Solve for Y(s), the Laplace transform of the solution y(t) to the initial value problem below. y" - 7y' +12y =5te³t, y(0) = 3, y'(0) = -5
Solve for Y(s), the Laplace transform of the solution y(t) to the initial value problem below. y" - 7y' +12y =5te³t, y(0) = 3, y'(0) = -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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![## Educational Content: Laplace Transforms
### Problem Statement
**Solve for \( Y(s) \), the Laplace transform of the solution \( y(t) \) to the initial value problem below.**
\[ y'' - 7y' + 12y = 5te^{3t}, \; y(0) = 3, \; y'(0) = -5 \]
- **Click [here](#) to view the table of Laplace transforms.**
- **Click [here](#) to view the table of properties of Laplace transforms.**
\[ Y(s) = \_\_\_\_ \]
### Properties of Laplace Transforms
The properties of Laplace transforms are provided in the table below. These properties are essential for solving differential equations using the Laplace transform.
1. \( \mathcal{L}\{f + g\} = \mathcal{L}\{f\} + \mathcal{L}\{g\} \)
2. \( \mathcal{L}\{cf\} = c\mathcal{L}\{f\} \) for any constant \( c \)
3. \( \mathcal{L}\{e^{at}f(t)\}(s) = \mathcal{L}\{f\}(s-a) \)
4. \( \mathcal{L}\{f'\}(s) = s\mathcal{L}\{f\}(s) - f(0) \)
5. \( \mathcal{L}\{f''\}(s) = s^2 \mathcal{L}\{f\}(s) - sf(0) - f'(0) \)
6. \( \mathcal{L}\{f^{(n)}\}(s) = s^n \mathcal{L}\{f\}(s) - s^{n-1}f(0) - \cdots - f^{(n-1)}(0) \)
7. \( \mathcal{L}^{-1}\{F_1 + F_2\} = \mathcal{L}^{-1}\{F_1\} + \mathcal{L}^{-1}\{F_2\} \)
8. \( \mathcal{L}^{-1}\{cF\} = c\mathcal{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F047a7e6a-f025-4b5b-ab83-4ffe14f69253%2Fa72ee698-fbbb-4081-8af6-f926a229704b%2Fdevt77_processed.png&w=3840&q=75)
Transcribed Image Text:## Educational Content: Laplace Transforms
### Problem Statement
**Solve for \( Y(s) \), the Laplace transform of the solution \( y(t) \) to the initial value problem below.**
\[ y'' - 7y' + 12y = 5te^{3t}, \; y(0) = 3, \; y'(0) = -5 \]
- **Click [here](#) to view the table of Laplace transforms.**
- **Click [here](#) to view the table of properties of Laplace transforms.**
\[ Y(s) = \_\_\_\_ \]
### Properties of Laplace Transforms
The properties of Laplace transforms are provided in the table below. These properties are essential for solving differential equations using the Laplace transform.
1. \( \mathcal{L}\{f + g\} = \mathcal{L}\{f\} + \mathcal{L}\{g\} \)
2. \( \mathcal{L}\{cf\} = c\mathcal{L}\{f\} \) for any constant \( c \)
3. \( \mathcal{L}\{e^{at}f(t)\}(s) = \mathcal{L}\{f\}(s-a) \)
4. \( \mathcal{L}\{f'\}(s) = s\mathcal{L}\{f\}(s) - f(0) \)
5. \( \mathcal{L}\{f''\}(s) = s^2 \mathcal{L}\{f\}(s) - sf(0) - f'(0) \)
6. \( \mathcal{L}\{f^{(n)}\}(s) = s^n \mathcal{L}\{f\}(s) - s^{n-1}f(0) - \cdots - f^{(n-1)}(0) \)
7. \( \mathcal{L}^{-1}\{F_1 + F_2\} = \mathcal{L}^{-1}\{F_1\} + \mathcal{L}^{-1}\{F_2\} \)
8. \( \mathcal{L}^{-1}\{cF\} = c\mathcal{
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