Example 1.  Use the Taylor method of order    to compute numerical solutions for the differential equation    with initial condition    over the interval  .

Solution  1.

First, enter the function    and create explicit formulas for    for  ,  respectively, which will involve  .  

Second, replace    with     and construct the implicit formulas    for  , respectively.  

Second, use the subroutine to compute the set of points and store them in the variable taylorset.
Then plot this set of points using the built in Mathematica subroutine ListPlot.  

(c) John H. Mathews 2003