Determine the form of a particular solution for the differential equation. Do not solve. y"-y=te 7t +6t² e 7t The form of a particular solution is y(t) = (Do not use d, D, e, E, i, or I as arbitrary constants since these letters already have defined meanings.)
Determine the form of a particular solution for the differential equation. Do not solve. y"-y=te 7t +6t² e 7t The form of a particular solution is y(t) = (Do not use d, D, e, E, i, or I as arbitrary constants since these letters already have defined meanings.)
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
Determine the form of a particular solution for the differential equation. Do not solve.
\[ y'' - y = t e^{7t} + 6 t^2 e^{7t} \]
### Question
The form of a particular solution is \( y_p(t) = \) [ \_\_\_\_\_\_\_\_\_ ]
**Note:** (Do not use \( d, D, e, E, i, \) or \( l \) as arbitrary constants since these letters already have defined meanings.)
### Guide
In order to find the form of a particular solution for the given non-homogeneous differential equation, analyze the non-homogeneous part of the equation, \( t e^{7t} + 6 t^2 e^{7t} \).
1. Identify the functions multiplying each term involving \( e^{7t} \):
- \( t e^{7t} \)
- \( 6 t^2 e^{7t} \)
2. The form of the particular solution must incorporate these terms and their derivatives. Since the highest power of \( t \) is 2 and the exponent \( e^{7t} \) is common in both terms, the form of the particular solution can be assumed to be:
\[ y_p(t) = (A t^2 + B t + C) e^{7t} \]
Here, \( A, B, \) and \( C \) are constants to be determined.
### Fill-In
The form of a particular solution is \( y_p(t) = (A t^2 + B t + C) e^{7t} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0bff7f51-4391-4187-9637-fbf171ec4a0e%2Ff47aba51-6e25-4024-9096-e2dc9ef80d83%2Fhyte7tk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Determine the form of a particular solution for the differential equation. Do not solve.
\[ y'' - y = t e^{7t} + 6 t^2 e^{7t} \]
### Question
The form of a particular solution is \( y_p(t) = \) [ \_\_\_\_\_\_\_\_\_ ]
**Note:** (Do not use \( d, D, e, E, i, \) or \( l \) as arbitrary constants since these letters already have defined meanings.)
### Guide
In order to find the form of a particular solution for the given non-homogeneous differential equation, analyze the non-homogeneous part of the equation, \( t e^{7t} + 6 t^2 e^{7t} \).
1. Identify the functions multiplying each term involving \( e^{7t} \):
- \( t e^{7t} \)
- \( 6 t^2 e^{7t} \)
2. The form of the particular solution must incorporate these terms and their derivatives. Since the highest power of \( t \) is 2 and the exponent \( e^{7t} \) is common in both terms, the form of the particular solution can be assumed to be:
\[ y_p(t) = (A t^2 + B t + C) e^{7t} \]
Here, \( A, B, \) and \( C \) are constants to be determined.
### Fill-In
The form of a particular solution is \( y_p(t) = (A t^2 + B t + C) e^{7t} \).
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