Monday Math 108

The Laplace Transform
Part 9: Derivatives and Integrals

Now, let’s examine the Laplace transform’s action on derivatives and integrals. Let us have function f(t) with Laplace transform:
.
What, then, is the Laplace transform of f’(t)?

Using the definition of the Laplace transform,
.
Now, we perform integration by parts, with , , giving us , , and so we have
,
but for our region of convergence, the limit must be zero, and we have that . Plugging into the above, we see
,

(Note, when there is a discontinuity in f(t) at t=0, then we replace f(0) in the above formula with ; contrast to here.)

Consider then the Laplace transform of the second derivative f”(t). We can apply the above rule twice; since ,
.
Extending on, we see the Laplace transform of the nth derivative of f(t) is
.

This gets us the prime use of the Laplace transform: it turns differential equations, particularly initial value problems, into algebraic equations. For example, the inhomogeneous differential equation , , where H(t) is the Heaviside step function (this corresponds to the charging of a previously uncharged RC circuit via a battery connected at t=0).
Naming the Laplace transform of y(t) as Y(s), we take the Laplace transform of both sides of our differential equation:
,
and our solution is simply a matter of finding the inverse transform of that last term. Using partial fractions,
,
and from here, we see , and so
.

Next, we consider integrals. Let us have, as before, f(t), t>0, with Laplace transform F(s). Now, let us define . Then g(0)=0 and g’(t)=f(t). Letting us write the Laplace transform of g(t) as G(s), we apply our derivative rule to the transform of g’(t) to get:
,
but g’(t)=f(t), so
. Solving for G(s), we then see
,
and we have the formula for integrals.

Note, we could also have found this by noting that , where the * denotes convolution, and by our result here regarding the Laplace transform of convolutions,
,
as above.

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6 Responses to “Monday Math 108”

  1. Dr.D Says:

    So, what is your point? Are you just copying all of this out of a book? Are you re-deriving it for your own interest? Where are you going with all of this?

    I have yet to see anything original here, so I am pretty puzzled about it all.

    • twistedone151 Says:

      If you’d read some of the other Monday Math (or Physics Friday) posts on this blog, you’d note that they are mostly informative showcases for problem solutions I find interesting, and methods I find useful, often with a certain degree of definitions, background, and supporting proofs. They’re generally not intended to be original in the concepts or methods displayed, only in the specifics of the presentation (they are my own words and phrasings). As to a book, I’m doing these pretty much from memory, without any particular book or source.

  2. Monday Math 109 « Twisted One 151's Weblog Says:

    […] Math 109 By twistedone151 The Laplace Transform Part 10: Initial and Final Value Theorems Last time, we found the formula for the Laplace transform of the derivative of a function; if we have f(t), […]

  3. Monday Math 110 « Twisted One 151's Weblog Says:

    […] By twistedone151 The Laplace Transforn Part 11: Frequency Derivatives and Integrals We found previously what happens when one takes the Laplace transform of the derivative of a function, , and what […]

  4. Monday Math 113 « Twisted One 151's Weblog Says:

    […] , we see , and so . We can also use this to find the transform of the error function, via our integration rule: (also with […]

  5. Monday Math 114 « Twisted One 151's Weblog Says:

    […] Part 15: Applying the Laplace Transform – Differential Equations As was briefly noted in this part, the primary use of Laplace transforms is in solving linear, ordinary differential equations, as […]

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