Examples of applications of Linear differential equations to physics.

Question

I wonder which other real life applications do exist for linear differential equations, besides harmonic oscillators and pendulums. I'm looking for examples to include in a document that talks about the topic. So basically I need things that are easy to model with a single differential equation.

Answer 1

I vote for Schroedinger equation - the cornerstone of our description of quantum world. Its particular cases cover all important 2nd order linear ODEs:

• Hermite $\rightarrow$ harmonic oscillator,
• Legendre $\rightarrow$ spherically symmetric potentials,
• Bessel $\rightarrow$ free particle in 2D,
• Airy $\rightarrow$ Stark effect in 1D,
• confluent hypergeometric $\rightarrow$ hydrogen atom,
• ...

I can go on.

Answer 2

I find it fascinating that the Schroedinger equation can represent a paraxial approximation to solutions of the wave equation. This is analogous to a parabola providing an approximation to a sphere near the axis, and in fact the derivation of the paraxial wave equation uses this very fact. That said, what I find interesting is that the solutions of one differential equation may be approximated by the solutions of an entirely different differential equation.

An interesting aside from this - apart from differential equations - is that, in geometrical optics, we make the opposite approximation: we frequently approximate a parabola with a sphere.