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wmboyles / Math-Summaries

Medium length summaries of math subjects like multivariable calculus. Longer than a formula sheet, shorter than a textbook.
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"Differential Equations" Laplace Transforms #27

Open Suzukazole opened 3 years ago

Suzukazole commented 3 years ago

More properties of the Laplace Transform like convolution, integration could be added in a future version so that the section is more complete. Sometimes, these properties come in handy while taking Laplace transforms of some standard functions that may appear in a Differential equation too.

wmboyles commented 3 years ago

Adding these would probably be a good idea. Looking through Paul's Notes on convolution integrals, it seems like they would extend some of the discussion on partial fractions well.

However, the class on which the diffeq text is based is is some sense part one of two: students who only need more basic diffeq might only take the first class, while students who need to know more (like electrical engineers I think) will take the second class. So, it's possible that things like convolutions are covered in this class.

I'll ask around to see if convolutions are covered in this next class (update: they don't). If not, I can start working on including them. Otherwise, I'll probably hold off because I'll likely take the next class and then be able to make a better judgement on where to include it.

In the meantime, could you link what you think are some good resources on the topics?

Suzukazole commented 3 years ago

In the meantime, could you link what you think are some good resources on the topics?

The text recommended for our intro to DEs class was "Differential Equations with Applications and Historical Notes" By G.F. Simmons, which had a nice chapter on Laplace Transforms and covered the topics that I had mentioned as well.

Apart from that I found these online,

  1. Short Intro to convolution by Prof.Strang.
  2. I highly recommend this playlist by Prof.Mattuck, in particular, Lectures 19 through 23 cover Laplace transforms and convolutions.
  3. Notes for convolution from the above course.
wmboyles commented 3 years ago

@Suzukazole, what do you think about the added section? Is it worth getting into topics like the delta and Heaviside step functions?

Suzukazole commented 3 years ago

Looks good! Delta and step functions do show up quite a bit in differential equations, so maybe a section for that could be added later.