Initial value problems, by definition, are problems where we have a differential equation and specified values of the solution (and its derivatives) at the same point. If we have conditions such as $y(t_0) = y_0$ and $y(t_1) = y_1$ (called boundary conditions), then this problem is called a boundary value problem. To solve these problems, the same process can be used to get the general solution, but you use the boundary conditions instead to find a particular solution. However, unlike initial value problems, where we only needed some continuity conditions for there to be a unique solution, boundary value problems may have infinite, one, or no solutions.