Steply

Inverse Laplace transform of (2s - 4)/(s(s - 1))

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

Step by step

The question

Question: find the inverse Laplace transform of (2s − 4)/(s(s − 1)).

ℒ⁻¹{2s − 4s(s − 1)} =

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

We read the table backwards.

ℒ⁻¹: back from s to t

1 Partial fractions

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

2s − 4s(s − 1) = As + Bs − 1

Times both sides by the whole bottom s(s − 1).

Each letter keeps the factors it does not have.

2s − 4 = A(s − 1) + Bs

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

−4 = −1 × A.

s = 0: −4 = −1 × A

So A = 4.

A = 4

Put s = 1: every other letter is multiplied by 0.

−2 = 1 × B.

s = 1: −2 = 1 × B

So B = −2.

B = −2

So it splits into 4/s − 2/(s − 1).

2s − 4s(s − 1) = 4s − 2s − 1

Careful!

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

Split it first.

Careful: not ℒ⁻¹{1(s − a)(s − b)} = e^(at) × e^(bt) ✗

2 Use the table backwards

Each piece is already a row of the table.

1 → 1s
e^(at) → 1s − a

Read the table backwards, piece by piece: 4 − 2e^(t).

f(t) = 4 − 2e^(t)

3 Check

Let's check: the table forwards turns our answer back into 4/s − 2/(s − 1).

Right!

forwards: 4 − 2e^(t) → 4s − 2s − 1

And the split is right: at s = 2, the question and the pieces both give 0.

Right!

s = 2: 0 and 0

Let's look back.

First: partial fractions.

Then: use the table backwards.

Last: check.

Answer
f(t) = 4 − 2e^(t)

Your turn

Try this one yourself, then check:

ℒ⁻¹{2s − 3s(s − 1)}
Show the answer
f(t) = 3 − e^(t)

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