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
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**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.
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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