Inverse Laplace transform of 3s/(s^2 + 25)

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 3s/(s² + 25).

ℒ⁻¹{3ss² + 25} =

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: 3cos(5t).

f(t) = 3cos(5t)

Let's check: the table forwards turns our answer back into 3s/(s² + 25).

Right!

forwards: 3cos(5t) → 3ss² + 25

Let's look back.

First: use the table backwards.

Last: check.

Answer

f(t) = 3cos(5t)

Try this one yourself, then check:

ℒ⁻¹{5ss² + 25}

Show the answer

f(t) = 5cos(5t)