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ii) Consider the following differential equation on (-1,1) (1-x²)y" - 2xy + n(n+1)y=0 a) Write this equation in Sturm-Liouville form. b) Denote the solutions to this equation, which are finite at x = ±1, as Pn(x) for given n (n = 0, 1, 2,...). State the orthogonality condition satisfied by the eigenfunctions.

ii) Consider the following differential equation on (-1,1) (1-x²)y" - 2xy + n(n+1)y=0 a) Write this equation in Sturm-Liouville form. b) Denote the solutions to this equation, which are finite at x = ±1, as Pn(x) for given n (n = 0, 1, 2,...). State the orthogonality condition satisfied by the eigenfunctions.

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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Needed part II ( A and B ) part
3. i) Consider the function f(x) = π = x, 0 < x≤ T. a) Extend the function in an odd manner, and sketch the Fourier Sine series representation on a € [-37, 37). b) Determine the Fourier sine series of the odd extension. c) Using the results above and Parseval's identity, show that ㅠ 6 ∞ 0, -{" n=1 ii) Consider the following differential equation on (-1,1) (1-x²)y" - 2xy + n(n+1)y=0 a) Write this equation in Sturm-Liouville form. b) Denote the solutions to this equation, which are finite at x = ±1, as Pn(r) for given n (n = 0, 1, 2,...). State the orthogonality condition satisfied by the eigenfunctions. OPERT 1 n² c) Given that Po(x) = 1 and P₁(x) = x, find the first two terms in the generalised Fourier series expansion in terms of Pn for -1 ≤ x < 0, 0 < x≤ 1. ASNE
Transcribed Image Text:3. i) Consider the function f(x) = π = x, 0 < x≤ T. a) Extend the function in an odd manner, and sketch the Fourier Sine series representation on a € [-37, 37). b) Determine the Fourier sine series of the odd extension. c) Using the results above and Parseval's identity, show that ㅠ 6 ∞ 0, -{" n=1 ii) Consider the following differential equation on (-1,1) (1-x²)y" - 2xy + n(n+1)y=0 a) Write this equation in Sturm-Liouville form. b) Denote the solutions to this equation, which are finite at x = ±1, as Pn(r) for given n (n = 0, 1, 2,...). State the orthogonality condition satisfied by the eigenfunctions. OPERT 1 n² c) Given that Po(x) = 1 and P₁(x) = x, find the first two terms in the generalised Fourier series expansion in terms of Pn for -1 ≤ x < 0, 0 < x≤ 1. ASNE
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