An Analytical Example

If   is a sufficiently simple function, it may be possible to solve the recurrence relation for as a function of and . Such an explicit solution is rarely of practical interest, because it can usually only be found in cases where the differential equation itself can be solved in closed form. However, it can be helpful in the study of theoretical properties of the method under consideration.

Let us find the explicit form of the Euler approximation..

Euler approximation to the solution of the IVP

Here we have and hence

In view of , we thus find

and generally

Since , the value approximating the solution at the point is thus given by

As this tends to . We thus have shown that by decreasing the mesh size, the exact solution of the IVP can be approximated arbitrarily well in the special example under consideration.