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5. A partial differential equation is azu 0x2 = Q2 Ou at> where a is a given constant and a > 0, 00. The boundary conditions for the equation are u(0,t)=0 and u(L,t)=0 . The initial condition is u(x,0) = b, where b is a given constant. (1) Use the method of separation of variables to transform the partial differential equation into ordinary differential equations. You must use the following notations: u(x, t) = f(x) g(t) (2) Derive the boundary conditions for function f(x). (3) Obtain the solutions for f (x). (4) Obtain a solution for u(x, t) such that the initial condition u(x, 0) = b is satisfied. 3 marks each. However, you will not receive more than 8 marks if other notations are used