= Complete basis
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>{child} for the <heat equation> as shown at <fourier basis is complete for l2> and <solving partial differential equations with the Fourier series>
* <Hermite functions>{child} for the <quantum harmonic oscillator>
* <Legendre polynomials>{child} for <Laplace's equation> in <spherical coordinates>
* <bessel function>{child} for the <2D wave equation on a circular domain> in <polar coordinates>