me180_quiz01_solutions

me180_quiz01_solutions - University of California Berkeley...

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University of California, Berkeley ME 180, Engineering Analysis Using the Finite Element Method Spring 2008 Instructor: T. Zohdi Quiz 1 Solutions Problem 1 Solve d dx A 1 du dx ! = k 2 sin 2 π k L x ! , analytically, where A 1 , k , and L are constants. Solution: (5 points) We know that the analytical solution for any linear, nonhomogeneous ODE can be written as u ( x ) = u h ( x ) + u p ( x ) , i.e., it can be broken up into homogeneous and particular parts. The equation corresponding to the homogeneous part of the solution is A 1 d 2 u dx 2 = 0 . Considering the proposed solution u = e λ x results in the algebraic equation A 1 λ 2 e λ x = 0 . Assuming that A 1 , 0, and knowing that the exponential function never equals zero, results in λ 1 = λ 2 = 0, i.e., repeated real roots. This results in a homogeneous solution u h = c 1 xe 0 x + c 2 e 0 x = c 1 x + c 2 . The particular solution can be solved various ways (undetermined coe cients, variation of pa- rameters). If one considers a proposed particular solution

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me180_quiz01_solutions - University of California Berkeley...

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