given the legendre equation $(1-x^2)y'' - 2xy' + by = f(x)$ why can the solution be a series of legendre polynomials $y(x) = \sum_{n=0}^{\infty}a_n P_n(x)$? i thought legndre solves the homogenous version of the equation so wouldn't a series of legendre polynomials all equal 0, or even more isn't there only 1 legendre polynomial a solution to the equation?
2026-03-26 12:42:48.1774528968
legendre solution for non homogenous equation
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