# Parker–Sochacki method

In mathematics, the Parker–Sochacki method is an algorithm for solving systems of ordinary differential equations (ODEs), developed by G. Edgar Parker and James Sochacki, of the James Madison University Mathematics Department. The method produces Maclaurin series solutions to systems of differential equations, with the coefficients in either algebraic or numerical form.

## Summary

The Parker–Sochacki method rests on two simple observations:

• If a set of ODEs has a particular form, then the Picard method can be used to find their solution in the form of a power series.
• If the ODEs do not have the required form, it is nearly always possible to find an expanded set of equations that do have the required form, such that a subset of the solution is a solution of the original ODEs.

Several coefficients of the power series are calculated in turn, a time step is chosen, the series is evaluated at that time, and the process repeats.

The end result is a high order piecewise solution to the original ODE problem. The order of the solution desired is an adjustable variable in the program that can change between steps. The order of the solution is only limited by the floating point representation on the machine running the program.