Step by step
The question
Question: find the Laplace transform of t²e^(−3t).
ℒ{t2e^(−3t)} =
1 The idea
The Laplace transform turns a function of t into a function of s.
It is an integral, but we almost never work it out: we use a table made from it.
ℒ{f(t)} = ∫ from 0 to ∞ of e^(−st) f(t) dt
The table rows we need.
tne^(at) → n!(s − a)n + 1
2 Use the table
Each part by its row: t²e^(−3t) gives 2/(s + 3)³.
ℒ{t2e^(−3t)} = 2!(s + 3)3
Careful!
The top is n!: 2!
= 2, not 1.
Careful: not ℒ{t²} = 1s3 ✗
3 Tidy up
Multiply the numbers: 2/(s + 3)³.
= 2(s + 3)3
4 Check
Let's check: read the table backwards.
2/(s + 3)³ comes from t²e^(−3t): the question.
Right!
backwards: 2(s + 3)3 → t2e^(−3t)
Let's look back.
First: the idea.
Then: use the table.
Then: tidy up.
Last: check.
Answer
= 2(s + 3)3