## Question

###### (20 points) A function u(z,t) satisfies the PDEUII0 <I < 2 t20and the homogeneous boundary conditionsu(0,t) =0,2(2,+) = 0,t20Setting u(z,t) F(r)G(t) implies the separated equations F" 2F" AF = 0 and G' + AG = 0_ where is a separation constant _(a) Find all real eigenvalues A and corresponding eigenfunctions F(z) of the eigenvalue problem con sisting of the separated equation for F(r) and the boundary conditions F (0) and F(2) implied by the homogeneous BCs for u(I,t) (Hi

(20 points) A function u(z,t) satisfies the PDE UII 0 <I < 2 t20 and the homogeneous boundary conditions u(0,t) =0, 2(2,+) = 0, t20 Setting u(z,t) F(r)G(t) implies the separated equations F" 2F" AF = 0 and G' + AG = 0_ where is a separation constant _ (a) Find all real eigenvalues A and corresponding eigenfunctions F(z) of the eigenvalue problem con sisting of the separated equation for F(r) and the boundary conditions F (0) and F(2) implied by the homogeneous BCs for u(I,t) (Hint: Iyou need only consider Solve the separated equation for G(t) with the values for obtained in part (a), and then SUm over all of the product solutions F(r)G(t) found to form the general solution satisfying the PDE and its BCs. Find the solution of the PDE and BCs subject to the initial condition u(z,0) = 3ef sin(Tr) 2u: