# Legendres dierential equation is 1 x2 d2 y dy pp 1y

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Unformatted text preview: a1 x − 2(p − 1) 3 22 (p − 1)(p − 3) 5 x+ x 3! 5! − 23 (p − 1)(p − 3)(p − 5) 7 x + ··· . 7! We can now write the general solution to Hermite’s equation in the form y = a0 y0 (x) + a1 y1 (x), where y0 (x) = 1 − 2p 2 22 p(p − 2) 4 23 p(p − 2)(p − 4) 6 x+ x− x + ··· 2! 4! 6! and y1 (x) = x − 2(p − 1) 3 22 (p − 1)(p − 3) 5 23 (p − 1)(p − 3)(p − 5) 7 x+ x− x +··· . 3! 5! 7! For a given choice of the parameter p, we could use the power series to construct tables of values for the functions y0 (x) and y1 (x). Tables of values for these functions are found in many ”handbooks of mathematical functions.” In the language of linear algebra, we say that y0 (x) and y1 (x) form a basis for the space of solutions to Hermite’s equation. When p is a positive integer, one of the two power series will collapse, yielding a polynomial solution to Hermite’s equation. These polynomial solutions are known as Hermite polynomials . Another Example. Legendre’s diﬀerential equation is (1 −...
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## This document was uploaded on 01/12/2014.

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