Step by step
The question
Question: find the Laplace transform of 5.
ℒ{5} =
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.
1 → 1s
2 Split it up
The transform of a sum is the sum of the transforms.
A number in front waits outside.
= 5ℒ{1}
ℒ{af + bg} = aℒ{f} + bℒ{g}
3 Use the table
Each part by its row: 1 gives 1 over s.
= 5 × 1s
4 Tidy up
Multiply the numbers: 5 over s.
= 5s
5 Check
Let's check: read the table backwards.
5/s comes from 5: the question.
Right!
backwards: 5s → 5
Let's look back.
First: the idea.
Then: split it up.
Then: use the table.
Then: tidy up.
Last: check.
Answer
= 5s