(6.33)  
has to be solved subject to the BCs (6.51)
 
(6.34) 
(6.35) 

Special care has to be given to the multiple BCs. An algorithm that calculates all fundamental solutions of the ODE system independently of the BCs and then projects these general solutions to the BCs could treat the N_{s} BCs simultaneously and thus no extra costs would result. At a first glance this idea seems to be strange because calculating all solutions first and then filtering out only the required ones seems to be superfluous since a firstorder N_{ODE}dimensional ODE system has N_{ODE} fundamental solution vectors and N_{s} is always very small in comparison to N_{ODE}. However, exactly this idea can be implemented in a very tricky way to result in a fundamental advantage. Before we work out this strategy in Section 6.4.3 some general information about numerical techniques suited for BVPs is provided in the following section.