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Unformatted text preview: 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 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 1 = 3 . 941, 2 = 15 . 824 and 3 = 35 . 629. 2. Find a formula for the general coefficient c k in the expansion f ( x ) = X k =1 c k k (2) where f ( x ) = x ( - x ) and the k = r 2 sin kx are the orthonormal basis functions on L 2 [0 , ]. 3. Consider the generalized boundary value/eigenvalue problem y 00 + q ( x ) y = 0 , y ( a ) = y ( b ) = 0 , q ( x ) > for x [ a, b ] . (3) Show that two distinct eigenfunctions u m and u n , with m 6 = n , are orthogonal with respect to the weight function q ( x ). (Originally posed as an exercise in Lecture 27 of lecture notes.))....
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This note was uploaded on 09/20/2010 for the course AMATH 351 taught by Professor Sivabalsivaloganathan during the Spring '08 term at Waterloo.
- Spring '08