Before we begin, we note that it’s very tempting to think that because we know the Laplace transforms of both
and
we can simply multiply those together to get the desired Laplace transform. However, this is not the case, just as similar statements were not true for finding the derivatives and integrals of the products of functions. Rather, we will need to use property
, with
and
We need to know what
is before we can proceed. Let’s go back to the naming system we have instituted. If we have a capital
that is the Laplace transform of a function lower case
We identified that function previously:
We use
to find its Laplace transform.
Then we continue finding by taking two derivatives (using the quotient rule for derivatives; details are omitted here).