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