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My question is in the title. Let me elaborate and give some context:

I'm teaching a first differential equations course, essentially for engineers, at the university. I'm developing the syllabus and content for the course. I'm at liberty to pretty much teach what I want; there's no departmental exam to teach to.

I will experiment with using computers - Mathematica by Wolfram, in particular - in order to bypass tedious computations and get to the meat of the subject, and my goal for the course is for every topic covered to have interesting, practical applications. This brings me to the Laplace Transform.

After studying mechanical vibration and resonance caused by a sinusoidal forcing function, it would be nice to also teach the students how to work with other periodic forcing functions - e.g. square waves & sawtooth waves - and Laplace Transforms are, to my knowledge, the best way to deal with these. Not to mention the dirac delta "impulse" functions, which are immensely practical.

However, after having tutored a student in a differential equations course earlier, and after having attempted to write notes on Laplace Transforms, I am face-to-face with the reality that this is a computationally-intense topic. Even worse, it is somewhat resilient to Mathematica. Mathematica can compute Laplace and Inverse Laplace Transforms, but the results are ugly and hard to interpret. I'm thinking that, perhaps, they are so hard to interpret that we might as well not even come up with the exact solution at all and just use a numerical ODE solver (Mathematica's NDSolve[] function).

I've been told the Laplace Transform is useful, but does its use justify the class time (out of a six week course, no less!) that would be necessary explaining the difficult-to-understand shifts between t-domain and s-domain and the staggering computations that accompany these shifts? Thoughts, experiences, references... all will be appreciated.

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    $\begingroup$ you should watch the MIT lectures on this topic. I don't have a link, but, it might make them seem easier... I would encourage you to not shy away from hand-to-hand combat with equations. Engineers need to be able to do such calculations and, honestly, the calculations involved in Laplace transforms are not hard. When you get to the algebra problem then they could use a CAS, but, to frame it, they have to think. If they are not willing to think, then, well, I'm out. Found the link: ocw.mit.edu/courses/mathematics/… $\endgroup$ – James S. Cook Jun 14 '16 at 23:58
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    $\begingroup$ Haha, "hand-to-hand combat with equations," I love it. Also, wow, I watched these videos years ago when I was learning it! Good memories! I forgot how cool Lecture 19 (Intro to Laplace) was. I will make it required viewing for my students. Thanks for your thoughts! $\endgroup$ – Jake Mirra Jun 16 '16 at 1:14
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    $\begingroup$ I have some notes at supermath.info/DifferentialEqns.pdf , if you want I have the source files and such in a google drive, you're welcome to cut and paste if anything is useful. Ideally, I'd like to create a more novel course which meaningfully uses differential forms and symmetries of differential equations. However, I'm a long ways from that goal for now... $\endgroup$ – James S. Cook Jun 16 '16 at 2:56
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Consider something besides an "all or nothing" approach.

Here's what I did a couple of times when the topic was optional and I didn't have much time, but I still wanted to give students an introduction to the method. Simply restrict yourself to introducing the method by defining the Laplace transform, computing it for some simple examples, explaining what the inverse Laplace transform is, and then showing students how to use the method to solve simple differential equations such as $y' – y = e^{-t}$ with $y(0) = 1,$ or $y'' + y = e^{-t}$ with $y(0) = y'(0) = 0.$ Incidentally, even when I covered the topic briefly in this manner, I still had time to discuss how the Laplace transform was a linear operator (all my students had either completed linear algebra or were currently taking linear algebra) and to emphasize how linearity allowed the method to work (e.g. take Laplace transform of both sides of the ODE, then apply linearity).

If you provide appropriate short handouts to your students that give them all the necessary basics, you should be able to cover this in 2 lectures. Incidentally, the two equations I gave above came directly off the last major test (from May 1999) in one of the classes that I covered Laplace transforms briefly in this way. I realize you are going to feel compelled to try and justify Laplace transforms by getting into sawtooth graphs and impulse functions and the like, but your students will likely see Laplace transforms later in an advanced undergraduate level mathematical methods for engineers course (and perhaps also in a later mechanics course). Even a brief introduction to Laplace transforms now will help them a lot when they see it later.

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    $\begingroup$ Thanks for your answer, this helps a great deal. I think I will essentially follow the advice to show them how it works for simple equations and test them on those so that they merely have an introduction to the topic - and let their more advanced engineering classes do the rest. Thanks so much! $\endgroup$ – Jake Mirra Jun 16 '16 at 1:17
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Also having a short period of time to introduce my DE students to Laplace Transforms I began with two 'Axioms':

(1) $\mathcal{L}\{c_1y_1(t)+c_2y_2(t)\}=c_1Y_1(s)+c_2Y_2(s)$

(2) $\mathcal{L}\{y^\prime\}=sY(s)-y(0)$.

From these two we derived the Laplace transforms for polynomial, exponential and sinusoidal functions and proceeded to use those results and their corresponding inverse transforms to solve a select assortment of differential equations.

Only after this did I introduce the 'definition' via the integral

\begin{equation} \mathcal{L}\{y(t)\}=\int_0^{\infty}e^{-st}y(t)\,dt \end{equation}

The DE course was a prerequisite for 'Engineering Mathematics I, II, III and IV' in which the students would get a more complete treatment of Laplace transforms.

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    $\begingroup$ I thought you missed \scriptL{1} = 1/s, but then noticed it follows from your axioms. Impressive! $\endgroup$ – Jake Mirra Jun 21 '16 at 21:38
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    $\begingroup$ When I first started teaching DE I realized that the integral definition was somewhat daunting to students so I adopted the "axiomatic" approach to show them that most of the properties could be derived using simple algebra and first derivatives. We found the corresponding inverse Laplace transforms as we went along and I gave them differential equations to solve appropriate to the rules we had proved so far. When we eventually got to the integral definition they had the confidence they needed to not be intimidated by it. I am now retired after 49 years in academia. $\endgroup$ – John Wayland Bales Jun 22 '16 at 0:14
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I think you are doing too much and too different.

  1. My advice is to do a standard diffyqs survey. the class will be mixed, but mostly engineers. They need to do some manual calculations so they really get a feel for things before disappearing into Mathematica world. This will help them in upper division engineering or physics courses where a lot of time spent on derivations on the blackboard or where they need to show work in homework. (If they end up being a very particular sort of design engineer, not most of them, or doing Ph.D. research, they may use more Mathematica then. But this is a good chance to have a feel for things. Every painter needs to grind his pigments once...)

  2. For Laplace, I would just show them the basics and use a table. It gives an exposure to all the engineers. Only the EEs (or maybe systems Es) need to go in depth on Laplace. But they will get that more detail in their courses. For this first exposure, I would keep it gentle. And the table helps to have some feel for things more than a computer black box. Yes, they may not get the conceptual blabla from the table. But they won't get that from the black box either. Won't get it even if you lecture on it. Keep things simple and make a basic exposure. For the few who need deeper understanding (on concepts, letalone methods), they can get that later in a more tailored fashion.

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