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1. Legendre functions for n = 0. Show that (6) with n = 0 gives Po(x) = 1 and (7) gives (use In (1 + x) = x − 1⁄x² + x³ + ...) 1,2 1 1 + x V2(x) = x + √√x³. + + In 2 X Verify this by solving (1) with n = 0, setting z = y' and separating variables. Y₁(x) = 1 - n(n + 1) 2! (n – 2)n(n + 1)(n+3) 4! .+... (n – 1)(n + 2) (n – 3)(n – 1)(n+2)(n+4) 3! 5! (1-x²)y" - 2xy' + n(n + 1)y = 0 (6) (7) 1₂(x) = x - (1) -x5 +.. (n constant)

1. Legendre functions for n = 0. Show that (6) with n = 0 gives Po(x) = 1 and (7) gives (use In (1 + x) = x − 1⁄x² + x³ + ...) 1,2 1 1 + x V2(x) = x + √√x³. + + In 2 X Verify this by solving (1) with n = 0, setting z = y' and separating variables. Y₁(x) = 1 - n(n + 1) 2! (n – 2)n(n + 1)(n+3) 4! .+... (n – 1)(n + 2) (n – 3)(n – 1)(n+2)(n+4) 3! 5! (1-x²)y" - 2xy' + n(n + 1)y = 0 (6) (7) 1₂(x) = x - (1) -x5 +.. (n constant)

Advanced Engineering Mathematics
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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1-5 LEGENDRE POLYNOMIALS AND FUNCTIONS 1. Legendre functions for n = 0. Show that (6) with n = 0 gives Po(x) = 1 and (7) gives (use In (1 + x) = x − 1⁄/ x² + x³ + ...) 3 - 1 + x Y2(x) = x + + In 2 1-x Verify this by solving (1) with n = 0, setting z = y' and separating variables. Y₁(x) = 1 n(n + 1) 2! (n – 2)n(n + 1)(n + 3) 4 4! - x² +... (n – 1)(n + 2) (n – 3)(n – 1)(n+2)(n+4) 3! 5! (1-x²)y" - 2xy' + n(n + 1)y = 0 (6) (7) y₂(x) = x - (1) -x5 +.... (n constant)
Transcribed Image Text:1-5 LEGENDRE POLYNOMIALS AND FUNCTIONS 1. Legendre functions for n = 0. Show that (6) with n = 0 gives Po(x) = 1 and (7) gives (use In (1 + x) = x − 1⁄/ x² + x³ + ...) 3 - 1 + x Y2(x) = x + + In 2 1-x Verify this by solving (1) with n = 0, setting z = y' and separating variables. Y₁(x) = 1 n(n + 1) 2! (n – 2)n(n + 1)(n + 3) 4 4! - x² +... (n – 1)(n + 2) (n – 3)(n – 1)(n+2)(n+4) 3! 5! (1-x²)y" - 2xy' + n(n + 1)y = 0 (6) (7) y₂(x) = x - (1) -x5 +.... (n constant)
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