As we saw in the last section
computing Laplace transforms directly can be
fairly complicated. Usually we just use
a table of transforms when actually computing Laplace transforms.
The table that is provided here is not an inclusive table, but does
include most of the commonly used Laplace transforms and most of the commonly
needed formulas pertaining to Laplace
transforms.
Before doing a couple of examples to illustrate the use of
the table let’s get a quick fact out of the way.
Fact
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Given f(t) and g(t) then,
for any constants a
and b.
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In other words, we don’t worry about constants and we don’t
worry about sums or differences of functions in taking Laplace
transforms. All that we need to do is
take the transform of the individual functions, then put any constants back in
and add or subtract the results back up.
So, let’s do a couple of quick examples.
Make sure that you pay attention to the difference between a
“normal” trig function and hyperbolic functions. The only difference between them is the “+ a2” for the “normal” trig
functions becomes a “- a2”
in the hyperbolic function! It’s very
easy to get in a hurry and not pay attention and grab the wrong formula. If you don’t recall the definition of the
hyperbolic functions see the notes for the table.
Let’s do one final set of examples.
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Example 2 Find
the transform of each of the following functions.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
Solution
(a)
This function is not in the table of Laplace
transforms. However we can use #30 in the table to compute its
transform. This will correspond to #30
if we take n=1.
So, we then have,
Using #30 we then have,
[Return to Problems]
(b)
This part will also use #30
in the table. In fact we could use #30
in one of two ways. We could use it
with .
Or we could use it with .
Since it’s less work to do one derivative, let’s do it the
first way. So using #9 we have,
The transform is then,
[Return to Problems]
(c)
This part can be done using either #6 (with ) or #32
(along with #5). We will use #32 so we can see an example of
this. In order to use #32 we’ll need
to notice that
Now, using #5,
we get the following.
This is what we would have gotten had we used #6.
[Return to Problems]
(d)
For this part we will use #24 along with the answer from the
previous part. To see this note that
if
then
Therefore, the transform is.
[Return to Problems]
(e)
This final part will again use #30 from the table as well as #35.
Remember that g(0)
is just a constant so when we differentiate it we will get zero!
[Return to Problems]
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As this set of examples has shown us we can’t forget to use
some of the general formulas in the table to
derive new Laplace transforms for functions that aren’t explicitly listed in
the table!