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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.

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    $\begingroup$ Schroedinger equation. $\endgroup$ – Start wearing purple Jun 24 '13 at 12:24
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    $\begingroup$ Maxwell's equations, Newton's seconds law. $\endgroup$ – tom Jun 24 '13 at 12:31
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    $\begingroup$ @tom Newton's law is in general a nonlinear equation (or a system of such). $\endgroup$ – Start wearing purple Jun 24 '13 at 12:56
  • $\begingroup$ Ok you are right. Btw. Can not be hamiltonian operator in Schroedinger equation be dependant in wave function? I don't have by no means any education in quantum physics. But I always thought about Schroedinger equation as about Newtons second law that $H$ and $F$ describes the system. So I don't see why $H$ cannot depend on wave function. $\endgroup$ – tom Jun 24 '13 at 13:10
  • $\begingroup$ @tom No, on the fundamental level $H$ does not depend on the wave function (it can depend on it in effective theories, once some approximations are made, but this is rather specialized discussion). In this sense, quantum mechanics is simpler than classical one. $\endgroup$ – Start wearing purple Jun 24 '13 at 13:21
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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.

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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.

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