Take the 2-minute tour ×  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!
share|improve this question Before someone tells you some
answers, it is rather instructive to work out examples yourself.
If you let $a$ be a constant (the sign matters!), you can find
the general solution to the ODE (without the boundary
conditions) and figure out when the boundary conditions can be
met or not. After you've done that, look up "Sturm-Liouville
theory" for self-adjoint second-order ODE's. –  Deane Yang May 6
'11 at 21:15 I don't think this is quite MO-level, though I'm
sure there could be a very interesting discussion to be had
around your 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.