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Quiz5 - Quiz 5 We consider the equation for all 0 < y < L...

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Quiz 5 We consider the equation for all 0 < y < L : - d 2 G dx 2 ( x, y ) = δ ( x - y ) , 0 < x < L, (1) dG dx (0 , y ) = 0 , G ( L, y ) = 0 . (2) 1. For fixed y , find the general solution of (1) on x (0 , y ) and x ( y, L ). 2. Use the boundary conditions (2) on G to simplify the expression found in 1. 3. Show directly from (1) that - dG dx ( x, y ) = α + H ( x - y ) , where α is a constant and H ( x ) is the Heaviside function defined by H ( x ) = 1 for x > 0 and H ( x ) = 0 for x < 0. 4. Deduce from 3. that G is a continuous function in x . 5. Solve for G ( x, y ) and draw the graph of G ( x, L 2 ). 6. Show that φ n ( x ) = cos( π L ( n + 1 2 ) x ) and λ n = ( π L ( n + 1 2 )) 2 , for n = 0 , 1 , · · · , are solutions of - d 2 φ n dx 2 ( x ) = λ n φ n ( x ) , 0 < x < L, n dx (0) = 0 , φ n ( L ) = 0 . Find the coefficients α n ( y ) such that G ( x, y ) = X n =0 α n ( y ) φ n ( x ) . Solutions. 1. Since δ ( x - y ) = 0 for x < y and x > y we have G 00 = 0 on each of these intervals. Thus G ( x ) = ax + b on (0 , y ) and G ( x ) = cx + d on ( y, L ). 2. G 0 (0) = 0 implies that a = 0 so that G ( x ) = b on (0 , y ). G ( L ) = 0 implies that cL + d = 0 so that d = - cL and G ( x ) = c ( x - L ) on ( y, L ).
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