the boundary value problem versus the initial value problem
Two ways of pinning down a moving rule lead to two very different worlds. An initial value problem is like launching a projectile: you fix everything at one instant — position and velocity right now — and the future unfolds, fully determined. A boundary value problem is like asking for a path that starts here and must end there: you fix one fact at each of two separate places, and you ask what curve threads between them. The first looks forward from a single point; the second reaches across an interval.
For a second-order equation both kinds use up the same two pieces of information, but they place them differently. An initial value problem gives y(a) and y'(a) — value and slope at the same point a. A boundary value problem gives y(a) and y(L) — values (or slopes) at two different points. This relocation changes everything. The initial value problem for a nice linear equation has, by the existence-and-uniqueness theorem, exactly one solution. The boundary value problem has no such guarantee: pushing the second condition to the far end can make the algebra for the two constants inconsistent (no solution) or degenerate (infinitely many).
The practical consequence is large. Initial value problems march forward step by step, and numerical methods like Euler or Runge-Kutta integrate them directly. Boundary value problems cannot simply march, because you do not know the starting slope; they require solving for the whole curve at once — by shooting (guessing the missing initial slope and correcting), by relaxation, or, in the special homogeneous case, by the eigenvalue methods of Sturm-Liouville theory. Knowing which type you face decides which tools apply.
Take y'' + y = 0. The IVP with y(0) = 0, y'(0) = 1 has the unique solution y = sin(x). But the BVP with y(0) = 0, y(pi) = 0 has infinitely many solutions y = c sin(x) for every c, since sin(pi) = 0 makes the second condition automatic. Same equation, opposite outcomes.
Same differential equation: the IVP has exactly one solution, while the BVP at these endpoints has infinitely many.
Do not carry over the comforting intuition from initial value problems. The clean 'one equation plus two conditions equals one solution' rule is an IVP fact; for a BVP, existence and uniqueness can both fail, and the failure is not a sign of a mistake.