# 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.

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).

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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