We're going to find the inverse Laplace
transform of big F of s in order to find
the function f of t.
So beginning with the beginning function, big F of s
equals two s divided by the quantity s squared plus 16
plus five divided by the quantity s squared plus 16.
We'll take the inverse Laplace transform of both sides
of the equation which will give us
the inverse Laplace transform of big F of s
equals the inverse Laplace transform
of two s divided by the quantity s squared plus 16
plus the inverse Laplace transform of
five divided by the quantity s squared plus 16.
In looking at our table of Laplace transforms,
because we're given the function big F of s,
and we're trying to find the function f of t,
we'll be using the table in this direction here.
So on the left side of our equation,
the inverse Laplace transform of big F of s
is equal to f of t, which is what we're looking for.
So we have f of t equals.
And when you figure out which formula to use
to find the inverse Laplace transforms on the right side,
notice how both the denominators are the quantity s squared
plus 16, and here we have a numerator of two s
and here we have a numerator of five.
So we'll be using these two rows in order to determine
the inverse Laplace transforms.
Notice how here we have a numerator of s and here we have
a numerator of a which is a constant.
So let's re-write the right side
as two times the inverse Laplace transform
of s divided by the quantity s squared plus 16
plus five times the inverse Laplace transform
of one divided by the quantity s squared plus 16.
So now looking at this inverse Laplace transform,
notice how it fits this form perfectly.
The numerator is s and the denominator
is s squared plus a squared, where a squared is equal to 16,
meaning a equals four.
But now looking at this inverse Laplace transform.
And again we know that s squared plus a squared
is equal to s squared plus 16, so because a equals four,
the numerator must be four as well.
And right now it's one.
So let's re-write one as the fraction four over four.
Now what we can do here is factor out one fourth,
so we just have four in the numerator.
So let's rewrite this one more time.
We have f of t equals two times
the inverse Laplace transform of s
divided by the quantity s squared.
Let's write 16 as plus four squared
so it fits the form perfectly.
Now here we're going to factor out 1/4, so we'll have
plus 5/4 times the inverse Laplace transform of.
Factor out 1/4 we're left with four,
divided by the quantity s squared.
Again, let's write 16 as four squared.
So again, notice how both inverse Laplace transforms
now fit these forms perfectly,
where in both cases a equals four.
Which means f of t is equal to two times
this inverse Laplace transform.
Which again, because a is equal to four
is equal to cosine four t.
And we have plus 5/4.
And this inverse Laplace transform is in this form
where a equals four so the inverse Laplace transform
is sin four t.
So we have 5/4 sine four t.
So by using inverse Laplace transforms,
we're able to find the function f of t,
which is what we enter here.
Two times cosine four t
plus 5/4 times sine four t.
I hope you found this helpful.