Problem 4. (10 points) Consider the three polynomials Po(x) = 1, P1(x) = x, and P2(x) = = (3x2 - 1). Examine whether the above polynomials are linearly independent, orthogonal to each other in the interval -1 0 is the corresponding Damkohler number. (a) Examine whether the linear operator _ is self-adjoint. (5 points) (b) Examine whether the EVP (i.e., the ODE along with the homogeneous BCs) is self- adjoint. (5 points) (c) Solve the EVP, i.e., calculate the eigenvalues, 1, and the corresponding normalized eigenfunctions. (15 points) Problem 6. (25 points) Consider the eigenvalue problem 40 (5) = -12 0 (5 ) in the domain 0 0. Calculate the eigenvalues and the corresponding normalized eigenfunctions. Hint: Examine whether _ is self-adjoint. If it is not, convert it to a self-adjoint operator