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I'm looking for applications of second-order linear differential equations aside from mechanics, economics and electrical circuits. I was given hints to look on Ecology or other physics fields but it seems rather difficult to find something.

Can you point me to some other examples?

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There are a lot of examples in physics. For example, the sound of a drum is given by the eigenvalue problem $\Delta u + \lambda u = 0$. Read a little about the Laplace operator (, it appears in an enormous number of physical contexts. – in_wolframAlpha_we_trust Jun 4 '12 at 9:35
Also Lotka-Volterra prey-predator system is a good example – Jon Jun 4 '12 at 10:11
Lookup any of these (not to give preference to wikipedia): - Maxwell's equations for electrodynamics - Schrodingers' equation - Fokker-Planck diffusion equation – Valentin Jun 4 '12 at 10:16
@Jon: I think the Lotka-Volterra system is neither linear nor second order in the given form. – Willie Wong Jun 4 '12 at 10:45
@WillieWong: Right, I was off the mark. – Jon Jun 4 '12 at 11:05

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