Trapezium or Midpoint Method

We know from (3.13) in Chap. 3, how that same derivative approximation is of higher order 0(8t2) when applied at the midpoint, and this leads to the trapezium method or midpoint rule, in which we must find an expression for the right-hand side of (4.1) at time t + 6t. This can be approximated as the average of the values at both ends  

This can be awkward to go on with, being implicit in yn+ p in our specific example (4.3), however, there is no problem, the above equation becoming 

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