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Math 257/316, Midterm 2, Section 101/102 November 20, 2009 Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed. Maximum score 100. 1. Solve the following inhomogeneous initial boundary value problem for the heat equation: ut = uxx − u, 0 < x < 1, t > 0 ux (0, t) = 0, ux (1, t) = sinh(1) u(x, 0) = cosh(x) − x Hint: You might find it helpful to know that Z 0 1 ( 2/n2 π 2 , n odd −x cos(nπx) dx = 0, n even. [50 marks] 2. Solve the following inhomogeneous initial boundary value problem for the heat equation: ut u(0, t) u(x, 0) = α2 uxx , 0 < x < π/2, t > 0 = 0, ux (π/2, t) = t2 /2 = 0 by using an appropriate expansion in terms of the eigenfunctions sin(2n+1)x corresponding to the eigenvalues λn = (2n + 1) where n = 0, 1, . . . Hint: You might find it helpful to know that Z π/2 −x sin ((2n + 1)x) dx = (−1)n+1 /(2n + 1)2 0 and that Z teγt dt = teγt /γ − eγt /γ 2 [50 marks]