What Distinguishes the Methods

For most of the methods used to solve odes (at least, those described here), the way in which the methods differ hinges on how the following three questions are answered  

To what value of t is that approximation intended to apply  

It is very important to be clear on these points in devising simulation strategies, especially (when going on to pdes) the boundary conditions, which must conform to these points as well. 

It will be seen that for the three methods Euler, BI and the trapezium method, the same approximation expression is used for the left-hand side of (4.1) but because of points made in questions (2) and (3) above, the methods are very different. 

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