m257_316_midterm1_sec103w2009

# m257_316_midterm1_sec103w2009 - x = 0 to determine two...

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Math 257/316, Midterm 1, Section 103 19 October 2009 Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed. Maximum score 80. 1. Consider the second order di f erential equation: Ly =9 x 2 y 00 +9 xy 0 (1 + x ) y =0 (1) (a) Classify the points x 0 (including the point at ) as ordinary points, regular singular points, or irregular singular points. [10 marks] (b) If you were given y (1) = 5 and y 0 (1) = 0 , what form of series expansion would you assume (you need not determine the expansion coe cients of this series)? What would be the minimal radius of convergence of this series? [5 marks] (c) Use the appropriate series expansion about the point
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Unformatted text preview: x = 0 to determine two independent solutions to (1). You only need to determine the f rst three non-zero terms in each case. [35 marks] 2. Apply the method of separation of variables to determine the solution to the one dimensional heat equation with the following homogeneous Dirichlet boundary conditions: u t = 2 u x 2 , &lt; x &lt; , t &gt; BC : u (0 , t ) = 0 = u ( , t ) IC : u ( x, 0) = x ( x ) Hint: It may be useful to know that R x ( x ) sin( nx ) dx = 2 n 3 (1 cos( n )) . [30 marks] 1...
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## This note was uploaded on 04/06/2012 for the course MATH 257 taught by Professor Peirce during the Fall '08 term at The University of British Columbia.

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