Examples of applications of Linear differential equations to physics. 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.
 A: 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.
A: 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.
