I have found a fine introduction to Heaviside’s methods in Spiegel’s “Applied Differential Equations”, 3rd ed, 1981, Prentice Hall. To be specific, pages 204-211. I can’t very well say, “Don’t buy the book just for this” – because that’s exactly what I did!

Let me emphasize that, as far as I know, Heaviside’s methods are now of primarily historical interest. I would not say that Laplace transforms make Heaviside’s methods rigorous – but that Laplace transforms provide a rigorous alternative which, like Heaviside’s, lets us do algebra instead of calculus.

Like Laplace transforms, the quick use of Heaviside’s methods takes advantage of shifting properties, linearity, and tables of known results to speed up calculations. I’m not going to take them that far. With Mathematica® or another symbolic system, I see no need to go beyond the introduction to Heaviside’s methods. What I wanted to see was: just what was Heaviside’s fundamental idea? It turns out that his fundamental idea suffices, given other tools available today.

Of course – as we have seen and as we’ll see below – Mathematica can quickly solve the differential equations to which Heaviside’s methods apply (linear, with constant coefficients). We don’t need Heaviside…

…but it turns out there’s at least one question Heaviside’s methods can answer very, very quickly: find a particular solution (rather than the general solution). I don’t know that I will ever use Heaviside for that, but I know that I could.

One last thing. For my present purposes, it suffices that I will find solutions to a differential equation – and I will confirm that my answers are solutions… but I’m not going to try to prove anything at all. I don’t know what the limitations of this method are – instead of confirming that the differential equation under consideration satisfies some set of conditions, I’ll simply confirm that the answer works.

With that, let me roll up my sleeves and show this to you.

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