Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem 6: Invert this Laplace transform.**
Given the function:
\[ F(s) = \frac{8}{s^3 + 4s} + \frac{e^{-3(s+1)}}{s + 1} \]
The task is to find the inverse Laplace transform of \( F(s) \).
**Explanation:**
1. **First Term:** \(\frac{8}{s^3 + 4s}\)
- This term suggests a partial fraction decomposition may be useful for inversion. The polynomial in the denominator can be factored, potentially assisting in finding the inverse transform using standard Laplace inverse tables.
2. **Second Term:** \(\frac{e^{-3(s+1)}}{s + 1}\)
- This term involves an exponential function, which indicates a time shift. The expression \( e^{-3(s+1)} \) incorporates a shift in the time domain. It will affect the inverse transformation by delaying the outcome.
In solving for the inverse Laplace transform, consider applying the partial fraction decomposition and use standard inverse Laplace transform formulas, such as those for exponential shifts and simple rational functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6cee183-215d-43ed-a508-06b044f7e9eb%2Fdb15bef1-3a4e-4f9f-a770-ef1fecd6004a%2Fru8h2qc_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 6: Invert this Laplace transform.**
Given the function:
\[ F(s) = \frac{8}{s^3 + 4s} + \frac{e^{-3(s+1)}}{s + 1} \]
The task is to find the inverse Laplace transform of \( F(s) \).
**Explanation:**
1. **First Term:** \(\frac{8}{s^3 + 4s}\)
- This term suggests a partial fraction decomposition may be useful for inversion. The polynomial in the denominator can be factored, potentially assisting in finding the inverse transform using standard Laplace inverse tables.
2. **Second Term:** \(\frac{e^{-3(s+1)}}{s + 1}\)
- This term involves an exponential function, which indicates a time shift. The expression \( e^{-3(s+1)} \) incorporates a shift in the time domain. It will affect the inverse transformation by delaying the outcome.
In solving for the inverse Laplace transform, consider applying the partial fraction decomposition and use standard inverse Laplace transform formulas, such as those for exponential shifts and simple rational functions.
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