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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2e672a1-54a8-4743-aad6-f3d1c4a0c4ab%2F69da446f-4998-4dac-bc75-84908875eb41%2F5o8hvlx_processed.jpeg&w=3840&q=75)
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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