# C find a nonzero polynomial solution to this

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Unformatted text preview: |x| < 1. However, we might suspect that the solutions to Legendre’s equation to exhibit some unpleasant behaviour near x = ±1. Experimentation with numerical solutions to Legendre’s equation would show that these suspicions are justiﬁed—solutions to Legendre’s equation will usually blow up as x → ±1. Indeed, it can be shown that when p is an integer, Legendre’s diﬀerential equation has a nonzero polynomial solution which is well-behaved for all x, but solutions which are not constant multiples of these Legendre polynomials blow up as x → ±1. Q(x) = Exercises: 1.2.1. We would like to use the power series method to ﬁnd the general solution to the diﬀerential equation d2 y dy − 4x + 12y = 0, dx2 dx which is very similar to Hermite’s equation. So we assume the solution is of the form ∞ an xn , y= n=0 a power series centered at 0, and determine the coeﬃcients an . a. As a ﬁrst step, ﬁnd the recursion formula for an+2 in terms of an . b. The coeﬃcients a0 and a1 will be determined by...
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## This document was uploaded on 01/12/2014.

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