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8.5.5 Solve the Legendre equation
(1 ? x2)y ? 2xy + n(n+ 1)y = 0
by direct series substitution and plot the solution for n = 0, 1, 2, 3.
(a) Verify that the indicial equation is
k(k ? 1) = 0.
(b) Using k = 0, obtain a series of even powers of x, (a1 = 0).
yeven = a0

1 ? n(n+ 1)
2!
x2 + n(n? 2)(n+ 1)(n+ 3)
4!
x4 + ·· ·

,
where
aj+2 = j( j + 1) ? n(n+ 1)
( j + 1)( j + 2)
aj .
(c) Using k = 1, develop a series of odd powers of x(a1 = 0).
yodd = a0

x ? (n? 1)(n+ 2)
3!
x3
+ (n? 1)(n? 3)(n+ 2)(n+ 4)
5!
x5 +· · ·

,
where
aj+2 = ( j + 1)( j + 2) ? n(n+ 1)
( j + 2)( j + 3)
aj .
(d) Show that both solutions, yeven and yodd, diverge for x= ±1 if the
series continue to infinity.
(e) Finally, show that by an appropriate choice of n, one series at a
time may be converted into a polynomial, thereby avoiding the
divergence catastrophe. In quantum mechanics this restriction
of n to integral values corresponds to quantization of angular
momentum.