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0910F-108-02-midterm2-questions

# 0910F-108-02-midterm2-questions - f x = x-L< x< L x =-L...

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Duke University Fall 2009 MATH 108-02 ODE and PDE Midterm #2 : November 10, 2009. No calculator allowed. No books or notes allowed. You are allowed to use the back of the sheets. You have 75 minutes to complete this exam. Document your work. Clarity will be considered in the grading of this exam. During this exam, you are allowed to use the following formulas if you find them useful. Z 1 0 x sin( nπx ) dx = ( - 1) n +1 , Z 1 0 x cos( nπx ) dx = ( - 1) n - 1 n 2 π 2 , Z 1 0 sin( nπx ) 2 dx = 1 2 , Z 1 0 cos( nπx ) 2 dx = 1 2 , (both if n > 0) 1. Find the Fourier series for f ( x ) = cos( x ) - cos(2 x ) seen as a periodic function of period 2 π . Justify your answer. 2. Find the Fourier series of f ( x ) = ( 1 , - π x < 0; 0 , 0 x < π ; f ( x + 2 π ) = f ( x ) . To what limit does the series converge at x = π and at x = 3 π/
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Unformatted text preview: f ( x ) = x,-L < x < L ; , x =-L or x = L ; f ( x-2 L ) , otherwise . Its Fourier series is 2 L π ∞ X n =1 (-1) n +1 n sin( nπx L ) . Use this information to ﬁnd a formula for π of the form π = ∑ a i where the a i do not include π . 4. Use separation of variables to replace the PDE tu xx + xu t = 0 by a pair of ODE. 5. Consider the heat conduction problem u xx = u t , < x < π, t > , u (0 ,t ) = π, u ( π,t ) = 0 , t > , u ( x, 0) = sin( x ) + π + π sin(3 x )-x, < x < π. Do all necessary steps. 1...
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