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.