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Question 2 (Unit 12) 17 marks The temperature distribution (x, t) along an insulated metal rod of length L is described by the differential equation 8²0 1 80 əx² Dat (0<x<L, t> 0), where D #0 is a constant. The rod is held at a fixed temperature of 0°C at one end and is insulated at the other end, which gives rise to the boundary conditions 20/0x = 0 when x = 0 for t> 0 together with 0 = 0 when x = L for t > 0. The initial temperature distribution in the rod is given by 0(x,0) = 0.3 cos (7) (0 ≤ x ≤L). 2L (a) Use the method of separation of variables, with 0(x, t) = X(x)T(t), to show that the function X(r) satisfies the differential equation X" - μX = 0 for some constant μ. Write down the corresponding differential equation that T(t) must satisfy. [3] (b) Find the two boundary conditions that X(z) must satisfy. [3] (c) Suppose that μ< 0, so μ = -k² for some k > 0. In this case the general solution of equation (*) is X(x) = A cos(kx) + B sin(kr). Find the non-trivial solutions of equation (*) that satisfy the boundary conditions, stating clearly what values k is allowed to take. [4] (d) Show that the function f(x, t) = exp(-Dk²t) cos(kx), satisfies the given partial differential equation for any constant k. [3]