Find f(t). &{ 4s - 1 3 \s² (s + 1)³√ f(t) = 7H(t)-t-7e --(61-5-12) 6t- - e X
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
![**Problem Statement:**
Find \( f(t) \).
\[
\mathcal{L}^{-1} \left\{ \frac{4s - 1}{s^2(s+1)^3} \right\}
\]
**Solution:**
\[
f(t) = 7H(t) - t - 7e^{-t} - e^{-t} \left( 6t - \frac{5e^{-t}t^2}{2} \right)
\]
**Explanation:**
1. **Notation:**
- \(\mathcal{L}^{-1}\) denotes the inverse Laplace transform.
- \(H(t)\) represents the Heaviside step function.
2. **Equations:**
- The initial expression is in the Laplace domain, which we transform into the time domain.
- The final time-domain function \(f(t)\) includes exponential decay and polynomial terms.
This exercise involves applying the inverse Laplace transform to a rational function in the complex frequency domain for time-domain analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccae5729-7489-41ed-a32a-24cdd49b7598%2F4c23c9e1-51fc-40bd-ab72-3ca8e7c43e6b%2Fqkmx10v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find \( f(t) \).
\[
\mathcal{L}^{-1} \left\{ \frac{4s - 1}{s^2(s+1)^3} \right\}
\]
**Solution:**
\[
f(t) = 7H(t) - t - 7e^{-t} - e^{-t} \left( 6t - \frac{5e^{-t}t^2}{2} \right)
\]
**Explanation:**
1. **Notation:**
- \(\mathcal{L}^{-1}\) denotes the inverse Laplace transform.
- \(H(t)\) represents the Heaviside step function.
2. **Equations:**
- The initial expression is in the Laplace domain, which we transform into the time domain.
- The final time-domain function \(f(t)\) includes exponential decay and polynomial terms.
This exercise involves applying the inverse Laplace transform to a rational function in the complex frequency domain for time-domain analysis.
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