Here is the page on Wikipedia:
So it says the fixed boundary conditions for the function itself as well as for the first $n-1$ derivatives. You can fix the boundary points physically say $y(a)=a'$ and $y(b)=b'$ but what about for the derivatives?
For example for this problem from my lectures-
We've fixed $y(0) =0$ for this diving board problem - where does $y'(0)=0$ come from?
Then in the variant where the swimmer holds the board up we fix the the other end of the board $x=L$ at a certain height so $y(L)=$ "whatever". We need 4 boundary condtions here where is the other? I assume by Wikipedia article that $y'(L)$ is something- is the other fixed boundary condtion but what is it- it's not mentioned in the notes...
Would appreciate any clarification of the ambiguity thanks...