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m257_316_midterm1_sec101w2009

# m257_316_midterm1_sec101w2009 - [5 marks(c Use the...

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Math 257/316, Midterm 1, Sections 101/102 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 =6 x 2 y 00 + 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) = 1 and y 0 (1) = 2 , 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?
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Unformatted text preview: [5 marks] (c) Use the appropriate series expansion about the point 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 Neumann boundary conditions: ∂ u ∂ t = ∂ 2 u ∂ x 2 , < x < π , t > BC : ∂ u ∂ x (0 ,t ) = 0 = ∂ u ∂ x ( π ,t ) IC : u ( x, 0) = 1 [30 marks] 1...
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