pset_4

# pset_4 - ACM 95/100b Problem Set 4 February 5 2009 Due by...

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February 5, 2009 Due by 5:00PM on 2/13/2008 Please deposit your problem set in the slot in 303 Firestone or upload it via Moodle. In either case please keep a copy of your problem set. Please remember to include your section number and section instructor. Collaboration is allowed on all problems but please write up the solutions yourself. Problem 1 (10 points) Using integration by parts, verify the Lagrange identity. That is show that if we deﬁne the operator L [ y ( x )] = - d dx ± p ( x ) d dx y ( x ) ² + q ( x ) y ( x ) , and examine the expression Z 1 0 L [ u ( x )] v ( x ) dx = Z 1 0 ³ - d dx ± p ( x ) d dx u ( x ) ² v ( x ) + q ( x ) u ( x ) v ( x ) ´ dx, then Z 1 0 { L [ u ( x )] v ( x ) - u ( x ) L [( v ( x )] } dx = - µ p ( x ) ³ du dx v ( x ) - u dv dx ´¶· · · · 1 0 . Show that the right hand side of the Lagrange identity does indeed vanish if one assumes the homogeneous separable boundary conditions at x = 0 , 1 discussed in class. Problem 2

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pset_4 - ACM 95/100b Problem Set 4 February 5 2009 Due by...

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