Editing Shooting method (section)

=== Standard boundary value problem ===
[[Image:Shooting method trajectories.svg|thumb|400px|Figure 1. Trajectories ''w''(''t'';''s'') for ''s'' = ''w''<nowiki>'</nowiki>(0) equal to −7, −8, −10, −36, and −40. The point (1,1) is marked with a circle.]][[Image:Shooting method error.svg|thumb|400px|Figure 2. The function ''F''(''s'') = ''w''(1;''s'') − 1.]]A [[boundary value problem]] is given as follows by Stoer and Bulirsch<ref name = "Stoer1980">Stoer, J. and Bulirsch, R. ''Introduction to Numerical Analysis''. New York: Springer-Verlag, 1980.</ref> (Section 7.3.1).
<math display="block"> w''(t) = \frac{3}{2} w^2(t), \quad w(0) = 4, \quad w(1) = 1 </math>

The [[initial value problem]]
<math display="block"> w''(t) = \frac{3}{2} w^2(t), \quad w(0) = 4, \quad w'(0) = s</math>
was solved for ''s'' = −1, −2, −3, ..., −100, and ''F''(''s'') = ''w''(1;''s'') − 1 plotted in the Figure 2.
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 Figure 1.

Stoer and Bulirsch<ref name = "Stoer1980"/> 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).{{clear}}