests that we use the method of undetermined coefficients.
**Steps to Solve:**
1. **Solve the Corresponding Homogeneous Equation:**
The homogeneous form of the given differential equation is:
\[ y'' - y' + y = 0. \]
2. **Find the General Solution of the Homogeneous Equation:**
Solve the characteristic equation obtained from the homogeneous differential equation.
3. **Guess the Particular Solution:**
For the non-homogeneous equation, make an educated guess (ansatz) for the particular solution based on the method of undetermined coefficients.
4. **Substitute and Determine Coefficients:**
Substitute the guessed particular solution into the original differential equation and adjust the coefficients to satisfy the equation.
5. **Combine Solutions:**
Combine the general solution of the homogeneous equation with the particular solution to get the general solution of the non-homogeneous equation.
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