The time-integration problem65 for ordinary differential equations (ODEs) is traditionally written as follows: We are given an integer (the number of ODEs to integrate), a “right-hand-side” function , and the value of a function satisfying the ODEs
This is a well-studied problem in numerical analysis. See, for example, Forsythe, Malcolm, and Moler [71, Chapter 6] or Kahaner, Moler, and Nash [92, Chapter 8] for a general overview of ODE integration algorithms and codes, or Shampine and Gordon , Hindmarsh , or Brankin, Gladwell, and Shampine  for detailed technical accounts.
For our purposes, it suffices to note that highly accurate, efficient, and robust ODE-integration codes are widely available. In fact, there is a strong tradition in numerical analysis of free availability of such codes. Notably, Table 3 lists several freely-available ODE codes. As well as being of excellent numerical quality, these codes are also very easy to use, employing sophisticated adaptive algorithms to automatically adjust the step size and/or the precise integration scheme used66. These codes can generally be relied upon to produce accurate results both more efficiently and more easily than a hand-crafted integrator. I have used the LSODE solver in several research projects with excellent results.
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