Laplace transform of 1

Step by step, with the reason for every step · The Laplace transform · Differential Equations

Step by step

Question: find the Laplace transform of 1.

ℒ{1} =

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

Each part by its row: 1 gives 1 over s.

ℒ{1} = 1s

Let's check: read the table backwards.

1/s comes from 1: the question.

Right!

backwards: 1s → 1

Let's look back.

First: the idea.

Then: use the table.

Last: check.

Answer

ℒ{1} = 1s

Try this one yourself, then check:

Find the Laplace transform of ℒ{t}

Show the answer

ℒ = 1s²