Ost<4, find F(s)=L{f (t)} using the defintion. e #1.) Given f (t) = 1 , t24
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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SUBJECT:
![**Problem #1:**
Given the piecewise function \( f(t) \):
\[
f(t) =
\begin{cases}
e^{-\frac{1}{2}t + 2}, & \text{for } 0 \leq t < 4, \\
1, & \text{for } t \geq 4
\end{cases}
\]
Find \( F(s) = \mathcal{L}\{f(t)\} \) using the definition of the Laplace Transform.
**Solution:**
To solve this problem, you need to apply the definition of the Laplace Transform to each piece of the function \( f(t) \).
Remember, the Laplace Transform of a function \( f(t) \) is given by:
\[
\mathcal{L}\{f(t)\} = \int_0^{\infty} e^{-st} f(t) \, dt
\]
You'll need to calculate the integral for each piece of the function over its respective interval and then sum the results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f015db8-6cb7-43c0-87e4-4ae6163fffe7%2F5c0084e5-853d-4f27-80b3-c8b6e52c19af%2F8ox5sh9_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem #1:**
Given the piecewise function \( f(t) \):
\[
f(t) =
\begin{cases}
e^{-\frac{1}{2}t + 2}, & \text{for } 0 \leq t < 4, \\
1, & \text{for } t \geq 4
\end{cases}
\]
Find \( F(s) = \mathcal{L}\{f(t)\} \) using the definition of the Laplace Transform.
**Solution:**
To solve this problem, you need to apply the definition of the Laplace Transform to each piece of the function \( f(t) \).
Remember, the Laplace Transform of a function \( f(t) \) is given by:
\[
\mathcal{L}\{f(t)\} = \int_0^{\infty} e^{-st} f(t) \, dt
\]
You'll need to calculate the integral for each piece of the function over its respective interval and then sum the results.
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