Several methods exists for finding better numerical solutions to first order differential equations of the formthan Euler’s simple formula

One method uses forwards and backwards Euler formulae to derive a ‘central’ formula.

Subtracting these gives

Theterm means that the error at each iteration is of the order ofThe error for the simple Euler forwards or backwards formulae are of the order ofso the central formula is more accurate.

Another method uses Euler’s simple formula to find an estimate forusingat then uses this estimate to find a second estimate forat the pointAn average of these two values ofis used to provide an improved estimate for

The process is

Use this value ofto find

Then find

Example. Ifwithestimatewith

The problem can be solved exactly by separation of variables.

hence

This is very accurate.