# Higher-order ODE's

1. May 15, 2012

### StationZero

Hello, new to the site and new to DE's. Quick question, I've started doing some self-study in linear ODE's and am currently studying solution methods for higher order equations. However, frustratingly, it is all pure math with no practical applications of the equations except for one second order equation relating to Hooke's law and the behavior of springs.

In any case, my question is: What is the practical application of 3rd, 4th, 5th, and higher order ODE's to the behavior of real-world physical systems or, in fact, any kind of system. It seems as though just the first and second order equations relating to velocity and acceleration would be adequate alone to cover real-world physical interactions.

Thanks in advance for the insight and sorry for the Newbie Naivete.

2. May 15, 2012

### chiro

Hey StationZero and welcome to the forums.

The short answer is basically that if a potential real-world problem has the structure of a particular DE that has a known algorithm to find said solution, or an explicitly known solution type, then it can be solved.

The key word here is potential: the mathematicians may not be solving this for a pre-specified specific purpose, but instead for the activity of solving a generic problem that may or may not appear.

Usually though the mathematicians end up always developing methods that are used in one way or another and from this point on, the math goes from pure to applied.

When results come that are able to solve the most general classes of problems, this will always be beneficial in the highest of probabilities for the future.

To put this into context think about say the work of people like Turing, Church, Von Neumann and others when they came up with theoretical models of computers (and practical like Von Neumann did): that was all just theoretical musings and was long before transistors were perfected to give the Von-Neumann Architecture that we use on a daily basis.

It's the same kind of thing and for this reason it's good that mathematicians have the patience and the desire to investigate things with no gauranteed payoff, which is funnily enough a rare attribute (for someone to spend large portions of their life to investigate something that may have absolutely no practical significance physically what-so-ever).

3. May 15, 2012

### AlephZero

You are right, most of the mathematical "laws" of physics generate first or second order ODEs or PDEs.

There are a few execptions. A fairly simple one is bending of a flexible beam, where the forces are related to the curvature of the beam. The curvature involves the second derivative of the displacements, and you end up with a 4th order ODE.
http://en.wikipedia.org/wiki/Eulerâ€“Bernoulli_beam_theory

4. May 15, 2012

### HallsofIvy

Problems in elasticity tend to produce fourth degree differential equations.

5. May 15, 2012

### theorem4.5.9

Instead of supplying examples of applications (I would fail miserably if I did) I'm going to suggest you pick up a book.

Nonlinear Dyanamics and Chaos - Steven Strogatz

Strogatz is one of the best expositors I have read, and this book is the best introduction to systems of ODE I know of. The work is motivated by real applications, NOT contrived examples the author invented to force the equations (like most low level math classes). Moreover, this book teaches powerful theory and methods in a manner which is pleasurable to read. The exercises are well thought out as well - some very simple, some extremely hard!

EDIT: I forgot to mention, ODEs of order 3 or more aren't handled until fairly late in the book. The reason is most of the machinery is easiest to understand in dimension 1 or 2. ODEs of order 3 or more are in a lot of ways all analogous, with the big difference being that their dimension is larger than 2. In 2 dimensions, you may have a single trajectory being a circle, which "traps" all trajectories in the interior. This can't happen in higher dimensions, which is a hint as to why one may be forced to work in higher dimensions. (By dimension I mean the dimension of the trajectory in the phase space, which is exactly the order of the ODE).

Last edited: May 15, 2012
6. May 15, 2012

### StationZero

Thanks to everyone who replied. I appreciate the intuition on these matters. In relation to Theorem4.5.9s reply, I am a little familiar with Strogatz's work in relation to small-world effects and scale-free architectures in relation to brain function. In fact, I am studying ODE's in order to model the chaotic neurodynamics of cortical function using Walter Freeman's K5 model. According to this model, cortical interactions can be modelled as a collection of coupled oscillators which, in turn, can be modelled as a collection of coupled, nonlinear, ordinary differential equations.

Of course, the solution to these sets of equations is highly complex and nonlinear and can only be approximated using numerical methods, but the results match pretty nearly actual EEG rhythms. Even so, they are still only second order equations and I couldn't really imagine a more complex system using higher order derivatives. I have a better perspective on that now so, thanks again.

As a non-related aside, what is the best Math App for solving and modelling ODE's? Matlab, Mathematica, or some other app?