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.

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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 P_n(x), where the index n = 0, 1, 2, · · · indicates the highest power of x in the polynomials.

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