Use undetermined coefficients to find the particular solution to y'" + 8y' + 15y = - 1040 sin(t) Y(t) = %D
Use undetermined coefficients to find the particular solution to y'" + 8y' + 15y = - 1040 sin(t) Y(t) = %D
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 Statement
Use the method of undetermined coefficients to find the particular solution to the following differential equation:
\[ y'' + 8y' + 15y = -1040 \sin(t) \]
Provide the particular solution in the form:
\[ Y(t) = \]
[Input field for the solution]
### Explanation
To solve this second-order non-homogeneous linear differential equation using the method of undetermined coefficients, follow these steps:
1. **Solve the Homogeneous Equation:**
- First, solve the associated homogeneous equation \( y'' + 8y' + 15y = 0 \).
- Find the characteristic equation: \( r^2 + 8r + 15 = 0 \).
- Solve the quadratic equation for \( r \).
2. **Guess the Particular Solution:**
- Since the non-homogeneous term is \(-1040 \sin(t)\), assume a particular solution of the form \( Y_p(t) = A \sin(t) + B \cos(t) \).
- Differentiate \( Y_p(t) \) to find \( Y_p'(t) \) and \( Y_p''(t) \).
3. **Substitute and Solve for Coefficients:**
- Substitute \( Y_p(t) \), \( Y_p'(t) \), and \( Y_p''(t) \) into the original differential equation.
- Equate the coefficients of \(\sin(t)\) and \(\cos(t)\) to solve for \( A \) and \( B \).
4. **Form the General Solution:**
- Combine the homogeneous solution with the particular solution to obtain the general solution of the differential equation.
### Goal
Determine the specific coefficients \( A \) and \( B \) for the strategic guess of \( Y(t) \) and input the particular solution into the form provided in the answer box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc700bc18-61e4-4e04-a31d-52f10b21c2d0%2Fee04ceb8-add5-48ca-b1ae-f76af6e7bc7e%2F2swdty_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Use the method of undetermined coefficients to find the particular solution to the following differential equation:
\[ y'' + 8y' + 15y = -1040 \sin(t) \]
Provide the particular solution in the form:
\[ Y(t) = \]
[Input field for the solution]
### Explanation
To solve this second-order non-homogeneous linear differential equation using the method of undetermined coefficients, follow these steps:
1. **Solve the Homogeneous Equation:**
- First, solve the associated homogeneous equation \( y'' + 8y' + 15y = 0 \).
- Find the characteristic equation: \( r^2 + 8r + 15 = 0 \).
- Solve the quadratic equation for \( r \).
2. **Guess the Particular Solution:**
- Since the non-homogeneous term is \(-1040 \sin(t)\), assume a particular solution of the form \( Y_p(t) = A \sin(t) + B \cos(t) \).
- Differentiate \( Y_p(t) \) to find \( Y_p'(t) \) and \( Y_p''(t) \).
3. **Substitute and Solve for Coefficients:**
- Substitute \( Y_p(t) \), \( Y_p'(t) \), and \( Y_p''(t) \) into the original differential equation.
- Equate the coefficients of \(\sin(t)\) and \(\cos(t)\) to solve for \( A \) and \( B \).
4. **Form the General Solution:**
- Combine the homogeneous solution with the particular solution to obtain the general solution of the differential equation.
### Goal
Determine the specific coefficients \( A \) and \( B \) for the strategic guess of \( Y(t) \) and input the particular solution into the form provided in the answer box.
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