MATH 257 Midterm: MATH 257+316 2008 Winter Test 1
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Students are not allowed to bring any notes into the exam. Rules governing examinations: each candidate must be prepared to produce, upon request, a. P n=0 anxn+r. (c) use the series expansion in (b) to determine two independent solutions of (1). You only need to calculate the rst three non-zero terms in each case. (d) determine the radius of convergence of the series in (c). Page 3 of 15 pages (question 1 continued) Consider the following initial boundary value problem for the heat equation: ut = uxx, 0 < x < 1, t > 0 ux(0, t) = 1 u(x, 0) = cos(3 x/2), and u(1, t) = 0. 0 < x < 1 (2) (a) determine a steady state solution to the boundary value problem. [5 marks] (b) use this steady state solution to determine the solution to the boundary value problem (2) by separation of variables.
