Here is an illustration of the shooting method.

A boundary value problem is given as follows by Stoer and Bulirsch (Section 7.3.1).

The initial value problem

was solved for s = -1, -2, -3, ..., -100, and F(s) = w(1;s) - 1 plotted in the first figure. Inspecting the plot of F, we see that there are roots near -8 and -36. Some trajectories of w(t;s) 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 w'(0) = -8 and w'(0) = -35.9 (approximately).

Reference: Josef Stoer and Roland Bulirsch, Introduction to Numerical Analysis. New York: Springer-Verlag, 1980.

External link: Brief Description of ODEPACK (at Netlib; contains LSODE)

The function F(s) = w(1;s) - 1.

Trajectories w(t;s) for s = w'(0) equal to -7, -8, -10, -36, and -40 (red, green, blue, cyan, and magenta, respectively). The point (1,1) is marked with a red diamond.