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m257_316_midterm1_sec103w2006

# m257_316_midterm1_sec103w2006 - = 0(1(a Classify the points...

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Math 257/316, Midterm 1, Section 103 11 October 2006 Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed. Maximum score 100. 1. Apply the method of separation of variables to determine a solution to the one dimensional heat equation with homogeneous Dirichlet boundary conditions, i.e. u t = α 2 2 u x 2 BC : u (0 , t ) = 0 = u ( L, t ) IC : u ( x, 0) = f ( x ) Evaluate the coe cients of the corresponding Fourier series for f ( x ) = x and L = 1 . Hence determine the solution of the above initial-boundary value problem. [40 marks] 2. Consider the second order di ff erential equation: Ly = 2 x 2 y 00 + xy 0 (1 + x ) y = 0
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Unformatted text preview: = 0 (1) (a) Classify the points x ≥ (do not include the point at in f nity) as ordinary points, regular singular points, or irregular singular points. [5 marks] (b) Explain how you would classify the point at in f nity (you need not carry out the calculations). [5 marks] (c) Explain how you would obtain a general solution of (1) about the point x = 1 . [10 marks] (d) Use the appropriate series expansion about the point x = 0 to determine two independent solutions to (1). Determine the radius of convergence of one of these series. [40 marks] 1...
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