# HW8 - UNIVERSITY OF CALIFORNIA Santa Barbara Department of...

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UNIVERSITY OF CALIFORNIA Santa Barbara Department of Electrical and Computer Engineering Problem Set No. 8 Issued: November 26, 2008 Fall 2008 Due: December 3, 2008 ECE 201A Final: Final exam will be held on December 9, 2008 between 8:00 and 11:00 a.m. All the material covered will be included in the final. Exam will be closed book. Formulas and equations too hard to remember or derive will be provided. Please bring a blue book to the exam. 1. Consider the following differential equation which is an eigenvalue equation. d 2 dx 2 y ( x ) + k 2 y ( x ) = 0 for 0 < x < 1 with the boundary condition y (0) = y (1) = 0. (a) Find an analytical solution for this differential equation. (b) Discretize the region 0 < x < 1 into N+1 equal segments choosing N points such that x i = i × Δ x for i = 1 ....... N where Δ x = 1 N + 1 This allows us to discretize y(x) such that y i = y ( x i ) . (c) Discretize the differential equation using the finite difference approximation to the second derivative with respect x , which is 2 1 1 1 1 2 2 2 ( ) 2 ( ) ( ) 2 ( ) i i i i i i y x y x y x y y y d y x dx x x + +

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