Step by step
The question
Question: find the inverse Laplace transform of 2/(s − 1)².
ℒ⁻¹{2(s − 1)2} =
The inverse transform goes back: which function of t has this transform?
We read the table backwards.
ℒ⁻¹: back from s to t
1 Use the table backwards
Each piece is already a row of the table.
tne^(at) → n!(s − a)n + 1
Read the table backwards, piece by piece: 2te^(t).
f(t) = 2te^(t)
2 Check
Let's check: the table forwards turns our answer back into 2/(s − 1)².
Right!
forwards: 2te^(t) → 2(s − 1)2
Let's look back.
First: use the table backwards.
Last: check.
Answer
f(t) = 2te^(t)