0, 5 sin(nt), 4 < t < 5 0, tく4 Find the Laplace transform F(s) of f(t) = t > 5 F(s) = %3D

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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Find the Laplace transform \( F(s) \) of \( f(t) \) given by

\[
f(t) = 
\begin{cases} 
0, & t < 4 \\
5 \sin(\pi t), & 4 \leq t < 5 \\
0, & t \geq 5 
\end{cases}
\]

Calculate \( F(s) = \) ____ 

To solve this problem, identify the intervals over which the function is non-zero and apply the Laplace transform piecewise where necessary. The function \( f(t) \) is non-zero only in the interval \( 4 \leq t < 5 \). For this interval, the term is \( 5 \sin(\pi t) \). Apply the Laplace transform to this portion while accounting for the step functions that define the interval.
Transcribed Image Text:Find the Laplace transform \( F(s) \) of \( f(t) \) given by \[ f(t) = \begin{cases} 0, & t < 4 \\ 5 \sin(\pi t), & 4 \leq t < 5 \\ 0, & t \geq 5 \end{cases} \] Calculate \( F(s) = \) ____ To solve this problem, identify the intervals over which the function is non-zero and apply the Laplace transform piecewise where necessary. The function \( f(t) \) is non-zero only in the interval \( 4 \leq t < 5 \). For this interval, the term is \( 5 \sin(\pi t) \). Apply the Laplace transform to this portion while accounting for the step functions that define the interval.
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