The classical second-order linear ordinary differential equation is that named after Sturm and Liouville: formally,
\begin{equation}
(pu')'=ru.
\end{equation}
It arises naturally in many physical situations, for example through radial considerations of Schrödinger's equation. A higher-order "analogue" of it is the Orr$-$Sommerfeld equation, which, after relabelling the coefficients, may be written as
\begin{equation}
(\phi u'')''=\psi u.
\end{equation}
It also arises naturally, namely from certain simplifications applied to Navier$-$Stokes' (in)famous equation, and describes to great accuracy the cross-stream behaviour of channel fluid flow.
I have been thinking long and hard about whether there are additional higher-order linear differential equations that emerge naturally from our mathematical models of the world. It even seems pretty much all of the linear partial differential equations (heat, wave, Schrödinger, etc.) are of second order. Does anybody know of higher-order examples?
Best Answer
One prominent example is the Abraham–Lorentz force, which depends on the derivative of the acceleration (AKA the jerk) of a charged particle. This leads to weird effects like pre-acceleration, since adding this into the force equation leads to a third-order equation, which can be integrated to show that the acceleration depends on the external force in a way that includes parts of it that are supposed to be in the particle's future.