The temperature distribution (x, t) along an insulated metal rod of length L is described by the differential equation a2e 1 ae ar2 D at (0 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 de/ax = 0 when r = 0 for t > 0 together with 8 = 0 when I = L for t > 0. The initial temperature distribution in the rod is given by 773 0(3,0) = 0.3cos (T≤ (0 0. In this case the general 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, t) = exp(-Dk²t) cos(kx), satisfies the given partial differential equation for any constant k. (e) Given that the general solution of the partial differential equation and boundary conditions may be expressed as D(2n-1) 0(x, t) = Cnexp (2n-1)72 cos 4L2 2L 1 find the particular solution that satisfies the given initial temperature distribution