Inverse Laplace transform of 8/(s(s^2 + 4))

Step by step

The question

Question: find the inverse Laplace transform of 8/(s(s² + 4)).

The inverse transform goes back: which function of t has this transform?

We read the table backwards.

1 Partial fractions

Split it into simple fractions, one for each factor at the bottom.

Over s² + b², the top can have an s.

Times both sides by the whole bottom s(s² + 4).

Each letter keeps the factors it does not have.

Put s = 0: every other letter is multiplied by 0.

8 = 4 × A.

So A = 2.

Now compare the s² parts on both sides.

So B = −2.

And the s parts.

So C = 0.

So it splits into 2/s − 2s/(s² + 4).

Careful!

A product of transforms is not the transform of a product.

Split it first.

2 Use the table backwards

Each piece is already a row of the table.

Read the table backwards, piece by piece: 2 − 2cos(2t).

3 Check

Let's check: the table forwards turns our answer back into 2/s − 2s/(s² + 4).

Right!

And the split is right: at s = 1, the question and the pieces both give 1.6.

Right!

Let's look back.

First: partial fractions.

Then: use the table backwards.

Last: check.