If
n = 0, then we have r^{2}
– 1/4 = 0 and r = 1/2 or r = – 1/2.

If
n = 1, then a_{1}
= 0 because (r+1)^{2}
– 1/4 ≠ 0. For n > 1 the coefficients can be found from the
relations

a_{n}
= – a_{n–2
}/[(r+n)^{2}
– 1/4] where r = 1/2 or r = – 1/2. Thus we get two
linearly independent solutions of the differential equation. Its
general solution can be obtained as their linear combination with
arbitrary constants C_{1}
and C_{2}.

Apr 3rd, 2015

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