Inverse Laplace transform of 3/(s^2 + 9)

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/(s² + 9).

ℒ⁻¹{3s² + 9} =

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.

sin(bt) → bs² + b²

Read the table backwards, piece by piece: sin(3t).

f(t) = sin(3t)

Let's check: the table forwards turns our answer back into 3/(s² + 9).

Right!

forwards: sin(3t) → 3s² + 9

Let's look back.

First: use the table backwards.

Last: check.

Answer

f(t) = sin(3t)

Try this one yourself, then check:

ℒ⁻¹{2s² + 4}

Show the answer

f(t) = sin(2t)