Here we want to find
the inverse Laplace transform of big F of s
which will give us the function f of t.
So beginning with the given function big F of s
equals six divided by s to the fourth,
we'll now take the inverse Laplace transform
of both sides of the equation to find f of t.
So the inverse Laplace transform of big F of s
equals the inverse Laplace transform
of six divided by s to the fourth.
And looking at our table of Laplace transforms,
notice how we're given big F of s,
and we're trying to find f of t,
which means we'll be using this table
in this direction here.
So notice how the inverse Laplace transform
of big F of s is equal to f of t.
So on the left side we do have f of t.
And now we need to work on determining
the inverse Laplace transform
of six divided by s to the fourth.
So looking at the column for big F of s,
we want to determine which form fits
six divided by s to the fourth.
Well because our denominator is s to the fourth,
we'll be using this row here
to determine f of t.
Focusing on the denominator,
notice how we have s to the fourth,
which should be s raised to the power of three plus one,
which means n is equal to three.
Which means we must have three factorial in our numerator.
And let's just say we don't recognize
that six is equal to three factorial.
We could rewrite this right side as six times
the inverse Laplace transform of one divided by,
let's write s to the fourth
as s to the power of three plus one.
In this form again, we know we need
three factorial in the numerator.
So what we could do is rewrite one
as three factorial divided by three factorial.
Let's go ahead and do that.
Three factorial divided by three factorial.
And then we'll factor out one over three factorial
so it fits this form perfectly.
So we'd have f of t is equal to
six divided by three factorial
times the inverse Laplace transform of three factorial
divided by s raised to the power of three plus one.
So again, now this fits the form perfectly.
And because n is equal to three,
this inverse Laplace transform is equal to t to the third.
So we have f of t equals, again three factorial is
three times two times one, which is six.
So we have six divided by six, which equals one.
So f of t is just t to the third.
Again we did do some extra work here
because six divided by s to the fourth
fit this form perfectly, but many times it won't.
So let's take a look at one more example.
Here we're given big F of s equals eight divided by s cubed.
We want to use inverse Laplace transforms to find f of t.
So again, we're given the function big F of s
equals eight divided by s to the third.
So to find f of t, we'll take
the inverse Laplace transform of both sides.
So the inverse Laplace transform of big F of s
equals the inverse Laplace transform
of eight divided by s to the third.
And once again, we are going to use
this row of the table to find f of t.
Well, on the left, the inverse Laplace transform
of big F of s is f of t.
And now on the right side, because we have s to the third,
we'd have s raised to the power of two plus one.
So n equals two, which means
the numerator must be two factorial.
Notice right now we have an eight,
so let's rewrite the right side as eight
times the inverse Laplace transform of one
divided by s raised to the power of two plus one.
And again, because we need a two factorial in the numerator,
let's write one
as two factorial divided by two factorial.
I know we could have written eight
as four times two factorial,
but this method always works
regardless of what the numerator might be.
And now we'll factor out one over two factorial,
which will leave us with just
two factorial in the numerator.
So f of t is equal to eight divided by two factorial
times the inverse Laplace transform of two factorial
divided by s raised to the power of two plus one.
Again now this fits the form perfectly, where n equals two.
So the inverse Laplace transform is just t squared.
So we have f of t equals,
two factorial is equal to two times one, or two.
Eight divided by two is equal to four.
So f of t equals four t squared.
I hope you found these two examples helpful.