A much better solution results if the time evolution through Dt is taken in two steps; first use the Euler method to evolve the solution to t + Dt/2, namely
Then use the "slope" at half time step to predict the solution at t + Dt in a second Euler like step
This method is often called the mid-point method, since the first step evolves the solution to a mid-point in time. Geometrically, the procedure corresponds to the following interpretation
The reason that this method should be more accurate than the simple Euler method is related to the fact that the "symmetric" form to compute the slope is more accurate than the "forward" or "backward" forms. The slope used to predict the next value of the function is calculated at half time step.
The better accuracy of the mid-point method can be illustrated by comparing this solution to that obtained by the Euler method for the radioactive decay problem, for which an exact solution exists.
The Maple worksheet (Radiactive_Euler_Mid_Point) is available.
Section 8.3 
Chapter 8 
Section 8.5
TOC
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