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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 wellbehaved 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.
 Winter '14
 Equations

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