In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to the solution of an initial value problem. The following exposition may be clarified by this illustration of the shooting method.
For a boundary value problem of a second-order ordinary differential equation, the method is stated as follows. Let

be the boundary value problem. Let
denote the solution of the initial value problem

Define the function
as the difference between
and the specified boundary value
.

If the boundary value problem has a solution, then
has a root, and that root is just the value of
which yields a solution
of the boundary value problem. The usual methods for finding roots may be employed here, such as the bisection method or Newton’s method.
Linear shooting method
The boundary value problem is linear if
has the form

In this case, the solution to the boundary value problem is usually given by
where
is the solution to the initial value problem

and
is the solution to the initial value problem

See the proof for the precise condition under which this result holds.
Remark
One can easily see that if
then
for all
. Thus
for all
. Besides, the formula
comes from the fact that we need to find
as a combination of
and
. In this manner,
should be of the form
for all
.
At
,
which implies that
. This is self-satisfied.
At
,
which implies that
.
Therefore,

Example
A boundary value problem is given as follows by Stoer and Bulirsch.
.
The initial value problem

was solved for
, and
plotted in the first figure. Inspecting the plot of
, we see that there are roots near
and
. Some trajectories of
are shown in the second figure.
Solutions of the initial value problem were computed by using the LSODE algorithm, as implemented in the mathematics package GNU Octave. Stoer and Bulirsch state that there are two solutions, which can be found by algebraic methods. These correspond to the initial conditions
and
(approximately).

The function
.

Trajectories
for
equal to
,
,
,
, and
(red, green, blue, cyan, and magenta, respectively). The point
is marked with a red diamond.
Source: http://en.wikipedia.org/wiki/Shooting_method