Let k> 0 be constant and let u(x, y) be a function satisfying the following partial differential equation du Ju = k t 0x2 for all x, t with 0 x 1 and t 0. boundary conditions: u Ju x x and u(x,0) = f(x) for 0 x 1, where f(x) is a given function. (0, t) Also suppose that u satisfies the following (1, t) = 0, (1) (a) Briefy state what kind of physical situation this boundary value problem could represent. 2 marks] (b) Using the method of separation of variables, find the set of ordinary differential equations that can be used to solve this PDE. [4 marks] (c) Show that the boundary conditions given above lead to homogeneous boundary conditions for one of the ODEs that you have found in part (b). [4 marks] (d) Show that the general solution to equation (1) and boundary conditions is u(x,t) = An cos(nna)e" n=0 nkt [20 marks]