Sol-HW8 - Math Methods Solutions for Assignment 8 Oct. 25,...

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Unformatted text preview: Math Methods Solutions for Assignment 8 Oct. 25, 2010 Fall 2010 Sturm-Liouville Theory, Legendre Functions Due Nov. 01, 2010 1. Find the eigenvalues and eigenfunctions of the boundary value problem y 00 + λ ( x + 1) 2 y = 0 on the interval 1 ≤ x ≤ 2 with boundary conditions y (1) = y (2) = 0. Write the equation in terms of a regular Sturm-Liouville eigenvalue problem, and find the coefficients in the expansion of an arbitrary function f ( x ) in a series of the eigenfunctions. Searching for solution as y = ( x + 1) α we get α ( α- 1) + λ = 0 α = 1 ± √ 1- 4 λ 2 A set of solutions is n ( x + 1) (1+ √ 1- 4 λ ) / 2 , ( x + 1) (1- √ 1- 4 λ ) / 2 o Solution satisfying the boundary conditions must be a periodic function, i.e. λ > 1 / 4. Hence, a set of solutions can be written as √ x + 1cos √ 4 λ- 1 2 ln x + 1 2 , √ x + 1sin √ 4 λ- 1 2 ln x + 1 2 The solution satisfying y (1) = 0 is y = c √ x + 1sin √ 4 λ- 1 2 ln x + 1 2 From the second boundary condition y...
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This note was uploaded on 09/28/2011 for the course PHYSICS 801 taught by Professor Ivanov during the Fall '10 term at Kansas State University.

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Sol-HW8 - Math Methods Solutions for Assignment 8 Oct. 25,...

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