Systems of ODE to High-Order converstion: Why?

Question

I am the TA for a course in ODE, and one of my students asked me a question yesterday: why in the world do we bother with converting between (constant-coefficient) systems and higher-order ODE?

I started to give what I think is the standard answer: we're really good at solving first order ODE but higher order stuff is hard, solution spaces, matrix exponentials, etc. etc.

But I found myself at a complete loss for why we might want to go in the other direction. Are there advantages to having a single equation, that outweigh the fact that it is of high order?

Answer 1

Strictly speaking, your question is not entirely well posed, for two main reasons:

1) there is no canonical way of transforming a higher-order equation into a higher-dimensional first-order equation, and in some applications it is crucial to use a transformation different from the usual one $x'=y$, $y'=z$, etc;

2) not all higher-dimensional first-order equations can be written as a higher-order equation, simply because not all matrices are invertible.

Having said this, one should really restrict ourselves to some canonical (probably "the" canonical transformation), and so one should consider only higher-order equations that are already in the expected form (that is, with a matrix in the canonical form).

Having done this, the answer to your question is that there is no "mathematical" advantage of going backwards, but there are potential "physical" advantages: higher-order linear equations tend to have terms that separately have some physical significance in the applications (they should!) and it is quite welcome to be able to interpret linearity of an equation as a possible superposition of "independent states".