Consider the following second order linear ODE with mixed boundary condition: 
$$\frac{d^2f}{dt^2}+a(t)f(t)=0,~\frac{df}{dt}(0)=u,~f(1)=0,$$
where $u\in R$ and $a\in C[0,1]$ are fixed.


Is the solution to this equation unique? If so, how to prove it? Thanks!



Second order linear ODE with mixed boundary condition
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.


