Derivation of The Midpoint Method
The midpoint method is a refinement of the Euler's method
and is derived in a similar manner. The key to deriving Euler's method is the approximate equality
which is obtained from the slope formula
and keeping in mind that
For the midpoint method, one replaces (3) with the more accurate
when instead of (2) we find
One cannot use this equation to find as one does not know at The solution is then to use a Taylor series expansion exactly as if using the Euler method to solve for :
which, when plugged in (4), gives us
and the midpoint method (1).
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