An initial value problem is how to aim my gun. A boundary value problem is how to aim my gun so that the bullet hits the target.

Qualitatively the methods of solution are sometimes different, because Taylor series approximate a function at a single point, i.e. at 0.

Here is an example:
For an ordinary 2nd order linear differential equation
y" + f(t) y' + g(t)y = 0,

- the initial value problem is to find the partial integral (solution) of the equation which satisfies the initial conditions
y(t0) = a, y'(t0) = b (t0, a and b are arbitrary constants);

- the boundary value problem is to find the partial integral which satisfies the boundary conditions
y(t0) = c, y(t1) = d (t0, t1, c and d are arbitrary constants, t0 < t1).

In other words: the initial value problem involves conditions in one point (the initial point of an interval), the boundary value problem involves conditions in various points (boundary points of the interval). 

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Probability sampling is a method for selecting choices on a completely random basis. Commonly, probability sampling is used to ensure that the selected sample is totally random, and not subject to any controls or rigging. This system works well, as long as all rules are followed, and the system is not violated in any way.

• Advantages
The principal advantage of probability sampling is fairness, as in contests where names are selected from a box full of entry forms. A selector will reach into the box, without looking, and pluck out the winner's name. This sort of selection process guarantees that every entrant has an equal chance at winning a prize.

• Disadvantages

The disadvantage of probability sampling is the possibility of flaws to the randomness model - in other words, people may cheat the system or interfere with the innate fairness of the probability sampling system. For example, in a contest, illegal or duplicate ballots may be added to a box, weighting the odds in someone's favor.

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(Rolle’s Theorem) Suppose f (x) is continuous on *a, b+ and dif- ferentiable on (a, b). If f (a) = 0 = f (b), then there exists a point c ∈ (a, b) such that f 0 (c) = 0.
Proof. Suppose f (x) is continuous on [a, b], differentiable on (a, b) and f (a) =
0 = f (b). We will prove the theorem using two cases. First, suppose that f (x) > 0 for some x ∈ (a, b). Since f (x) is continuous on [a, b], there exists a point c ∈ [a, b] for which f (c) is the maximum value of f on [a, b]. Furthermore, f (c) > 0 implies c = a and c = b, so c ∈ (a, b) and so f 0 (c) = 0 because f (x) is differentiable on (a, b).
Now suppose f (x) ≤ 0 for all x ∈ (a, b). Then either f (x) = 0 for all x ∈ (a, b) in which case f 0 (x) = 0 for all x ∈ (a, b), or else f (x) < 0 for some x ∈ (a, b). Since f (x) is continuous on [a, b], we know that there is a point c ∈ [a, b] for which f (c) is the minimum value of f (x) on [a, b]. Since f (c) is the minimum on [a, b] and f (x) < 0 for some x ∈ (a, b), f (c) < 0. Consequently, c = a and c = b, so c ∈ (a, b) and therefore f 0 (c) = 0 because f (x) is differentiable on (a, b). This proves the theorem.

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