Solve y'' + 9y = 18, y(0) = 1, y'(0) = 0 using Laplace

Step by step

The question

Question: solve y″ + 9y = 18, with y(0) = 1 and y′(0) = 0, using the Laplace transform.

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.

Take the transform of both sides, term by term.

Use the rules with y(0) = 1 and y′(0) = 0.

On the right, the table: 18 over s.

Careful!

Don't forget the starting value: ℒ{y′} is sY minus y(0).

2 Solve for Y

Put the Y terms together: Y(s² + 9).

Move the starting part to the right.

One fraction on the right: times s by s over itself.

Divide by s² + 9.

3 Partial fractions

Split it into simple fractions, one for each factor at the bottom.

Over s² + b², the top can have an s.

Times both sides by the whole bottom s(s² + 9).

Each letter keeps the factors it does not have.

Put s = 0: every other letter is multiplied by 0.

18 = 9 × A.

So A = 2.

Now compare the s² parts on both sides.

So B = −1.

And the s parts.

So C = 0.

So it splits into 2/s − s/(s² + 9).

4 Use the table backwards

Each piece is already a row of the table.

Read the table backwards, piece by piece: 2 − cos(3t).

5 Check

Let's check the starting values: y(0) = 1 and y′(0) = 0.

Right!

And put y into the equation: it gives 18, the right side.

Right!

Let's look back.

First: transform both sides.

Then: solve for Y.

Then: partial fractions.

Then: use the table backwards.

Last: check.