  This study introduces the reader to the theory of approximating the
solution(s) of a non-linear, second order, ordinary differential equation (ODE)
with piecewise polynomial functions by using the collocation method. It then
focuses on the application of the method to generate collocation approximations
for the Van der Pol equation with initial conditions and with varying equation
parameter, {\mu}. It is shown that the collocation method in its original form
is impractical for generating these approximations due to the numerical costs
of producing such an approximation, particularly as the stiffness of the
equation increases with parameter, {\mu}. An adaptation to the method, termed
segmented collocation, is proposed, reliant upon on the specific structure and
convergence for initial value problems, and shown theoretically and numerically
to be capable of generating equivalent approximations for much superior costs.
Further study is proposed; to compare the segmented method to other numerical
methods traditionally employed with equations of this type such as the
Runge-Kutta family; and to develop and test the method in order to ascertain
its complete usefulness and competitive standing in the context of
approximating second order, ODEs with initial conditions.
