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In this assignment the boldface n is a parameter based on the last digit of your student number. n is obtained by taking the last digit in your student number and adding 2. Eg. If your student number is 3216385, then take the last digit 5 and add 2 to get n=5+2. 1. The temperature u(x,t) within a particular rod of length n with insulated ends is governed by the following boundary value problem ∂u ∂2u ∂u(0,t) ∂u(n,t) 􏰀 x, 0<x<1 ∂t =n∂x2, ∂x =0= ∂x and u(x,0)= 0, 1<x<n Use the separation of variables method with u(x,t) = X(x)T(t) to determine the temperature along the rod for t > 0. (7 marks) 2. The vibrations along a particular beam u(x, t) with simply supported ends is governed by the following boundary value problem, u(0,t) = 0 = ∂2u(0,t) ∂x2 u(n,t) = 0 = ∂2u(n,t) ∂x2 ∂2u + n2 ∂4u = 0, ∂t2 ∂x4 u(x,0) = f(x) Use the separation of variables method with u(x,t) = X(x)T(t) to find an expression for the eigenvlaues λn. (8 marks)

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