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# LOOK AT THIS! - Instructions Print your name and student ID...

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Instructions: Print your name and student ID number in the spaces below. Name: ________________________________________ Student ID:__________________________________ Exam 1, PDE I (Math 4347) 9/23/2009, Wednesday All problems have equal weigh! You are not allowed to use your notes and calculators. Show your work clearly and completely! If you continue on the back, please indicate it on the front page. Question Points 1) 2) 3) 4) 1

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2 Problem 1: Solve the equation xu x + u y = 1 with the condition u ( x; 0) = sin x: Solution: The characteristic equations starting at ( x 0 ;y 0 ) are x t = x; y t = 1; x (0) = x 0 ; y (0) = y 0 : The solution of above equations gives the characteristic curve x ( t ) = x 0 e t ; y ( t ) = y 0 + t: At time t = y 0 , the characteristic curve intersects the x axis at the point ( x 0 e ± y 0 ; 0) . Let v ( t ) = u ( x ( t ) ;y ( t )) , then by the equation xu x + u y = 1 at ( x ( t ) ;y ( t )) , we have d dt v ( t ) = 1 : An integration of above from 0 to t yields v ( t ) v
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LOOK AT THIS! - Instructions Print your name and student ID...

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