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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
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.
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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