Step by step
The question
Question: find the Laplace transform of t².
ℒ{t2} =
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.
tn → n!sn + 1
2 Use the table
Each part by its row: t² gives 2/s³.
ℒ{t2} = 2!s3
Careful!
The top is n!: 2!
= 2, not 1.
Careful: not ℒ{t²} = 1s3 ✗
3 Tidy up
Multiply the numbers: 2/s³.
= 2s3
4 Check
Let's check: read the table backwards.
2/s³ comes from t²: the question.
Right!
backwards: 2s3 → t2
Let's look back.
First: the idea.
Then: use the table.
Then: tidy up.
Last: check.
Answer
= 2s3