Determine the form of a particular solution Yp of Ур y" - 9y' + 14y = 3x² - 5 sin2x + 8xe6x. Do not solve the equation.
Determine the form of a particular solution Yp of Ур y" - 9y' + 14y = 3x² - 5 sin2x + 8xe6x. Do not solve the equation.
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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help please can you show nice neat work to understand the work thanks.
![**Problem Statement:**
Determine the form of a particular solution \( y_p \) of the following differential equation:
\[ y'' - 9y' + 14y = 3x^2 - 5 \sin(2x) + 8xe^{6x}. \]
**Note:** Do not solve the equation.
### Explanation:
1. **Polynomial Component \( 3x^2 \):**
- The corresponding particular solution form would be \( Ax^2 + Bx + C \), where \( A \), \( B \), and \( C \) are constants to be determined.
2. **Trigonomentric Component \( -5 \sin(2x) \):**
- The corresponding particular solution form would be \( D \cos(2x) + E \sin(2x) \), where \( D \) and \( E \) are constants to be determined.
3. **Exponential Component \( 8xe^{6x} \):**
- The corresponding particular solution form for the exponential component would be \( (Fx + G)e^{6x} \), where \( F \) and \( G \) are constants to be determined.
### General Form of the Particular Solution:
The overall form of the particular solution \( y_p \) considering all components would be the sum of each part mentioned above:
\[ y_p = Ax^2 + Bx + C + D \cos(2x) + E \sin(2x) + (Fx + G)e^{6x} \]
This expression represents the assumed form of \( y_p \) for the given differential equation, combining polynomial, trigonometric, and exponential terms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1d700aa-072a-4ff7-be1d-cdca0f10aa9d%2F00f87b24-5866-435e-b5da-7605b7041329%2Frfefu4h_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine the form of a particular solution \( y_p \) of the following differential equation:
\[ y'' - 9y' + 14y = 3x^2 - 5 \sin(2x) + 8xe^{6x}. \]
**Note:** Do not solve the equation.
### Explanation:
1. **Polynomial Component \( 3x^2 \):**
- The corresponding particular solution form would be \( Ax^2 + Bx + C \), where \( A \), \( B \), and \( C \) are constants to be determined.
2. **Trigonomentric Component \( -5 \sin(2x) \):**
- The corresponding particular solution form would be \( D \cos(2x) + E \sin(2x) \), where \( D \) and \( E \) are constants to be determined.
3. **Exponential Component \( 8xe^{6x} \):**
- The corresponding particular solution form for the exponential component would be \( (Fx + G)e^{6x} \), where \( F \) and \( G \) are constants to be determined.
### General Form of the Particular Solution:
The overall form of the particular solution \( y_p \) considering all components would be the sum of each part mentioned above:
\[ y_p = Ax^2 + Bx + C + D \cos(2x) + E \sin(2x) + (Fx + G)e^{6x} \]
This expression represents the assumed form of \( y_p \) for the given differential equation, combining polynomial, trigonometric, and exponential terms.
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