Determine the inverse Laplace transform of the function below. 11 (2s+3)4 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. 11 (2s+3)4 -0 Properties of Laplace Transforms L{f+g} = L{f} + L{g} L{cf} c{f} for any constant c = Leatf(t)} (s) = L{f}(s-a) L {f') (s) = s£{f}(s)-f(0) L {f'') (s) = s²L{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 £{tf(t)} (s) = (-1)^ (L{f}(s)) dsn -S
Determine the inverse Laplace transform of the function below. 11 (2s+3)4 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. 11 (2s+3)4 -0 Properties of Laplace Transforms L{f+g} = L{f} + L{g} L{cf} c{f} for any constant c = Leatf(t)} (s) = L{f}(s-a) L {f') (s) = s£{f}(s)-f(0) L {f'') (s) = s²L{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 £{tf(t)} (s) = (-1)^ (L{f}(s)) dsn -S
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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![## Inverse Laplace Transform Example:
### Problem:
Determine the inverse Laplace transform of the function below:
\[ \frac{11}{(2s + 3)^4} \]
- **Links:**
- [Table of Laplace Transforms](#)
- [Properties of Laplace Transforms](#)
### Laplace Transform Notation:
\[ \mathcal{L}^{-1} \left\{ \frac{11}{(2s + 3)^4} \right\} = \square \]
---
## Properties of Laplace Transforms:
### Key Properties:
1. \(\mathcal{L}\{f + g\} = \mathcal{L}\{f\} + \mathcal{L}\{g\}\)
2. \(\mathcal{L}\{cf\} = c\mathcal{L}\{f\}\) for any constant \(c\)
3. \(\mathcal{L}\{e^{at}f(t)\} = \mathcal{L}\{f(s - a)\}\)
4. \(\mathcal{L}\{f'(t)\}(s) = s\mathcal{L}\{f\}(s) - f(0)\)
5. \(\mathcal{L}\{f''(t)\}(s) = s^2\mathcal{L}\{f\}(s) - sf(0) - f'(0)\)
6. \(\mathcal{L}\{f^{(n)}(t)\}(s) = s^n\mathcal{L}\{f(s)\} - s^{n-1}f(0) - \dots - f^{(n-1)}(0)\)
7. \(\mathcal{L}\{t^n f(t)\}(s) = (-1)^n \frac{d^n}{ds^n} (\mathcal{L}\{f(s)\})\)
8. \(\mathcal{L}^{-1} \{F_1 + F_2\} = \mathcal{L}^{-1} \{F_1\} + \mathcal{L}^{-1} \{F_2\}\)
9. \(\mathcal{L}^{-1} \{cf\} = c](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71da004f-23dc-47ac-9c54-84cf84b3724a%2F9eef7bc8-6d4b-43bd-a5ef-13a0183ecf72%2F52lp6oi_processed.png&w=3840&q=75)
Transcribed Image Text:## Inverse Laplace Transform Example:
### Problem:
Determine the inverse Laplace transform of the function below:
\[ \frac{11}{(2s + 3)^4} \]
- **Links:**
- [Table of Laplace Transforms](#)
- [Properties of Laplace Transforms](#)
### Laplace Transform Notation:
\[ \mathcal{L}^{-1} \left\{ \frac{11}{(2s + 3)^4} \right\} = \square \]
---
## Properties of Laplace Transforms:
### Key Properties:
1. \(\mathcal{L}\{f + g\} = \mathcal{L}\{f\} + \mathcal{L}\{g\}\)
2. \(\mathcal{L}\{cf\} = c\mathcal{L}\{f\}\) for any constant \(c\)
3. \(\mathcal{L}\{e^{at}f(t)\} = \mathcal{L}\{f(s - a)\}\)
4. \(\mathcal{L}\{f'(t)\}(s) = s\mathcal{L}\{f\}(s) - f(0)\)
5. \(\mathcal{L}\{f''(t)\}(s) = s^2\mathcal{L}\{f\}(s) - sf(0) - f'(0)\)
6. \(\mathcal{L}\{f^{(n)}(t)\}(s) = s^n\mathcal{L}\{f(s)\} - s^{n-1}f(0) - \dots - f^{(n-1)}(0)\)
7. \(\mathcal{L}\{t^n f(t)\}(s) = (-1)^n \frac{d^n}{ds^n} (\mathcal{L}\{f(s)\})\)
8. \(\mathcal{L}^{-1} \{F_1 + F_2\} = \mathcal{L}^{-1} \{F_1\} + \mathcal{L}^{-1} \{F_2\}\)
9. \(\mathcal{L}^{-1} \{cf\} = c
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