Heaviside’s Operational Calculus

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.

Ordinary Differential Equations and the Laplace Transform

Introduction

Down the road, I expect to be using Laplace transforms to set up and solve electric circuits, and for transfer functions in control theory. An obvious starting point is to remind you just what a Laplace transform is.

So I should show you at least one example of solving a differential equation using Laplace transforms.

But if I do that, I really should remind you of the alternative solution, the one you almost certainly learned 1st.

On top of that, I really should show you what Mathematica® can do.

As if all that weren’t enough – though it really won’t take very long – I have seen a nice approach to Heaviside’s operational calculus, and I want to show that to you, too. Ah, by the time I explain it, this will justify a post of its own.

So, I propose to take a typical equation for these methods – linear, with constant coefficients – and I am going to

1. let Mathematica solve it symbolically
2. check the symbolic answer
3. let Mathematica solve it numerically
4. solve it using Laplace transforms
5. solve the homogeneous equation and then find a particular solution to the inhomogeneous equation

and in a subsequent post I will

• solve it using Heavisisde’s operational calculus