The Legendre’s differential equation is  (1 − x2)y'' − 2xy' + l(l + 1)y = 0, where l is a constant, and y = y(x). Show that the series solutions can terminate if l is a positive integer. In particular, if l is an even integer, show that the series terminate if k = 0, and if l is an odd integer, they terminate if k = 1. These terminating solutions are the Legendre polynomials Pn(x), where the index n = 0, 1, 2, · · · indicates the highest power of x in the polynomials.

Advanced Engineering Mathematics
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ISBN:9780470458365
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Chapter2: Second-order Linear Odes
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The Legendre’s differential equation is  (1 − x2)y'' − 2xy' + l(l + 1)y = 0, where l is a constant, and y = y(x).

Show that the series solutions can terminate if l is a positive integer. In particular, if l is an even integer, show that the series terminate if k = 0, and if l is an odd integer, they terminate if k = 1. These terminating solutions are the Legendre polynomials Pn(x), where the index n = 0, 1, 2, · · · indicates the highest power of x in the polynomials.  

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