Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.
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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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