Inverse Laplace transform of 3/(s + 4)

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

Step by step

Question: find the inverse Laplace transform of 3/(s + 4).

ℒ⁻¹{3s + 4} =

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

We read the table backwards.

ℒ⁻¹: back from s to t

Each piece is already a row of the table.

e^(at) → 1s − a

Careful!

s + 4 means a = −4: the opposite sign of the number in the bracket.

Careful: not 3/(s + 4) → e^(4t) ✗

Read the table backwards, piece by piece: 3e^(−4t).

f(t) = 3e^(−4t)

Let's check: the table forwards turns our answer back into 3/(s + 4).

Right!

forwards: 3e^(−4t) → 3s + 4

Let's look back.

First: use the table backwards.

Last: check.

Answer

f(t) = 3e^(−4t)

Try this one yourself, then check:

ℒ⁻¹{2s + 8}

Show the answer

f(t) = 2e^(−8t)