[39] (a) u(t-3)(t-8) +2u(t-7) cos(xt) (b) (i-8 t-8+2 cos(xt) t<3, 3
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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Hello, the first image has the questions and the second image has the provided answers. Could you please walk me through this problem? The answers attached have been confirmed to be correct, so I just need to know how to get from the questions to the answers. Thank you!
![**Transcription for Educational Website:**
**Problem 39**
**(a)** \( u(t - 3)(t - 8) + 2u(t - 7) \cos(\pi t) \)
**(b)**
\[
\begin{cases}
0 & t < 3, \\
t - 8 & 3 \le t < 7, \\
t - 8 + 2 \cos(\pi t) & t \ge 7.
\end{cases}
\]
**Explanation:**
This problem consists of two parts: (a) and (b).
- **Part (a)**
- The expression involves a unit step function \( u(t) \) applied to \( (t - 3) \) and \( (t - 7) \), multiplied by \((t - 8)\) and an oscillating function \( 2 \cos(\pi t) \).
- **Part (b)**
- This piecewise function defines different expressions for \( t \) in various intervals:
- For \( t < 3 \), the function evaluates to 0.
- For \( 3 \le t < 7 \), the function is linear, \( t - 8 \).
- For \( t \ge 7 \), the function combines both linear and oscillatory components, \( t - 8 + 2 \cos(\pi t) \).
These problems often appear in contexts involving piecewise functions and unit step functions, which are useful for modeling signals and systems in mathematics and engineering disciplines.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F752c80f7-6a34-4004-b4db-3d79d0db3666%2F51ec6f76-3111-4ae3-acde-70c0c1df3841%2F3fv0niy_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
**Problem 39**
**(a)** \( u(t - 3)(t - 8) + 2u(t - 7) \cos(\pi t) \)
**(b)**
\[
\begin{cases}
0 & t < 3, \\
t - 8 & 3 \le t < 7, \\
t - 8 + 2 \cos(\pi t) & t \ge 7.
\end{cases}
\]
**Explanation:**
This problem consists of two parts: (a) and (b).
- **Part (a)**
- The expression involves a unit step function \( u(t) \) applied to \( (t - 3) \) and \( (t - 7) \), multiplied by \((t - 8)\) and an oscillating function \( 2 \cos(\pi t) \).
- **Part (b)**
- This piecewise function defines different expressions for \( t \) in various intervals:
- For \( t < 3 \), the function evaluates to 0.
- For \( 3 \le t < 7 \), the function is linear, \( t - 8 \).
- For \( t \ge 7 \), the function combines both linear and oscillatory components, \( t - 8 + 2 \cos(\pi t) \).
These problems often appear in contexts involving piecewise functions and unit step functions, which are useful for modeling signals and systems in mathematics and engineering disciplines.
![[39] Find the inverse Laplace transform of \( G(s) = e^{-3s}(-5s^{-1} + s^{-2}) - 2e^{-7s} \frac{s}{(s^2 + \pi^2)} \).
(a) Express the answer using the symbols of unit step functions.
(b) Express the answer piecewise.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F752c80f7-6a34-4004-b4db-3d79d0db3666%2F51ec6f76-3111-4ae3-acde-70c0c1df3841%2Fncj64mv_processed.png&w=3840&q=75)
Transcribed Image Text:[39] Find the inverse Laplace transform of \( G(s) = e^{-3s}(-5s^{-1} + s^{-2}) - 2e^{-7s} \frac{s}{(s^2 + \pi^2)} \).
(a) Express the answer using the symbols of unit step functions.
(b) Express the answer piecewise.
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