I am dealing with a vector valued second order homogeneous BVP:


$\ddot u(t) = A(t)\dot u(t) + B(t)u(t)$ with $u(0)=u(1)=0.$


where $A$ and $B$ are $n \times n$ matrices with smooth coefficients and $u$ is a vector valued smooth function on $[0,1]$.


Are there any results that say something about the non-trivial solutions of this equation(their existence, their number, a priori estimates) without too "stringent" conditions on $A$ and $B$. Or is there any criteria implying the non-existence of non-trivial solutions?. 


Any help or reference is highly appreciated. 



vector valued BVP for ODE's
There are many different ways to obtain sufficient criteria for the nonexistence of nontrivial solutions. Here are some rather simple ways:


1. You can solve the equation explicitly when the coefficients are constant. It is rather straightforward in this situation to identify when nontrivial solutions exist or not.
2. You can use Sturm comparison theory to compare the equation with variable coefficients to a model equation with constant coefficients. This is an easy way to extend what you learned in #1 to variable coefficients. To learn more about this, look for descriptions of Sturm-Liouville theory in ODE books or of Jacobi fields on Riemannian manifolds in books on Riemannian geometry. There are papers and monographs by Jurgen Jost and/or Hermann Karcher that explain this very well in the context of Jacobi fields.
3. Rewrite the equation by moving all the terms to one side, leaving zero on the other. Take the dot product of the equation with $u$ and integrate over the interval. Look for conditions on $A$ and $B$ such that if the integral vanishes, so must $u$.
4. There are probably relatively simple conditions on $A$ and $B$ that allow you to use the maximum principle to infer that $u$ must be zero.


