math39100fall2005

# math39100fall2005 - Department of Mathematics Math 391...

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Department of Mathematics Math 391 Final Examination Fall 2005 Part I. Answer ALL questions. Total 70 points. 1 . [6 Points] Find the general solution to y (4) - 4 y 00 + 4 y = 0. 2 . [10 Points] Solve ˆ e y 2 - 5 y sin( xy ) + 1 x ! dx + ˆ 2 xye y 2 - 5 x sin( xy ) - 1 y ! dy = 0. 3 . [5 Points] Suppose u ( x,t ) satisﬁes the partial diﬀerential equation: tu xx + 2 xu xt + xtu x = 0 . Use the separation of variables method to replace the partial diﬀerential equation by two ordinary diﬀerential equations. 4. (a) [5 Points] Using only the deﬁnition, ﬁnd the Laplace Transform Y ( s ) of y ( t ) = e at where a is a constant. For what values of s does the Laplace Transform Y ( s ) exist? (b) [10 Points] Solve, using Laplace Transforms, the initial value problem: y 00 - 4 y 0 + 4 y = 3 , y (0) = 0 , y 0 (0) = 1 . (See table at end of exam.) 5 . [6 Points] Find two linearly independent solutions of 2 x 2 y 00 + 3 xy 0 - y = 0 for x > 0 and compute their Wronskian. 6

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## This note was uploaded on 02/16/2011 for the course MATH 390 taught by Professor Popvassilev during the Spring '11 term at CUNY City.

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math39100fall2005 - Department of Mathematics Math 391...

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