Find the Laplace transform of the following function, 0, t<1 1st<2 t-1, f(t) = -t+3, 2
Find the Laplace transform of the following function, 0, t<1 1st<2 t-1, f(t) = -t+3, 2
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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![**Problem Statement:**
Find the Laplace transform of the following function:
\[
f(t) =
\begin{cases}
0, & t < 1 \\
t - 1, & 1 \leq t < 2 \\
-t + 3, & 2 \leq t < 3 \\
0, & t \geq 3
\end{cases}
\]
**Explanation:**
This piecewise function \(f(t)\) is defined over four different intervals of the variable \(t\):
1. For \(t < 1\), the function is \(f(t) = 0\).
2. For \(1 \leq t < 2\), the function is a linear equation \(f(t) = t - 1\).
3. For \(2 \leq t < 3\), the function changes to another linear equation \(f(t) = -t + 3\).
4. For \(t \geq 3\), the function returns to \(f(t) = 0\).
The task is to derive the Laplace transform for this piecewise function across its defined intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F773711d9-15ab-4ebd-b211-4dc06ccf5abd%2F7e6bc009-2be0-4e0a-ae65-685534f8ec75%2Ffz2rw12_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the Laplace transform of the following function:
\[
f(t) =
\begin{cases}
0, & t < 1 \\
t - 1, & 1 \leq t < 2 \\
-t + 3, & 2 \leq t < 3 \\
0, & t \geq 3
\end{cases}
\]
**Explanation:**
This piecewise function \(f(t)\) is defined over four different intervals of the variable \(t\):
1. For \(t < 1\), the function is \(f(t) = 0\).
2. For \(1 \leq t < 2\), the function is a linear equation \(f(t) = t - 1\).
3. For \(2 \leq t < 3\), the function changes to another linear equation \(f(t) = -t + 3\).
4. For \(t \geq 3\), the function returns to \(f(t) = 0\).
The task is to derive the Laplace transform for this piecewise function across its defined intervals.
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