Step by step
The question
Question: find the Laplace transform of 2t⁴.
ℒ{2t4} =
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 Split it up
The transform of a sum is the sum of the transforms.
A number in front waits outside.
= 2ℒ{t4}
ℒ{af + bg} = aℒ{f} + bℒ{g}
3 Use the table
Each part by its row: t⁴ gives 24/s⁵.
= 2 × 4!s5
Careful!
The top is n!: 4!
= 24, not 1.
Careful: not ℒ{t⁴} = 1s5 ✗
4 Tidy up
Multiply the numbers: 48/s⁵.
= 48s5
5 Check
Let's check: read the table backwards.
48/s⁵ comes from 2t⁴: the question.
Right!
backwards: 48s5 → 2t4
Let's look back.
First: the idea.
Then: split it up.
Then: use the table.
Then: tidy up.
Last: check.
Answer
= 48s5