- [Instructor] Solving a differential equation
using the Laplace transform,
we find big Y of s equals the Laplace transform of y
to be big Y of s.
We want to find y of t, which means you want to find
the inverse Laplace transform of big Y of s.
To do this, our first step is
to take the inverse Laplace transform
of both sides of this equation.
So I'd have the inverse Laplace transform
of big Y of s equals,
let's write this first Laplace transform
as the opposite of the inverse Laplace transform
of one divided by quantity s minus eight,
and then minus the inverse Laplace transform
of 72 divided by s to the fifth
minus the inverse Laplace transform of four
divided by the quantity s minus eight squared plus 16.
Now let's look at our table of Laplace transforms.
Notice how we have y of t in the first column,
and big Y of s in the second column.
So because we're determining inverse Laplace transforms,
we'll use this table in this direction here.
Let's see if we can define the formulas we'll use
to determine these inverse Laplace transforms.
In our cell here we have
one over the quantity s minus eight,
which fits the form of one over the quantity s minus a,
well in our case a equals eight,
so we'll use this formula
for this first inverse Laplace transform.
Here we have a denominator of s to the fifth
which fits the form of s raised to power of n
plus one in the denominator,
so we'll use this formula
for the second inverse Laplace transform.
And for this last inverse Laplace transform,
notice how we have a denominator
that is the quantity s minus eight squared plus 16.
So looking at these two formulas here,
notice how because our numerator is four,
which is a constant,
we'll be using this formula here
to find the last inverse Laplace transform.
Notice this formula has the quantity
s minus a in the numerator.
Now let's work on making these inverse Laplace transforms
fit the form of these formulas.
So to begin, the inverse Laplace transform
of big Y of s equals y of t,
which is what we're looking for.
So y of t is equal to,
notice how this inverse Laplace transform
fits this formula perfectly where a equals eight,
so we'll go ahead and leave this in its current form.
So we'll have the opposite of the inverse Laplace transform
of one divided by the quantity s minus eight.
Now here, let's go ahead and factor out the 72.
Let's write this as minus 72
times the inverse Laplace transform of,
and looking at our formula, notice how the denominator
is s raised to power of n plus one.
Let's write this as s raised to power of four plus one
so we can see n equals four,
and notice how our numerator must be n factorial,
and right now it's actually one,
so let's write one as four factorial
divided by four factorial.
In this form, we do have a four factorial in the numerator,
but in our next step we'll have to factor out
one over four factorial
so the numerator is just four factorial, not equal.
And now for the last inverse Laplace transform,
looking at the denominator, notice how the form
is the quantity s minus a squared, so a equals eight,
and then we have plus b squared where b squared is 16.
So let's write the denominator
as the quantity s minus eight square plus four squared.
So again, we can easily see a equals eight
and b equals four.
Notice how the numerator
must be equal to b which in this case it
is because the numerator is already four.
So we can leave this inverse Laplace transform
in its current form.
So now our next step,
let's leave this
first and third inverse Laplace transform the same,
and here we'll factor out one over four factorial.
If we factor out one over four factorial,
we have minus 72 divided by four factorial
times the inverse Laplace transform of,
and now we have a numerator of four factorial
and a denominator of s raised to power of four plus one,
and now it fits the formula perfectly.
And this terms stays the same.
So we have y of t equals.
Here we'll have the opposite
of this inverse Laplace transform
where a equals eight is going to be
e raised to power of eight t.
So we have the opposite of e raised to power of eight t.
Let's go ahead and simplify 72 divided by four factorial.
That'd be 72 divided by four
times three times two times one,
or 72 divided by 24, which equals three.
So we have minus three, the inverse Laplace transform
where n equals four is t the fourth,
so we have minus three t to the fourth
and then finally we have
minus this inverse Laplace transform
where again a equals eight and b equals four
would be e raised to power of eight t times sine of four t.
So this is our function y of t,
which is the inverse Laplace transform of big Y of s.
Going back to our first slide,
this is what they're asking for here.
We have the opposite of e raised to power of eight t
minus three t to the fourth minus e to the eight t
times sine four t.
I hope you found this helpful.