Thus far we have only discussed initial value problems.
In the projectile problems, for example, we fixed the initial position
and velocity and integrated over the variables up to a final position
and velocity.
But there are many ODE problems in which two or more points
are fixed and solutions of the ODE must be found that satisfy these constraints.
Such boundary value problems can be solved with two approaches:
 Shooting Method
 Finite difference method
The finite difference method works only with linear ODEs. A grid of points
is defined over the intervals between the initial and final values of
the independent variable. Then derivatives are substituted with difference
equations between the grid points, resulting in a set of algebraic linear
equations. The end points are known from the boundary conditions while
the values at the inner points must be solved . The equations can be solved
with relaxation methods. (See the references
for more about the finite difference methods.)
Here we will focus on the shooting method
since it works with either linear or nonlinear equations.
References
& Web Resources
