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# ass7 - AMATH 351 Assignment No 7 Fall 2005 Sturm-Liouville...

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AMATH 351 Assignment No. 7 Fall 2005 Sturm-Liouville BVPs, Laplace Transforms Questions 1,3,6,7,9 and 10 are due Tuesday, November 29, 2005, 1:30 p.m. (under my door) 1. Find numerical values of the upper and lower bounds to the first three eigenvalues λ k , k = 1 , 2 , 3, for the eigenvalue problem y 00 + (2 + x ) y = 0 , y (0) = y (1) = 0 . (1) The approximate numerical values of these eigenvalues are λ 1 = 3 . 941, λ 2 = 15 . 824 and λ 3 = 35 . 629. 2. Find a formula for the general coefficient c k in the expansion f ( x ) = X k =1 c k φ k (2) where f ( x ) = x ( π - x ) and the φ k = r 2 π sin kx are the orthonormal basis functions on L 2 [0 , π ]. 3. Consider the generalized boundary value/eigenvalue problem y 00 + λq ( x ) y = 0 , y ( a ) = y ( b ) = 0 , q ( x ) > 0 for x [ a, b ] . (3) Show that two distinct eigenfunctions u m and u n , with λ m 6 = λ n , are orthogonal with respect to the weight function q ( x ). (Originally posed as an exercise in Lecture 27 of lecture notes.) 4. Do the same as in the above question for distinct eigenfunctions u m and u n of the generalized Sturm-Liouville eigenvalue problem: d dx p ( x ) d ∂x

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ass7 - AMATH 351 Assignment No 7 Fall 2005 Sturm-Liouville...

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