I've always thought there aren't many applications of ODE's as opposed to PDE's due to it's simplicity in comparison, but I've only really dealt with problems up to the third order, and so I was wondering if ther are any insightful applications for when $n>3$ of a (linear) ode?
Take for example:
- When $n=1$, we can model the rate of decay Arsenal's fanbase.
- When $n=2$ we can use Airy's equation to model diffraction of light.
- When $n=3$ we can model the jerk in a transport vehicle.
It always seemed to me that ODE's beyond second/third order are studied just to understand the theory, but what does it mean to have a fourth/fifth order ODE in real life? What is the highest order ODE that is readily used today?
Maybe the most direct example is the design of roller coasters. Given a finite set of points in space through which the car must travel, the engineer needs to design a trajectory. Of course the trajectory has to be continuous. And of course its tangent has to be continuous. But less obvious is that the 2nd-derivative also should be continuous, or the jerk at the discontinuities will eventually damage the track. It turns out that the more continuous derivatives you have, the smoother the ride (so less stress on the track.) I understand that modern roller coaster designers will go so far as to ensure that the 7th derivative is continuous along the track. So that's at least a 7th-order DE.