Without solving for the undetermined coefficients, find the correct form of the particular solution of the differential equation y(4) +81y" = 4x+ sin(9x). Clearly state Y, and Yp.
Without solving for the undetermined coefficients, find the correct form of the particular solution of the differential equation y(4) +81y" = 4x+ sin(9x). Clearly state Y, and Yp.
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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![### Determining the Form of the Particular Solution of the Differential Equation
#### Problem Statement:
**Without solving for the undetermined coefficients,** find the correct form of the particular solution of the differential equation:
\[ y^{(4)} + 81y'' = 4x + \sin(9x). \]
**Clearly state \( Y_c \) and \( Y_p \).**
#### Explanation:
In this problem, we are given a fourth-order differential equation and asked to find the form of the particular solution without determining the coefficients. The right-hand side of the equation consists of terms that suggest the particular solution should address polynomial and trigonometric functions.
1. **Homogeneous Equation and Complementary Solution:**
- The complementary solution (\(Y_c\)) is derived from the homogeneous part of the equation:
\[ y^{(4)} + 81y'' = 0. \]
2. **Particular Solution:**
- The non-homogeneous part of the equation, \(4x + \sin(9x)\), indicates that the particular solution (\(Y_p\)) should match the structure of these terms.
- For the polynomial term \(4x\), the particular solution should include a polynomial of the same degree:
\[ Y_p = Ax + B. \]
- For the trigonometric term \(\sin(9x)\), the particular solution should match the form of sinusoidal functions:
\[ Y_p = C\sin(9x) + D\cos(9x). \]
Hence, the combined correct form of the particular solution would be:
\[ Y_p = Ax + B + C\sin(9x) + D\cos(9x). \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F336090ec-d2ca-4d15-a2ae-85a71ff5580d%2F486aa9ff-0be5-441b-a96f-2c3052531a17%2F070hp7_processed.png&w=3840&q=75)
Transcribed Image Text:### Determining the Form of the Particular Solution of the Differential Equation
#### Problem Statement:
**Without solving for the undetermined coefficients,** find the correct form of the particular solution of the differential equation:
\[ y^{(4)} + 81y'' = 4x + \sin(9x). \]
**Clearly state \( Y_c \) and \( Y_p \).**
#### Explanation:
In this problem, we are given a fourth-order differential equation and asked to find the form of the particular solution without determining the coefficients. The right-hand side of the equation consists of terms that suggest the particular solution should address polynomial and trigonometric functions.
1. **Homogeneous Equation and Complementary Solution:**
- The complementary solution (\(Y_c\)) is derived from the homogeneous part of the equation:
\[ y^{(4)} + 81y'' = 0. \]
2. **Particular Solution:**
- The non-homogeneous part of the equation, \(4x + \sin(9x)\), indicates that the particular solution (\(Y_p\)) should match the structure of these terms.
- For the polynomial term \(4x\), the particular solution should include a polynomial of the same degree:
\[ Y_p = Ax + B. \]
- For the trigonometric term \(\sin(9x)\), the particular solution should match the form of sinusoidal functions:
\[ Y_p = C\sin(9x) + D\cos(9x). \]
Hence, the combined correct form of the particular solution would be:
\[ Y_p = Ax + B + C\sin(9x) + D\cos(9x). \]
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