Determine £¹{F}. F(s) = 7s² - 18s +7 s(s-7) (s-5) Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. £¹{F} =[ Properties of Laplace Transforms L{f+g} = L{f} + L{g} L{cf} cL{f} for any constant c L{eat f(t)} (s) = L{f}(s-a) L {f'} (s) = s£{f}(s) – f(0) L {t''}(s) = s² L{f}(s) - sf(0) - f'(0) L {f(n)} (s) = s^L{f}(s) - sn-1f(0) -sn-2f'(0) ----f(n-1) (0) dn ds -(L{f}(s)) {F₁+F₂} = £²1 {F₁} £¹{CF} = c£¯¹{F} L-1 {tf(t)} (s) = (-1)^- {F2} X Table of Laplace Transforms f(t) 1 eat t", n=1, 2, ... sin bt cos bt eat, n= 1, 2,.... eat sin bt eat cos bt F(s) = £{f}(s) 1 SS>0 1 s-a nl Sn+11 b s²+ b² S> 0 S> 0 S> 0 S s² + b² nl (s-a)+1/ b (s-a)² + b² s-a (s-a)² + b² S>0 ,s> a s> a s> a
Determine £¹{F}. F(s) = 7s² - 18s +7 s(s-7) (s-5) Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. £¹{F} =[ Properties of Laplace Transforms L{f+g} = L{f} + L{g} L{cf} cL{f} for any constant c L{eat f(t)} (s) = L{f}(s-a) L {f'} (s) = s£{f}(s) – f(0) L {t''}(s) = s² L{f}(s) - sf(0) - f'(0) L {f(n)} (s) = s^L{f}(s) - sn-1f(0) -sn-2f'(0) ----f(n-1) (0) dn ds -(L{f}(s)) {F₁+F₂} = £²1 {F₁} £¹{CF} = c£¯¹{F} L-1 {tf(t)} (s) = (-1)^- {F2} X Table of Laplace Transforms f(t) 1 eat t", n=1, 2, ... sin bt cos bt eat, n= 1, 2,.... eat sin bt eat cos bt F(s) = £{f}(s) 1 SS>0 1 s-a nl Sn+11 b s²+ b² S> 0 S> 0 S> 0 S s² + b² nl (s-a)+1/ b (s-a)² + b² s-a (s-a)² + b² S>0 ,s> a s> a s> a
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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![### Laplace Transforms and Their Properties
#### Determine the Inverse Laplace Transform
Given:
\[ F(s) = \frac{7s^2 - 18s + 7}{s(s - 7)(s - 5)} \]
#### Clickable Links
- [Click here to view the table of Laplace transforms.](#)
- [Click here to view the table of properties of Laplace transforms.](#)
#### Inverse Laplace Transform
\[ \mathcal{L}^{-1}\{F\} = \square \]
### Properties of Laplace Transforms
The following properties help in understanding and manipulating Laplace transforms:
- \( \mathcal{L}\{f + g\} = \mathcal{L}\{f\} + \mathcal{L}\{g\} \)
- \( \mathcal{L}\{cf\} = c\mathcal{L}\{f\} \), for any constant \( c \)
- \( \mathcal{L}\{e^{at}f(t)\} = \mathcal{L}\{f(s - a)\} \)
- \( \mathcal{L}\{f'\} = s\mathcal{L}\{f\}(s) - f(0) \)
- \( \mathcal{L}\{f''\} = s^2\mathcal{L}\{f\}(s) - sf(0) - f'(0) \)
- \( \mathcal{L}\{f^{(n)}\} = s^n\mathcal{L}\{f\}(s) - s^{n-1}f(0) - s^{n-2}f'(0) - \ldots - f^{(n-1)}(0) \)
- \( \mathcal{L}\{t^n f(t)\}(s) = (-1)^n \frac{d^n}{ds^n}(\mathcal{L}\{f\}(s)) \)
- \( \mathcal{L}^{-1}\{F_1 + F_2\} = \mathcal{L}^{-1}\{F_1\} + \mathcal{L}^{-1}\{F_2\} \)
- \( \mathcal{L}^{-1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71da004f-23dc-47ac-9c54-84cf84b3724a%2F7c12080e-7c9e-41fa-a2f0-a47a17708fd0%2Fbb6ugrq_processed.png&w=3840&q=75)
Transcribed Image Text:### Laplace Transforms and Their Properties
#### Determine the Inverse Laplace Transform
Given:
\[ F(s) = \frac{7s^2 - 18s + 7}{s(s - 7)(s - 5)} \]
#### Clickable Links
- [Click here to view the table of Laplace transforms.](#)
- [Click here to view the table of properties of Laplace transforms.](#)
#### Inverse Laplace Transform
\[ \mathcal{L}^{-1}\{F\} = \square \]
### Properties of Laplace Transforms
The following properties help in understanding and manipulating Laplace transforms:
- \( \mathcal{L}\{f + g\} = \mathcal{L}\{f\} + \mathcal{L}\{g\} \)
- \( \mathcal{L}\{cf\} = c\mathcal{L}\{f\} \), for any constant \( c \)
- \( \mathcal{L}\{e^{at}f(t)\} = \mathcal{L}\{f(s - a)\} \)
- \( \mathcal{L}\{f'\} = s\mathcal{L}\{f\}(s) - f(0) \)
- \( \mathcal{L}\{f''\} = s^2\mathcal{L}\{f\}(s) - sf(0) - f'(0) \)
- \( \mathcal{L}\{f^{(n)}\} = s^n\mathcal{L}\{f\}(s) - s^{n-1}f(0) - s^{n-2}f'(0) - \ldots - f^{(n-1)}(0) \)
- \( \mathcal{L}\{t^n f(t)\}(s) = (-1)^n \frac{d^n}{ds^n}(\mathcal{L}\{f\}(s)) \)
- \( \mathcal{L}^{-1}\{F_1 + F_2\} = \mathcal{L}^{-1}\{F_1\} + \mathcal{L}^{-1}\{F_2\} \)
- \( \mathcal{L}^{-1
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