What's the difference between an initial value problem and a boundary value problem?

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.

For a simple example (second order ODE), an initial value problem would say $y(a)=p$, $y'(a)=q$.

A boundary value problem would specify $y(a)=p$, $y(b)=q$.

Initial Value Problems:

In initial value problems, we are given the value of function $y(x)$ and its derivative $y'(x)$ at the same point ( initial point ) sy at $x = 0$ i.e $y(0)= xi1$ and $y'(0)= x_2$.

Boundary Value Problems:

In boundary value problem, we are given the value of function $y(x)$ at two different points, i.e $y(a)= x_1$ and $y(b)= x_2$.