Step by step
The question
Question: solve y″ + 9y = 0, with y(0) = 0 and y′(0) = 3, using the Laplace transform.
y″ + 9y = 0
y(0) = 0, y′(0) = 3
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)
ℒ{y″} = s2Y − sy(0) − y′(0)
Take the transform of both sides, term by term.
ℒ{y″} + 9ℒ{y} = 0
Use the rules with y(0) = 0 and y′(0) = 3.
(s2Y − 0 − 3) + 9Y = 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(s² + 9).
Y(s2 + 9) − 3 = 0
Divide by s² + 9.
Y = 3(s² + 9)
3 Use the table backwards
Each piece is already a row of the table.
sin(bt) → bs² + b²
Read the table backwards, piece by piece: sin(3t).
y = sin(3t)
4 Check
Let's check the starting values: y(0) = 0 and y′(0) = 3.
Right!
y(0) = 0, y′(0) = 3
And put y into the equation: it gives 0, the right side.
Right!
put in: −9sin(3t) + 9 × sin(3t) = 0
Let's look back.
First: transform both sides.
Then: solve for Y.
Then: use the table backwards.
Last: check.
Answer
y = sin(3t)