The Legendre’s differential equation is (1 − x2)y'' − 2xy' + l(l + 1)y = 0, where l is a constant, and y = y(x). Consider a series solution about the point x = 0 of the form provided, where k is a constant and the an are coefficients that need to be determined. Show that the recurrence relation is given by an+2 (n + k + 2)(n + k + 1) = an [(n + k)(n + k + 1) − l(l + 1)].
The Legendre’s differential equation is (1 − x2)y'' − 2xy' + l(l + 1)y = 0, where l is a constant, and y = y(x). Consider a series solution about the point x = 0 of the form provided, where k is a constant and the an are coefficients that need to be determined. Show that the recurrence relation is given by an+2 (n + k + 2)(n + k + 1) = an [(n + k)(n + k + 1) − l(l + 1)].
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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The Legendre’s differential equation is (1 − x2)y'' − 2xy' + l(l + 1)y = 0, where l is a constant, and y = y(x).
Consider a series solution about the point x = 0 of the form provided, where k is a constant and the an are coefficients that need to be determined.
Show that the recurrence relation is given by an+2 (n + k + 2)(n + k + 1) = an [(n + k)(n + k + 1) − l(l + 1)].
Transcribed Image Text:y = xk
An
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