The method of separation of variables is to be used to solve and analyze solutions of the partial differential equation Ox² 8t² with the boundary conditions, u(0,t) = 0, -(x,t)=0. The solutions can be expressed as a product of functions of one variable in the form u(x,t) = X(x) T(t). (a) (b) (c) Write down the ordinary differential equation which X(x) must satisfy and state its boundary conditions (5 marks) Write down the corresponding ordinary differential equation which T(t) must satisfy. (2 marks) Assuming that the separation constant μ = R², where k > 0, solve the non- trivial solutions for X (x) that must satisfy the boundary conditions. State clearly the values that u can take. Answer: (a) d²x(x) 2 dx² -μX(x) = 0 with X(0)=X'(x) = 0. d²T(t) (b) = μT (t) dt² (c) X(x)=B sin (2x-1), 27° x where =- (6 marks)