Laplace Transform piecewise function with domain from 1 to inff(t)={t−11≤t&lt;23−t2≤t&lt;30t≥3

Step 1Set&nbsp;b(t)=u(t)−u(t−1)&nbsp;as the unit box or square function with support&nbsp;[0,&nbsp;1]. Then your function can be represented asf(t)=(b×b)(t−1),check it that the convolution of b with itself gives the hat function. Now apply standard Laplace rulesL{f(t)}(s)=e-sL{(b×b)(t)}=e−s(L{b(t)})2etc.

Solved: "Laplace Transform piecewise function with domain from 1..."

High School
Calculus and Analysis
Calculus 2
Differential equations

Willie Kelley

Answered question

2022-04-15

Laplace Transform piecewise function with domain from 1 to inf

f (t) = {\begin{cases} t - 1 1 \leq t < 2 \\ 3 - t 2 \leq t < 3 \\ 0 t \geq 3 \end{cases}

Answer & Explanation

cadhail6n1t

Beginner2022-04-16Added 14 answers

Step 1
Set $b (t) = u (t) - u (t - 1)$ as the unit box or square function with support $[0, 1]$ . Then your function can be represented as
$f (t) = (b \times b) (t - 1),$
check it that the convolution of b with itself gives the hat function. Now apply standard Laplace rules
$L {f (t)} (s) = e^{- s} L {(b \times b) (t)}$
$= e^{- s} (L {b {(t)})}^{2}$
etc.

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