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Consider the equation d 2 dy + dx -y = 0, x2 -1 < x < 1. a) Decide if a Sturm-Liouville problem consisting of this equation would be regular or singular. b) Show that To(x) = 1, T¡(x) = x, and T2(x) = 2x² – 1 are eigenfunctions of such Sturm-Liouville problem, and find the corresponding eigenvalues Ao, 11, and 12. c) Using the appropriate weight function, show that Tị is orthogonal to T2 on the interval (-1, 1).

Consider the equation d 2 dy + dx -y = 0, x2 -1 < x < 1. a) Decide if a Sturm-Liouville problem consisting of this equation would be regular or singular. b) Show that To(x) = 1, T¡(x) = x, and T2(x) = 2x² – 1 are eigenfunctions of such Sturm-Liouville problem, and find the corresponding eigenvalues Ao, 11, and 12. c) Using the appropriate weight function, show that Tị is orthogonal to T2 on the interval (-1, 1).

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter13: Conic Sections
Section13.1: Circles
Problem 48PS
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Consider the equation (Vi-2)+ =0, -1 <*<1. d -1 < x < 1. dx dx V1 – x2 a) Decide if a Sturm-Liouville problem consisting of this equation would be regular or singular. b) Show that To(x) = 1, T1(x) = x, and T2(x) = 2x2 – 1 are eigenfunctions of such Sturm-Liouville problem, and find the corresponding eigenvalues 1o, 11, and å2. c) Using the appropriate weight function, show that T1 is orthogonal to T2 on the interval (-1, 1).
Transcribed Image Text:Consider the equation (Vi-2)+ =0, -1 <*<1. d -1 < x < 1. dx dx V1 – x2 a) Decide if a Sturm-Liouville problem consisting of this equation would be regular or singular. b) Show that To(x) = 1, T1(x) = x, and T2(x) = 2x2 – 1 are eigenfunctions of such Sturm-Liouville problem, and find the corresponding eigenvalues 1o, 11, and å2. c) Using the appropriate weight function, show that T1 is orthogonal to T2 on the interval (-1, 1).
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