To obtain an approximate solution of order **p = 2** to the IVP (17), let
and generate the sequences

where , , .

Euler's method is the special case, , and has order 1; the improved Euler method has and the Euler-Cauchy method has .

Approximate the solution to , at using the Euler-Cauchy method with . The recurrence relation for is

and the resulting approximations are given in Table 4. The IVP has the solution . The approximate and exact solutions are represented graphically in Figure 3 where the approximating values have been joined by straight line segments.

Table 4: Approximate solution using the Euler-Cauchy method.