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ps7 - Problem Set 7 March 2 2007 ACM 95b/100b Niles A...

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Problem Set 7 March 2, 2007 Due March 9, 2007 ACM 95b/100b 3pm in Firestone 303 Niles A. Pierce (2 pts) Include grading section number 1. (15 pts) Show that the second order linear differential equation A ( x ) u + B ( x ) u + C ( x ) u = 0 , A ( x ) = 0 can be transformed to Sturm-Liouville form d dx p ( x ) du dx + q ( x ) u = 0 via multiplication by an integrating factor h ( x ). 2. Consider the regular Sturm-Liouville eigenvalue problem d dx p ( x ) du dx + q ( x ) u + λr ( x ) u = 0 , a < x < b α 1 u ( a ) + α 2 u ( a ) = 0 β 1 u ( b ) + β 2 u ( b ) = 0 with -∞ < a < b < , p ( x ) , p ( x ) , q ( x ) , r ( x ) continuous with p ( x ) > 0, r ( x ) > 0 for a x b , and α 1 , α 2 , β 1 , β 2 real with | α 1 | + | α 2 | > 0, | β 1 | + | β 2 | > 0. a) (10 pts) Use the fact that each eigenvalue corresponds to exactly one linearly independent eigenfunction to show that the eigenfunctions can be chosen to be real. b) (10 pts) Derive the Rayleigh quotient λ = - p ( x ) u du dx b a + b a p ( x ) du dx 2 - q ( x ) u 2 dx b a u 2 r ( x ) dx R ( u ) . Note: It can be shown that the smallest eigenvalue, λ 1 , is the minimum value of the Rayleigh quotient R ( u ) over all continuous functions u (
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