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.
Figure 3: Analytical solution y(t) = tan t vs. numerical
solution given at