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(a) Consider the general Sturm-Liouville equation with eigenvalue \ dx de (pdy) +qy+oy=0, a

(a) Consider the general Sturm-Liouville equation with eigenvalue \ dx de (pdy) +qy+oy=0, a

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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(a) Consider the general Sturm-Liouville equation with eigenvalue \ d dr (pdy) +qy+loy = 0, a<x<b. dx Suppose we have a sequence of eigenfunctions yn, corresponding to distinct eigen- values. Derive the formula d (An - Am)σymyn dz (P(YnYm — YmYn)). - (b) Show that Chebyshev's equation - (1 − x²)y″ − xy' + λy = 0, -1 < x <1 can be written in the Sturm-Liouville form − x²) ¹³ y′ ) ' + \( 1 − x²) ¯ ½³ y = 0, - −1 < x < 1. (c) It can be shown that Chebyshev's equation with the boundary conditions y(x), y(x) remain bounded as x → ±1 has eigenvalues \n = n², n = 0, 1, 2, .... The corresponding eigenfunctions yn are Chebyshev's polynomials T(x) satisfying the recurrence relation Tn+1(x) = 2xTn(x) — Tn−1(x), To(x) = 1, T₁(x) = = x. Find T3(x) and T₁(x) and verify that they satisfy Chebyshev's equation. (d) By using the result of (a) and (b) show that [ In (x) Tm(x) (1 - x²) dx 0, for nm.
Transcribed Image Text:(a) Consider the general Sturm-Liouville equation with eigenvalue \ d dr (pdy) +qy+loy = 0, a<x<b. dx Suppose we have a sequence of eigenfunctions yn, corresponding to distinct eigen- values. Derive the formula d (An - Am)σymyn dz (P(YnYm — YmYn)). - (b) Show that Chebyshev's equation - (1 − x²)y″ − xy' + λy = 0, -1 < x <1 can be written in the Sturm-Liouville form − x²) ¹³ y′ ) ' + \( 1 − x²) ¯ ½³ y = 0, - −1 < x < 1. (c) It can be shown that Chebyshev's equation with the boundary conditions y(x), y(x) remain bounded as x → ±1 has eigenvalues \n = n², n = 0, 1, 2, .... The corresponding eigenfunctions yn are Chebyshev's polynomials T(x) satisfying the recurrence relation Tn+1(x) = 2xTn(x) — Tn−1(x), To(x) = 1, T₁(x) = = x. Find T3(x) and T₁(x) and verify that they satisfy Chebyshev's equation. (d) By using the result of (a) and (b) show that [ In (x) Tm(x) (1 - x²) dx 0, for nm.
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