# How "practical" is the Laplace transform method for constant coefficient ODE?

I just finished teaching a chapter on using Laplace transform to solve constant coefficient second order linear differential equations. I touted how amazing the method was because it incorporates the initial data from the start, works for strange forcing terms, and reduces the problem of solving an ODE to computing the unit impulse response $$e(t)$$ (by taking the easy inverse Laplace of the reciprocal of the characteristic function) and the convolution $$e*g$$, where $$g$$ is the forcing term--notice how we do not need to compute Laplace of $$g$$.

I know theoretically this is significant, and that for discontinuous and non-standard forcing terms this is one of the best methods. However, I was left with the feeling that if we cannot really calculate the convolution (closed form) then this is not as impressive after all! So:

1. How commonly is this method actually used in practice for solving ODE -- say by engineers?

2. Are there ways to compute the convolution for a considerably large collection of pairs of functions?

3. If we cannot find convolution in closed form, is this method used to produce numerical solutions, e.g., by estimating the integral in the definition of the convolution?

References will be appreciated (over heuristics)!

• This is nice but I do not see any relevance to my specific questions. Jul 2 '21 at 12:27

[I answer it here, since a comment did not enough room]

As with most 'classic' theory, I think that engineers do not really use these fundamental calculations. I worked at a few engineering companies (in the electronics and mechatronics departments, so a lot of signal processing, control engineering and stuff like that), and what you see there is that these sort of problems are often solved using rules of thumb (in terms of interpretation of data) or using advanced simulators as Matlab Simulink.

However, I believe that in order to use and understand these method, you need to know the fundamentals, and what is going on 'under the hood', as this theory is basically the corner stone of the highly advanced methods implemented by the simulator-software providers.

Therefore, I think that in order to use the 'rules of thumb' and simulators, not only with Laplace related applications but with every kind of development in engineer you need to understand the fundamentals and know how things relate in order to use them properly.

Sidenote: In control engineering (in companies) the $$s$$-domain is used quite often for back-of-the-envelope calculations, so not perse to describe responses, but more in terms of transfer functions. For example every mechatronic system can be "approximated" by a mass-spring-damper system as initial guess, from which you can easily (with some mass-spring-damper characteristics) analyze the dominant behavior of a system.

So, no references, but I hope this helps for Q1...

• Thanks, it does add some insight. That was my guess too: that none of this is really done as it is in textbooks out there in field. Jul 2 '21 at 12:58
• A similar example is the solving of a convex optimization problem. You can easily implement a highly robust ellipsoidal method, but because it converges super slow, all of the common solvers are interior-point method based. Still the ellipsoidal method is taught first as this is simple, elegant, intuitive, robust etc. and can lead to insights when further developing more advanced methods. Jul 2 '21 at 15:38