Find a particular solution yp of the following equation using the Method of Undetermined Coefficients. Primes denote the derivatives with respect to t. y" +5y' +10y=510 e²t cos 6t
Find a particular solution yp of the following equation using the Method of Undetermined Coefficients. Primes denote the derivatives with respect to t. y" +5y' +10y=510 e²t cos 6t
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Finding a Particular Solution using the Method of Undetermined Coefficients
**Objective**: Find a particular solution \( y_p \) of the following differential equation using the Method of Undetermined Coefficients. In this context, primes denote the derivatives with respect to \( t \).
#### Equation:
\[ y'' + 5y' + 10y = 510 e^{2t} \cos 6t \]
---
**Solution**: A particular solution is \( y_p(t) = \) [input box for user to enter the function].
In the differential equation given above:
- \( y'' \) is the second derivative of \( y \) with respect to \( t \),
- \( y' \) is the first derivative of \( y \) with respect to \( t \),
- The right-hand side of the equation contains \( 510 e^{2t} \cos 6t \), which indicates the form of the non-homogeneous term that needs to be used when assuming a particular solution using the Method of Undetermined Coefficients.
---
To solve this, one should typically assume a particular solution of a similar form to the non-homogeneous term. Given the non-homogeneous term includes \( e^{2t} \cos 6t \), the assumed form for \( y_p \) could be:
\[ y_p(t) = (Ae^{2t}\cos 6t + Be^{2t}\sin 6t) \]
Where \( A \) and \( B \) are constants to be determined.
By substituting this assumed solution back into the differential equation and solving for the coefficients, one can determine the values of \( A \) and \( B \).
Finally, substitute the calculated values back into the assumed form to get the particular solution \( y_p(t) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F896deee6-4ebc-4afd-8502-502eb7aa6712%2Ff8e33a55-1267-4c67-924c-4d8d41937dd1%2Fojx22ap_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding a Particular Solution using the Method of Undetermined Coefficients
**Objective**: Find a particular solution \( y_p \) of the following differential equation using the Method of Undetermined Coefficients. In this context, primes denote the derivatives with respect to \( t \).
#### Equation:
\[ y'' + 5y' + 10y = 510 e^{2t} \cos 6t \]
---
**Solution**: A particular solution is \( y_p(t) = \) [input box for user to enter the function].
In the differential equation given above:
- \( y'' \) is the second derivative of \( y \) with respect to \( t \),
- \( y' \) is the first derivative of \( y \) with respect to \( t \),
- The right-hand side of the equation contains \( 510 e^{2t} \cos 6t \), which indicates the form of the non-homogeneous term that needs to be used when assuming a particular solution using the Method of Undetermined Coefficients.
---
To solve this, one should typically assume a particular solution of a similar form to the non-homogeneous term. Given the non-homogeneous term includes \( e^{2t} \cos 6t \), the assumed form for \( y_p \) could be:
\[ y_p(t) = (Ae^{2t}\cos 6t + Be^{2t}\sin 6t) \]
Where \( A \) and \( B \) are constants to be determined.
By substituting this assumed solution back into the differential equation and solving for the coefficients, one can determine the values of \( A \) and \( B \).
Finally, substitute the calculated values back into the assumed form to get the particular solution \( y_p(t) \).
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