Step by step
The question
Question: solve y′ − 3y = 0, with y(0) = 2, using the Laplace transform.
y′ − 3y = 0
y(0) = 2
1 Transform both sides
Call the transform of y, Y.
The derivative rules bring in the starting values: that is why Laplace is so handy here.
ℒ{y} = Y
ℒ{y′} = sY − y(0)
Take the transform of both sides, term by term.
ℒ{y′} − 3ℒ{y} = 0
Use the rules with y(0) = 2.
(sY − 2) − 3Y = 0
Careful!
Don't forget the starting value: ℒ{y′} is sY minus y(0).
Careful: not ℒ{y′} = sY ✗
2 Solve for Y
Put the Y terms together: Y(s − 3).
Y(s − 3) − 2 = 0
Divide by s − 3.
Y = 2(s − 3)
3 Use the table backwards
Each piece is already a row of the table.
e^(at) → 1s − a
Read the table backwards, piece by piece: 2e^(3t).
y = 2e^(3t)
4 Check
Let's check the starting values: y(0) = 2.
Right!
y(0) = 2
And put y into the equation: it gives 0, the right side.
Right!
put in: 6e^(3t) − 3 × 2e^(3t) = 0
Let's look back.
First: transform both sides.
Then: solve for Y.
Then: use the table backwards.
Last: check.
Answer
y = 2e^(3t)