Our O.D.Es professor had the "amazing" idea of heavily introducing advanced linear algebra material (which is not an official prerequisite for the course) along with boundary value problems. Not being trained in these types of exercises I am facing quite a few difficulties. If anybody would be willing to help, it would be most appreciated.
For the following sets of boundary conditions, consider the equation:
$$u''(x) + \lambda u(x) = 0, \ \ \ 0 < x < 1$$
and find the spectrum of eigenvalues, in essence the set of values $\lambda$ in the complex plane for which a nontrivial solution exists, and give the eigenfunction (or eigenfunctions) for each such eigenvalue.
1) $u(0) + u(1) = 0$ and $u'(0) + u'(1) = 0$ 2) $u(0) + u(1) = 0$ and $u'(0) - u'(1) = 0$