Finding a complete basis such that each vector solves a given differential equation is the basic method of solving partial differential equation through separation of variables.
The first example of this you must see is solving partial differential equations with the Fourier series.
Notable examples:
- Fourier series for the heat equation as shown at Fourier basis is complete for and solving partial differential equations with the Fourier series
- Hermite functions for the quantum harmonic oscillator
- Legendre polynomials for Laplace's equation in spherical coordinates
- Bessel function for the 2D wave equation on a circular domain in polar coordinates
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