Partial differential equations

3* Consider the following initial and boundary-value problem Ut - Uxx = e- + 2 cos(x), -T 0 u(x, 0) = X, - IT 0 ux (-7, t) = Ux(7, 1), t> 0. Suppose we know that the homogeneous solution of the above problem is given by un(x, t) : e n't( An cos(nx) + Bn sin(nx)) with An = 0 and Bn = 4 n 2(-1)" for any integer n 2 0. Use the Duhamel principle to find the inhomogeneous solution