Inverse Laplace transform of s/(s^2 + 16)

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 s/(s² + 16).

ℒ⁻¹{ss² + 16} =

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.

cos(bt) → ss² + b²

Read the table backwards, piece by piece: cos(4t).

f(t) = cos(4t)

Let's check: the table forwards turns our answer back into s/(s² + 16).

Right!

forwards: cos(4t) → ss² + 16

Let's look back.

First: use the table backwards.

Last: check.

Answer

f(t) = cos(4t)

Try this one yourself, then check:

ℒ⁻¹{ss² + 25}

Show the answer

f(t) = cos(5t)