I'm not sure whether this is really appropriate for MathOverflow or not. Still, let me say a little more: As I've mentioned above, you can work out completely the case where $a$ is constant using explicit solutions. If $a$ is not constant, I'm not aware of a definitive answer but you can get separate necessary conditions and sufficient conditions, involving upper or lower bounds on $a$ using the Sturm comparison theorem. Last, I believe that it is possible to find an integral condition on $a$ that is sufficient for there to be a unique solution to the boundary value problem.

The vector-valued version of this problem is used to analyze how geodesics on a Riemannian manifold behave given assumptions on the curvature.

aroundyour question. My advice would be to assume that the solution is not unique in general, because boundary conditions seldom lead to uniqueness. So I would look for a counterexample. A much trickier question will be: are there easy conditions to check for which wedohave uniqueness. This might be what Deane has in mind, but I can't be sure. – Thierry Zell May 7 '11 at 1:19Sturm-Liouville Theory. amazon.com/dp/0821839055 – Igor Khavkine May 7 '11 at 8:59