It is only valid for so-called "natural boundary conditions". So these are boundary conditions that are fixed, free, roller supported, hinged, and so on. As soon as we have elements on the boundary that can store potential or kinetic energy, this will be slightly different. For instance when you have a mass or a spring connected to an end, the term between [brackets] will have a nonzero effect.
but when we considering boundary condition, it lost the generality at 5:12, right? so it is only valid for particular B.C.?
It is only valid for so-called "natural boundary conditions". So these are boundary conditions that are fixed, free, roller supported, hinged, and so on. As soon as we have elements on the boundary that can store potential or kinetic energy, this will be slightly different. For instance when you have a mass or a spring connected to an end, the term between [brackets] will have a nonzero effect.