Steply

Inverse Laplace transform of (s + 3)/((s + 1)(s + 2))

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 (s + 3) over the product (s + 1)(s + 2).

ℒ⁻¹{s + 3(s + 1)(s + 2)} =

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.

s + 3(s + 1)(s + 2) = As + 1 + Bs + 2

Times both sides by the whole bottom (s + 1)(s + 2).

Each letter keeps the factors it does not have.

s + 3 = A(s + 2) + B(s + 1)

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

2 = 1 × A.

s = −1: 2 = 1 × A

So A = 2.

A = 2

Put s = −2: every other letter is multiplied by 0.

1 = −1 × B.

s = −2: 1 = −1 × B

So B = −1.

B = −1

So it splits into 2/(s + 1) − 1/(s + 2).

s + 3(s + 1)(s + 2) = 2s + 1 − 1s + 2

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.

e^(at) → 1s − a

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

f(t) = 2e^(−t) − e^(−2t)

3 Check

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

Right!

forwards: 2e^(−t) − e^(−2t) → 2s + 1 − 1s + 2

And the split is right: at s = 1, the question and the pieces both give 0.6667.

Right!

s = 1: 0.6667 and 0.6667

Let's look back.

First: partial fractions.

Then: use the table backwards.

Last: check.

Answer
f(t) = 2e^(−t) − e^(−2t)

Your turn

Try this one yourself, then check:

ℒ⁻¹{s + 8(s + 1)(s + 2)}
Show the answer
f(t) = 7e^(−t) − 6e^(−2t)

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