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2.2.1 Taylor Series Methods     continued...

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.