  Purely numerical methods do not always provide an accurate way to find all
the global solutions to nonlinear ODE on infinite intervals. For example,
finite-difference methods fail to capture the asymptotic behavior of solutions,
which might be critical for ensuring global existence. We first show, by way of
a detailed example, how asymptotic information alone provides significant
insight into the structure of global solutions to a nonlinear ODE. Then we
propose a method for providing this missing asymptotic data to a numerical
solver, and show how the combined approach provides more detailed results than
either method alone.
