Step by step
The question
Question: solve y′ + y = 0, with y(0) = 5, using the Laplace transform.
y′ + y = 0
y(0) = 5
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′} + ℒ{y} = 0
Use the rules with y(0) = 5.
(sY − 5) + Y = 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 + 1).
Y(s + 1) − 5 = 0
Divide by s + 1.
Y = 5(s + 1)
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: 5e^(−t).
y = 5e^(−t)
4 Check
Let's check the starting values: y(0) = 5.
Right!
y(0) = 5
And put y into the equation: it gives 0, the right side.
Right!
put in: −5e^(−t) + 5e^(−t) = 0
Let's look back.
First: transform both sides.
Then: solve for Y.
Then: use the table backwards.
Last: check.
Answer
y = 5e^(−t)