Application for differential equation of higher order

Question

We found some interesting insights in differential equations of the form

$y^{(n)}(x)+F_\lambda(y(x),y'(x),...,y^{(n-1)}(x))=0$,

i.e. for ordinary differential equations of $n$-th order with $n\geq2$. The function $F$ is polynomial which can include a set of parameters $\lambda$.

We know, that in physics usually the highest derivative is of order two(?), but we are searching for applications of this kind of differential equations for $n\geq3$ in physics, engineering, or in any other area. If you have an idea or know models or theories in which such equations occur, you input would be appreciated very much.

Related: http://physics.stackexchange.com/q/52024/2451 , http://physics.stackexchange.com/q/189000/2451 and links therein. — Qmechanic, Sep 30 '16 at 12:07
Vibrating plates: https://en.wikipedia.org/wiki/Vibration_of_plates#Rectangular_plates — Gert, Sep 30 '16 at 12:30
The KdV equation is quite famous and interesting: https://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation — DelCrosB, Sep 30 '16 at 14:41
Many equations in elasticity theory are fourth-order in space. — Emilio Pisanty, Sep 30 '16 at 17:20

score 1 · Answer 1 · answered Sep 30 '16 at 19:39

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That's just not true. If a linear system has $n$ independent ways in which energy can be stored as states, and energy can flow between these states, then you can model the system with an nth order polynomial.

Granted some systems can be approximated by a linear 2nd order rational polynomial function, but a closer look shows higher order fits might work better. For example you might assume a spring mass system as a lumped parameter 2nd order system, but find the spring has torsional as well as bending modes, and the rigid mass perhaps not so rigid. But we tend to model the states that more matter to our application, and so we might neglect the higher order modes.

And look carefully, the real world is hardly ever linear. Nonlinear systems are more the norm.

answered Sep 30 '16 at 19:39

docscience

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Thank you, this is helpful. "That's just not true. If a linear system has n independent ways in which energy can be stored as states, and energy can flow between these states, then you can model the system with an nth order polynomial." Can you please provide a minimal example? – Oct 02 '16 at 20:22
@Thanks A simple RC ladder network with more than two capcitors comes to mind. Each capacitor in the network adds a state and increases order. And a mechanical equivalent: any spring mass system with more than two masses and springs. – docscience Oct 02 '16 at 20:45

score 0 · Answer 2 · answered Mar 12 '17 at 03:42

The Dirac equation, which is a system of four first-order equations for four components of the Dirac spinor, is generally equivalent to one fourth-order equation for one component of the spinor (http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf , published in the Journal of Mathematical Physics, https://arxiv.org/abs/1502.02351)

Application for differential equation of higher order

2 Answers2