Some thoughts on the Laplacian method for solving differential equations

In the summer months, since the conclusion of my exams and work experience, I have been at a loss of normal summery things to do. Instead, I have taken this opportunity to work on some extra-curricular techniques and subject areas within maths and physics, just for a bit of fun. Clearly in general, since work on transforms, matrices and quantum probability are very much new topics for me, it has taken a large amount of read-learn-practice before I can come up with some useful projects to demonstrate my new skills. In the meantime, I will have a short discussion about something I have recently learned, called the Laplace Transform.

It is a name that I have heard every now and again from my maths teacher, and across my wider reading on the internet, yet only with nothing else to study for have I been brave enough to delve into the actual processes involved. In essence, the transform is often described in the physical world as taking a problem in the "time domain" and translating it into the "frequency domain"; conversely on a more abstract mathematical level it is merely a way to simplify differential problems into contrived algebraic ones, which may suit the number-cruncher better as it is a generally more consistent method for solving a range of differential equations, as opposed to the more guesswork-orientated method of characteristic-equation-then-particular-integral.

So how does it work? The Laplace transform of a function f(t) is found by multiplying it by the function e^-st, then finding the definite integral of this product with respect to t, between the limits of positive infinity and zero. It seems strange when put as bluntly as that but, as soon as some examples are performed, it becomes clear how convenient such a multiplication is by the seemingly arbitrary exponential term. This is for two reasons: integration by parts is straightforward where ^dv/_dt is substituted as e^-st; the exponential of a negative independent variable reduce to 0 or 1 in the limits. Using these key understandings, and a few examples from the internet to guide the method, I set out to prove a table of such transforms - with the exception of the convolution integrals at the very bottom I was able to do so, and took great pleasure and satisfaction in doing so! Here are a couple of good examples, which also introduced new functions to me which will come in useful in the future:

The second proof I did was perhaps blasé to choose as an example - in my black book where I have recorded all the written work from my work experience and personal study, I sequentially worked through simpler proofs before the moderately-challenging ones such as tcos(at), ones such as sin(at) and cos(at). Yet I felt it would be unsatisfying to reference results such as "given that the Laplace transform of sin(at) is ...", so I have effectively completed three proofs in one during the second example (it is not difficult to infer from the working that L{sin(at)} = ^a/_(s²+a²) and that, with a fulfilling sense of symmetry, L{cos(at)} =  ^s/_(s²+a²)). This symmetry does indeed somewhat extend into the mathematics of the order-one-polynomial-trig-function product, where L{tsin(at)} =  ^2as/_(s²+a²)².

Moving swiftly on... One of the more important conceptual implications of the Laplace Transform, which helps to link mathematical abstraction with the physical world, is to see what happens when the operator is applied to f(t) = t. One finds that ¹/_s is obtained, nicely satisfying the equation that frequency ∝ time^-1. Hence it is clear to see how useful such a relation might be, for example, in the field of electrical engineering as one tracks electromagnetic oscillations over time in a differential perspective.

Okay so now I know how Laplace Transforms work, a little bit of context on the kind of subject areas they may be useful in, and furthermore how to compute them. Now it's time to actually put them to their main use, which is for solving differential equations. Here is an example of such a differential equation - notice how I first solve it using the method more familiar to me, that of the characteristic equation and particular integral amalgamation, then contrast it with the Laplacian approach. The new method certainly seems more contrived in this simplistic case:

In this case the two solutions are very similar in length, but the algebra is undoubtedly more involved in the Laplacian version - the partial fraction decomposition in particular adds unnecessary complexity to the working. But observe how the initial conditions are worked nicely into the solution, making it more coherent, whereas in the familiar method they are used to solve for unknown constants which travel through the solution unresolved until the very end. So overall there is no real improvement using the transforms. But let's attempt a more complicated example. First, the auxiliary method...

(note that I say 'displaced from equilibrium' but I mean 'displaced from natural length of the spring - this also applies to the first example I did)

Then, the Laplacian method:

It is very difficult to see if these two solutions are equivalent, since they use different combinations of overarching constants. However I plotted them on Desmos, as can be found here. From there is is clear that they do not agree, yet one is considerably more likely to be correct - that is the blue line, the one that passes through the point (0,x₀) whereas the other line does not; interestingly this is the solution from the Laplace transforms! I could go back through the first method and find the mistake but I am satisfied that the relative ease of the second method has been demonstrated, so it would be pretty unnecessary.

The point of that was to show that as problems get more complicated, the Laplace Transform comes into its own. But complicated in a very specific way - the auxiliary method does not become much more difficult as the degree of the differential equation increases, as long as the characteristic equation can be factorised without too much trouble, yet the Laplace method will since each expansion of a transformed derivative increases the number of terms in the algebraic result (L{x''} produces 3 terms, L{x'''} produces 4 terms etc.). No, what matters is that the non-homogeneous f(t) element of the equation becomes more exotic - this is what causes so much trouble in finding the particular integral, but it hardly makes the Laplace method more difficult because only some partial fraction manipulation is required to separate it off into various sines, cosines, exponentials and combinations of those. Therefore Laplace Transforms are most effective, at the level I understand them, for solving low-degree non-homogeneous ODEs with polyfunctional f(t).

But there is more to the Laplace Transform than this, which is but an entry level understanding of how to apply them in linear situations. In the first example there was no non-conservative air resistance to waste mechanical energy, so oscillation extended indefinitely towards t = +∞, yet in the second there was an exponential decay in the amplitude of the oscillation. This appeared in the s-domain as the given trigonometric function transform, but s-shifted by a constant - and hence was a use of the s-shift Laplace Transform, one of the most useful ones in the table. There is a very similar t-shift, where the exponential function appears as a multiplier in the s-domain - in the t-domain this takes the form of u_c·f(t-c), where u_c is the unit step function at c.

But the versatility doesn't stop there. The issue I have at the moment is that I cannot progress beyond dealing with linear ODEs, since I have not yet learned how do perform convolution integrals (these produce a product of functions in the s-domain) or transforms of a product of two functions. All things to look forward to learning in the future I suppose!