I would like to know if there are any models that make use of higher-order (5th, 6th and so on...) linear differential equations. The highest that I know is the Euler-Bernoulli beam equation. Do you know any? Or how to model with higher-order ones?
Thanks
I wasn't able to find any high-order linear differential equations but was able to find some examples of nonlinear DEs. A lot of the times when these are used, they are used when approximations of other functions are used, such as in this paper, which looks at the Fermi-Pasta-Ulam model. Equation $(6)$ on page $4$ is: $$\begin{align}\frac{1}{4}\frac{d^2y}{dT^2}&=y_{\xi\xi}+y_\xi y_{\xi\xi}-\mu y^2_{\xi}y_{\xi\xi}+\delta y_{\xi\xi\xi\xi}+2\delta y_{\xi\xi}y_{\xi\xi\xi}+\delta y_{\xi}y_{\xi\xi\xi\xi}-\\ &-4\mu \delta y_{\xi}y_{\xi\xi}y_{\xi\xi\xi}-\mu \delta y_{\xi\xi}^3- \mu \delta y_{\xi}^2y_{\xi\xi\xi\xi}+\frac{2}{5}\delta^2y_{\xi\xi\xi\xi\xi\xi}\end{align}$$
There's also this paper, which models the diffusion of a fluid using the equation $$\frac{\partial h}{ \partial t} = \frac{\partial}{\partial x}\left( h^n \frac{\partial^5h}{\partial x^5} \right)$$
The Korteweg-De Vries equation is a model of "waves on shallow water surfaces", and there's higher order generalizations of it, such as equation $(2)$ on page $3$ of this paper: $$u_t+uu_x+u_{xxxxx}=0$$
That paper also references the Kawahara equation (equation $(4)$). I don't have access to the original paper, but in the last linked one, it says it is $$u_t + uu_x + \alpha u_{xxx}+u_{xxxxx}=0$$ which is also used for modeling shallow water waves.
These last two aren't models per se, but more so examples of when higher-order DEs pop up in physics.
I don't have access to this paper either, but the abstract says that a fifth order equation must be satisfied in order for the equation modeling the fluid to admit a "Lie point symmetry" (I don't understand what that means though).
The last one I'll put here came up in the study of black holes. I understood virtually nothing in this paper, but the equation $H(x)$ , which is the "conformal factor of the two dimensional space" (which has something to do with black holes, I'm guessing), must satisfy a sixth order DE in order to "generate a five dimensional solution". This differential equation (equation $6.19$) is $$\left(H^2H''''\right)'' = 0$$
although it trivially reduces to a $4$th order DE.