Determine the inverse Laplace transform of the function below. 1
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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Question
![### Determine the Inverse Laplace Transform
To determine the inverse Laplace transform of the function below:
\[
\frac{1}{s^4}
\]
Click the links to view relevant tables:
- [Table of Laplace Transforms](#)
- [Table of Properties of Laplace Transforms](#)
#### Inverse Laplace Transform Expression
\[
\mathcal{L}^{-1}\left\{\frac{1}{s^4}\right\} =
\]
### Properties of Laplace Transforms
The properties are outlined in a table as follows:
1. **Linear Combination:**
\[
\mathcal{L}\{f + g\} = \mathcal{L}\{f\} + \mathcal{L}\{g\}
\]
2. **Scaling by a Constant:**
\[
\mathcal{L}\{cf\} = c\mathcal{L}\{f\} \quad \text{for any constant } c
\]
3. **Exponential Scaling:**
\[
\mathcal{L}\{e^{at}f(t)\} = \mathcal{L}\{f(t)\}(s - a)
\]
4. **Derivative in Time:**
\[
\mathcal{L}\{f'(t)\} = s\mathcal{L}\{f(t)\} - f(0)
\]
5. **Second Derivative in Time:**
\[
\mathcal{L}\{f''(t)\} = s^2\mathcal{L}\{f(t)\} - sf(0) - f'(0)
\]
6. **nth Derivative in Time:**
\[
\mathcal{L}\{f^{(n)}(t)\} = s^n\mathcal{L}\{f(t)\} - s^{n-1}f(0) - ... - f^{(n-1)}(0)
\]
7. **Integrals Transform:**
\[
\mathcal{L}\left\{\int_0^t f(u) du\right\} = \frac{1}{s}\mathcal{L}\{f(t)\}
\]
8. **Frequency Differentiation:**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F047a7e6a-f025-4b5b-ab83-4ffe14f69253%2F5d6513fd-8092-4dbc-9c8b-2fde1e60db2e%2Fm1xe0p_processed.png&w=3840&q=75)
Transcribed Image Text:### Determine the Inverse Laplace Transform
To determine the inverse Laplace transform of the function below:
\[
\frac{1}{s^4}
\]
Click the links to view relevant tables:
- [Table of Laplace Transforms](#)
- [Table of Properties of Laplace Transforms](#)
#### Inverse Laplace Transform Expression
\[
\mathcal{L}^{-1}\left\{\frac{1}{s^4}\right\} =
\]
### Properties of Laplace Transforms
The properties are outlined in a table as follows:
1. **Linear Combination:**
\[
\mathcal{L}\{f + g\} = \mathcal{L}\{f\} + \mathcal{L}\{g\}
\]
2. **Scaling by a Constant:**
\[
\mathcal{L}\{cf\} = c\mathcal{L}\{f\} \quad \text{for any constant } c
\]
3. **Exponential Scaling:**
\[
\mathcal{L}\{e^{at}f(t)\} = \mathcal{L}\{f(t)\}(s - a)
\]
4. **Derivative in Time:**
\[
\mathcal{L}\{f'(t)\} = s\mathcal{L}\{f(t)\} - f(0)
\]
5. **Second Derivative in Time:**
\[
\mathcal{L}\{f''(t)\} = s^2\mathcal{L}\{f(t)\} - sf(0) - f'(0)
\]
6. **nth Derivative in Time:**
\[
\mathcal{L}\{f^{(n)}(t)\} = s^n\mathcal{L}\{f(t)\} - s^{n-1}f(0) - ... - f^{(n-1)}(0)
\]
7. **Integrals Transform:**
\[
\mathcal{L}\left\{\int_0^t f(u) du\right\} = \frac{1}{s}\mathcal{L}\{f(t)\}
\]
8. **Frequency Differentiation:**
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