If we are willing to do more work, we can obtain the results given in Table 1.
Table 1: Approximate solution using Euler's method.
We note from the last column of Table 1 that ; in fact, for , . This is consistent with the fact that Euler's method is of order p = 1.
If we repeat the above calculations using a Taylor method of order 2, we obtain
and the results in Table 2. From Table 2, we see that for , .
Table 2: Approximate solution using the Taylor series algorithm of order 2.