Step by step
The question
Question: find the inverse Laplace transform of 4/s + 2/(s + 3).
ℒ⁻¹{4s + 2s + 3} =
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 Split it up
One piece at a time.
= 4s + 2s + 3
2 Use the table backwards
Each piece is already a row of the table.
1 → 1s
e^(at) → 1s − a
Careful!
s + 3 means a = −3: the opposite sign of the number in the bracket.
Careful: not 2/(s + 3) → e^(3t) ✗
Read the table backwards, piece by piece: 4 + 2e^(−3t).
f(t) = 4 + 2e^(−3t)
3 Check
Let's check: the table forwards turns our answer back into 4/s + 2/(s + 3).
Right!
forwards: 4 + 2e^(−3t) → 4s + 2s + 3
Let's look back.
First: split it up.
Then: use the table backwards.
Last: check.
Answer
f(t) = 4 + 2e^(−3t)