(a) Determine the adjoint of the differential operator u = L[u] = u' + 2xu with respect to the L2 inner products on [0, I] when subject to the fixed boundary conditions u(0) = u(1) = 0.
(b) Is the self-adjoint operator K = L* °L is positive definite? Explain your answer.
(c) Write out the boundary value problem represented by K[u] = f.
(d) Find the solution to the boundary value problem when f(x) = ex2. Hint: To integrate the differential equation, work with the factored form of the differential operator.
(e) Why can't you impose the free boundary conditions u(0) = u'(1) = 0 in this situation?