Consider the eigenvalue problem on 0 S a: 3 7r, (1245 595 _ E+8E+(16+A)u_0, #00 + 4(0) = 0, 42%) + 4(7r) = 0- (a) Assuming A to be a non-negative constant, what is the general solution of the homogeneous ODE. Consider the cases A > 0 and A = 0 separately. (b) Apply the boundary conditions to determine the eigenvalues and eigenfunctions. (c) What is the adjoint problem? Obtain the adjoint eigenfunctions. (d) ) (6 Using (a) and (b) write the simplest forms for the integrals for the stande inner product (k,1j)j)2 and the weighted inner product ;(k, j>a orthogonality conditions and show that they are equivalent. What are the functions 1), q, a that put this problem in stande SturmLiouville form? (see Prob 1) 00 (f) Obtain the coefcients ck in the expansion of the function u(:c) = Z cknk(3:) that solves the BVP: k={} 2 :77: + 8%: + 121:. = 1734\" sin(3=/2)a it"(O) + 4u(0) = 5000 it"(fr) + 4u(7r) = 4. Hint: If (Mm) is a solution of L4; 2 A45 then it is also a solution of Lq at = 5\\ where 1 = A + c