**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.

ft=0+u1-0+t-1+u2-t+1-t+3+u3t-3+0where unt=0, t<n1, t≥nft=0+u1t-1+u2-2t+4+u3t-3Lft=L0+Lu1t-1+Lu2-2t+4+Lu3t-3Since Luaft=e-asLft+a, therefore:Lft=L0+e-1sLt+1-1+e-2sL-2t+2+4+e-3sLt+3-3=L0+e-1sLt+e-2sL-2t+e-3sLtSince Ltn=n!sn+1 and Laft=aLft, therefore:Lft=00!s0+1+e-s1!s1+1+e-2s-21!s1+1+e-3s1!s1+1Lft=e-ss2-2e-2ss2+e-3ss2Lft=e-s-2e-2s+e-3ss2

Answered: Find the Laplace transform of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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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.

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