SL - / 2) 20 . 191 sin n x 3 7 . 725 = 0 . 984 (5 / 2) 59 ....

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Eigenvalues and eigenfunctions of a Sturm-Liouville problem In class on November 13 we discussed the problem y ′′ + λy = 0 for 0 < x < 1; y (0) = 0 , y (1) - y (1) = 0 . We found the eigenvalues to be λ 1 = 0, λ n = κ 2 n for n 2, where κ n is the ( n - 1) st positive root of the equation κ = tan κ . Graphically, the κ n are determined as the intersections of the curves z = κ and z = tan κ : k 2 4 6 8 10 12 14 16 18 z 0 2 4 6 8 10 12 14 16 18 Maple Fnds the Frst 6 eigenvalues as given in the table below; note that as expected, κ n is very close to (2 n - 1) π/ 2 for n large: n κ n λ n φ n ( x ) 1 0 x 2 4 . 493 = 0 . 954 (3
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Unformatted text preview: / 2) 20 . 191 sin n x 3 7 . 725 = 0 . 984 (5 / 2) 59 . 680 sin n x 4 10 . 904 = 0 . 992 (7 / 2) 118 . 900 sin n x 5 14 . 066 = 0 . 995 (9 / 2) 197 . 858 sin n x 6 17 . 221 = 0 . 997 (11 / 2) 296 . 554 sin n x Here are the Frst four eigenfunctions 1 ( x ) , . . . , 4 ( x ); with a little good will one can believe that they indeed satisfy n (1) = n (1): x 0.2 0.4 0.6 0.8 1.0 K 1.0 K 0.5 0.5 1.0...
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This note was uploaded on 09/29/2009 for the course 642 527 taught by Professor Speer during the Fall '07 term at Rutgers.

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