Use undetermined coefficients to find the particular solution to y'' – 3y' + y = 4t2 + 6t + 5
Use undetermined coefficients to find the particular solution to y'' – 3y' + y = 4t2 + 6t + 5
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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![### Using Undetermined Coefficients to Find the Particular Solution
To find the particular solution to the second-order non-homogeneous differential equation using the method of undetermined coefficients, follow the instructions below:
The given differential equation is:
\[ y'' - 3y' + y = 4t^2 + 6t + 5 \]
We need to find the particular solution \( y_p(t) \).
1. **Identify the form of the non-homogeneous term:**
- The non-homogeneous term on the right-hand side is \( 4t^2 + 6t + 5 \).
2. **Guess the form of the particular solution \( y_p(t) \):**
- Since the non-homogeneous term \( 4t^2 + 6t + 5 \) is a polynomial of degree 2, guess the particular solution in the form of a polynomial of degree 2:
\[ y_p(t) = At^2 + Bt + C \]
where \( A \), \( B \), and \( C \) are constants to be determined.
3. **Find the derivatives of \( y_p(t) \):**
\[ y_p'(t) = 2At + B \]
\[ y_p''(t) = 2A \]
4. **Substitute \( y_p(t) \), \( y_p'(t) \), and \( y_p''(t) \) into the original differential equation:**
\[ 2A - 3(2At + B) + (At^2 + Bt + C) = 4t^2 + 6t + 5 \]
5. **Collect like terms and equate coefficients of corresponding powers of \( t \):**
\[ 2A - 6At - 3B + At^2 + Bt + C = 4t^2 + 6t + 5 \]
Simplifying, we get:
\[ At^2 + (-6A + B)t + (2A - 3B + C) = 4t^2 + 6t + 5 \]
6. **Equate coefficients:**
- Coefficient of \( t^2 \): \( A = 4 \)
- Coefficient of \( t \): \( -6A + B =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc700bc18-61e4-4e04-a31d-52f10b21c2d0%2F01bb2b14-653d-408d-8a66-2de2e01cb654%2F4ub5ls_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Using Undetermined Coefficients to Find the Particular Solution
To find the particular solution to the second-order non-homogeneous differential equation using the method of undetermined coefficients, follow the instructions below:
The given differential equation is:
\[ y'' - 3y' + y = 4t^2 + 6t + 5 \]
We need to find the particular solution \( y_p(t) \).
1. **Identify the form of the non-homogeneous term:**
- The non-homogeneous term on the right-hand side is \( 4t^2 + 6t + 5 \).
2. **Guess the form of the particular solution \( y_p(t) \):**
- Since the non-homogeneous term \( 4t^2 + 6t + 5 \) is a polynomial of degree 2, guess the particular solution in the form of a polynomial of degree 2:
\[ y_p(t) = At^2 + Bt + C \]
where \( A \), \( B \), and \( C \) are constants to be determined.
3. **Find the derivatives of \( y_p(t) \):**
\[ y_p'(t) = 2At + B \]
\[ y_p''(t) = 2A \]
4. **Substitute \( y_p(t) \), \( y_p'(t) \), and \( y_p''(t) \) into the original differential equation:**
\[ 2A - 3(2At + B) + (At^2 + Bt + C) = 4t^2 + 6t + 5 \]
5. **Collect like terms and equate coefficients of corresponding powers of \( t \):**
\[ 2A - 6At - 3B + At^2 + Bt + C = 4t^2 + 6t + 5 \]
Simplifying, we get:
\[ At^2 + (-6A + B)t + (2A - 3B + C) = 4t^2 + 6t + 5 \]
6. **Equate coefficients:**
- Coefficient of \( t^2 \): \( A = 4 \)
- Coefficient of \( t \): \( -6A + B =
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