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.
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.
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.