There is rarely a geodesic from the genesis of an idea to its conclusion.
I am currently struggling to solve this issue. I want to figure out a way to use history to help motivate Laplace Transforms. Teaching this concept has been a struggle for decades. In 1963, S. T. R. Hancock wrote of this challenge in a very contemporary way:
In technical colleges one is often called upon to introduce advanced mathematical techniques to students whose background is not very extensive. Such a method is the use of the Laplace Transform for the solution of linear differential equations with constant coefficients. Textbooks which deal with this topic, even those specifically written for engineers, derive the transform from the Fourier Integral, or from Heaviside’s Operational Calculus, or just brusquely define the process [Hancock 1963, p. 3].
Figure 11. Statement of Laplace Transform. Wikipedia.
Laplace Transforms are obviously named after Pierre-Simon Laplace (1749–1827). But that is the only simple thing about their evolution; for instance:
For years, I’ve struggled to find the right way to use the history of this topic to help with the learning. And, I’ve not figured it out. Yet. But, what is good for the goose is good for the struggling mathematician. I’ve . . .
Each of these steps have clarified my path—a bit. But as I’ve not yet cracked the problem, I clearly don’t have all the answers. Nonetheless, hopefully you now know a few common pitfalls to avoid along with some possibly novel potential solutions to help you as you fight through them. And, knowing is half the battle. Good luck with the other half!