Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem Statement:**
Find the Laplace transform of the given function:
\[
f(t) =
\begin{cases}
0, & t < 2 \\
t^2 - 4t + 8, & t \geq 2
\end{cases}
\]
\[
\mathcal{L}\{f(t)\} = \, ? , \quad s > 0
\]
**Explanation:**
The problem requires finding the Laplace transform of a piecewise function \( f(t) \), which is defined as follows:
- \( f(t) = 0 \) for \( t < 2 \)
- \( f(t) = t^2 - 4t + 8 \) for \( t \geq 2 \)
The Laplace transform, denoted by \( \mathcal{L} \{ f(t) \} \), is a tool in mathematics and engineering used to transform a function of time \( f(t) \) into a function of a complex variable \( s \).
The solution will involve dealing with the piecewise nature of \( f(t) \) and calculating its Laplace transform for the specified condition \( s > 0 \).](https://content.bartleby.com/qna-images/question/d5655175-49d2-4db6-a7fd-785d70a02cda/bd000c7d-a973-40e7-8b53-878950fac4af/mpe3lui_processed.png)
Transcribed Image Text:**Problem Statement:**
Find the Laplace transform of the given function:
\[
f(t) =
\begin{cases}
0, & t < 2 \\
t^2 - 4t + 8, & t \geq 2
\end{cases}
\]
\[
\mathcal{L}\{f(t)\} = \, ? , \quad s > 0
\]
**Explanation:**
The problem requires finding the Laplace transform of a piecewise function \( f(t) \), which is defined as follows:
- \( f(t) = 0 \) for \( t < 2 \)
- \( f(t) = t^2 - 4t + 8 \) for \( t \geq 2 \)
The Laplace transform, denoted by \( \mathcal{L} \{ f(t) \} \), is a tool in mathematics and engineering used to transform a function of time \( f(t) \) into a function of a complex variable \( s \).
The solution will involve dealing with the piecewise nature of \( f(t) \) and calculating its Laplace transform for the specified condition \( s > 0 \).
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