Consider the problem: y″ + 2y′ + (1 − \)y = 0, y'(0) = y' (π) = 0 1. Find the eigenvalues and eigenfunctions. 2. Verify that the eigenfunctions of the problem are orthogonal on [0,7] to the weight function w(x) = −1. 3. Find an orthonormnal set of eigenfunction on [0, 7].

4. Solve given that x' = Ax + A has eigenvalues -2 and -6. -3 3 1 5

Find the eigenvalues and eigenfunctions of the boundary-value problemx2y′′+xy′+ 9λy= 0, y′(1) = 0,y(e) = 0.   Put the equation in self-adjoint form, and give an orthogonality relation   Show that if ym and yn are two eigenfunctions corresponding to two different eigenvalues λm and λn respectively, then ym and yn are linearly independent