Given the partial differential equation: au(x, t) at (a) By showing all possible cases, use separable method to find the general solutions, u(x, t), of the problem. (b) Given that the boundary and initial conditions are as below: Boundary conditions: au(x, t) x Initial conditions: u(0,t)= u(1,t)=0, t>0 u(x,0) = x(1-x), 0x1. du(x, 0) at 1 u(x, t) = (mm) n=1 = 0, 0x 1. Show that the solution for the nontrivial case, which is for k = -2, is: [1- (-1)"] cos nt sin nx (12 marks) (13 marks)