What physical phenomena are modelled by Chebyshev equation?
The Legendre polynomials arise naturally when solving the Poisson equation for a system with spherical symmetry (such as the hydrogen atom).
The Bessel functions arise naturally when solving the Poisson equation for a system with cylindrical symmetry.
In essentially the same way, the Chebyshev equation and its solutions arise when you consider a problem using an elliptical coordinate system.
I remember a side comment (perhaps in Arfken & Weber?) that all the named "special functions" arise from solving the Poisson equation in different coordinate systems. (I forget which one is toroidal coordinates.)