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math39100spring2005

# math39100spring2005 - Department of Mathematics Math 391...

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Department of Mathematics Math 391 Final Examination Date: May, 2005 Part I. Answer ALL questions. Total 64 points. 1 . [13 Points] Solve the initial value problem: y 00 - 4 y 0 + 4 y = x 2 + 12 e 2 x , y (0) = 1 , y 0 (0) = 0 . 2 . [8 Points] Solve y cos( xy ) + y 2 x dx + x cos( xy ) + 1 2 ln( x ) + 1 e y dy = 0 . 3 . [9 Points] Find the general solution to y 00 - 2 y 0 + y = e x x . 4 . [7 Points] Solve xy 0 - 2 y = xy + xe x . 5 . [13 Points] For the equation 2 xy 00 - y 0 + y = 0, (a) Show x = 0 is a regular singular point. (b) Find the indicial equation and the recurrence relation corresponding to the larger root. (c) Find the first four terms of the series solution valid near x > 0 correcponding to the larger root. 6 . [4 Points] Use separation of variables to replace the partial differential equation: xtu xx + u xt + tu x = 0 , where u is a function of x and t , by two ordinary differential equations. 7 . [10 Points] Use the Laplace Transform method to solve: y 00 + 4 y = 2 , y (0) = 1 , y 0 (0) = 3 . Note that: L{ e at } = 1 s - a , L{ sin at } = a s 2 + a 2 L{ cos at } = s s 2 + a 2 . Part II begins on the back.

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Part II. Answer any THREE (3) COMPLETE questions. Total: 36 points. 8 . [12 Points] Find the Fourier series for f ( x ) = ( x + 2 if - 2 < x 0; 2 - x if 0 < x 2, where f ( x + 4) = f ( x ) for all
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