Third order ordinary differential equations

Most applied differential equations are second order. This probably has something to do with the fact that Newton’s laws are second order differential equations.

Higher order equations are less common in application, and when they do pop up they usually have even order, such as the 4th order beam equation.

What about 3rd order equations? Third order equations are rare in application, and third order linear equations are even more rare.

This post will focus on ordinary differential equations (ODEs), but similar remarks apply to PDEs.

Third order nonlinear equations

One example of a third order nonlinear ODE is the Blasius boundary layer equation from fluid dynamics

y''' + yy'' = 0

and its generalization, the Falkner-Skan equation

y''' + yy'' + \beta\left(1 - (y')^2\right) = 0.

Third order linear equations

I’ve seen two linear third order ODEs in application, but neither one is very physical.

The first [1] is an equation satisfied by the product of Airy functions:

 y''' - 4xy' - 2y = 0.

Here is a short proof that the product of Airy functions satisfy this equation.

Airy functions come up in applications such as quantum mechanics, optics, and probability. I suppose products of Airy functions may come up in those areas, but the equation above seems like something discovered after the fact. It seems someone thought it would be useful to find a differential equation that products of Air functions satisfy. I doubt someone wrote down the equation because it came up in applications, then discovered that products of Airy functions were the solutions.

The second [2] is a statistical model for analyzing handwriting:

x'''(t) = \beta_1(t) x'(t) + \beta_2(t) x''(t) + f(t).

Here someone decided to try modeling handwriting movements as a function of velocity, acceleration, and “jerk” (third derivative of position). It may be a useful model, but it wasn’t derived in the same physical sense as say the equations of mechanical vibrations. You could also object that since x(t) does not appear in the equation, this is a second order differential equation for y(t) = x′(t).

Related posts

[1] Abramowitz and Stegun, Handbook of Mathematical Functions, equation 10.4.57.

[2] Ransay and Silverman. Applied Functional Data Analysis: Methods and Case Studies. Equation 12.1.

One thought on “Third order ordinary differential equations

  1. Bruce Postlethwaite

    Most applied differential equations come from continuity balances (mass, energy, moles and momentum). Of these only momentum balances produce fundamentally second-order equations. Real world dynamics are represented by very high-order differential equations that result from the interaction of low-order fundamental balances

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