The temperature distribution 0(x, t) along an insulated metal rod of length L is described by the differential equation 2²0 1 20 əx² D Ət (0 < x < L, t > 0), where D0 is a constant. The rod is held at a fixed temperature of 0°C a one end and is insulated at the other end, which gives rise to the boundary 0 for t> 0 together with 0 = 0 when x = L for conditions = 0 when x 20 Əx 1 t> 0. The initial temperature distribution in the rod is given by condo (b) (c) = 7 0(x,0) = 0.3 cos (2T) (0≤x≤1). L (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. Write down the corresponding differential equati that T(t) must satisfy. salam d 1 Find the two boundary conditions that X(x) must satisfy. dineob ba Suppose that μ< 0, so μ = -k² for some k > 0. In this case the gene solution of equation (*) is X(x) = A cos(kx) + B sin(kx). Find the non-trivial solutions of equation (*) that satisfy the boundary conditions, stating clearly what values k is allowed to take. (d) Show that the function f(x, y) = exp(-Dk²t) cos(kx), satisfies the given partial differential equation for any constant k.