We're asked to find
the inverse Laplace transform of eight
divided by the product of s cubed and s plus two.
In order to use the table below
to determine the inverse Laplace transform,
we first need to perform partial fraction decomposition
on the given function.
When doing this, remember s cubed
is a repeated linear factor.
To set this up, we have eight
divided by the product of s cubed and s plus two
equals the constant A divided by one factor of s,
plus the constant B divided by two factors of s,
plus C divided by three factors of s,
and then finally, plus the constant D
divided by the last linear linear factor of s plus two.
The next step is to multiply
both sides of the equation by the LCD
which is s cubed times a quantity s plus two.
This gives us the equation eight equals A
times s squared times s plus two.
Notice when multiplying A over s
by s cubed times s plus two,
one factor of s simplifies out
and then we have plus Bs times the quantity s plus two.
When multiplying B over s squared
times s cubed times s plus two,
notice two factors of s simplify out.
Then we have plus C times the quantity s plus two.
Finally, plus D times s cubed.
Now we expand the right side of the equation
and equate the coefficients.
This gives us eight equals As cubed
plus two As squared
and then plus Bs squared plus two Bs,
plus Cs plus 2C plus Ds cubed.
Now let's gathered the s cubed terms,
s squared terms, s terms,
and constants on the right.
This gives us eight equals As cubed plus Ds cubed,
plus 2As squared plus Bs squared,
and then we have plus two Bs plus Cs plus 2C.
And now let's factor s cubed from the s cubed terms,
s squared from the s squared terms,
and s from the s terms.
This gives us eight equals
the quantity A plus D times s cubed,
plus the quantity 2A plus B times s squared,
plus the quantity 2B plus C times s, then plus 2C.
And now we equate the coefficients,
but notice on the left side of the equation
there are no s cubed terms,
s squared terms, or s terms
which indicates that A plus D must equal zero,
2A plus B must equal zero,
2B plus C must equal zero,
and finally, eight must equal 2C
indicating C is equal to four.
And then working our way back up.
If C is equal to four,
2B is equal to negative four
and therefore B is equal to negative two.
And if B is equal to negative two,
2A must equal positive two
and therefore A is equal to one.
And if A is equal to one
and A plus D is equal to zero,
D is equal to negative one.
And now performing substitution back into the top equation,
we now know the given function,
eight divided by the product of s cubed
and s plus two is equal to one divided by s
because A is equal to one.
And since B is equal to negative two,
we have minus two divided by s squared.
And because C is equal to four,
we have plus four divided by s cubed.
And because D is equal to negative one,
we have minus one divided by the quantity s plus two.
Now that we have the function in this form,
we can determine the inverse Laplace transform.
Let's do this on the next slide.
The inverse Laplace transform
of the given function eight divided by the product
of s cubed and s plus two
is equal to the inverse Laplace transform
of one divided by s minus two divided by s squared,
plus four divided by s cubed,
minus one divided by the quantity s plus two.
Using the property of linearity,
this is equal to the inverse Laplace transform
of one divided by s
minus two times the inverse Laplace transform
of one divided by s squared,
plus the inverse Laplace transform
of four divided by s cubed.
But looking at the table below,
the inverse Laplace transform of two divided by s cubed
is equal to t squared.
So I'm going to write this as minus two
times the inverse Laplace transform of two divided by cubed
and then finally, minus the inverse Laplace transform
of one divided by the quantity s plus two.
So again, now using our table below,
the inverse Laplace transform of one divided by s
where C is equal to one is equal to one
and then minus two times the inverse Laplace transform
of one divided by s squared
which is equal to t.
And then plus two times the inverse Laplace transform
of two divided by s cubed which is t squared
and then finally, minus the inverse Laplace transform
of one divided by the quantity s plus two.
Looking at the last formula in the first column,
the inverse Laplace transform of one
divided by the quantity s plus A
is equal to e to the power of negative at,
and therefore the inverse Laplace transform
of one divided by the quantity s plus two
is e to the power of negative 2t
which indicates the inverse Laplace transform,
the given function eight divided by the product
of s cubed and s plus two
is equal to one minus 2t plus 2t squared
minus e to the negative 2t.
I hope you found this helpful.