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## Homework Statement

Consider the following implicit scheme:

[tex]y_{n+1}=y_{n}+\frac{\Delta t}{2}\left [f(y_{n+1})+f(y_{n})][/tex]

By linearization one can obtain an explicit scheme which is an approximation to this - with approximation error [tex]O(\Delta t^{3})[/tex]

## Homework Equations

The solution is:

[tex]y_{n+1}=y_{n}+\Delta t\left [1-\frac{1}{2}\Delta t f'(y_{n}) \right ]^{-1}f(y_{n})[/tex]

And the notation: [tex]y'(x)=f(x,y(x))[/tex]

## The Attempt at a Solution

I think I have to take the Taylor expansion of [tex]f(y_{n+1})[/tex]

I get:

[tex]f(y_{n+1})=f(y_{n})+\Delta tf'(y_{n})+O(\Delta t^{2})[/tex]

Substituting in the main scheme:

[tex]y_{n+1}=y_{n}+\frac{\Delta t}{2}\left [f(y_{n})+\Delta tf'(y_{n})+O(\Delta t^{2})+f(y_{n})]f(y_{n+1})=y_{n}+\frac{\Delta t}{2}\left [f(y_{n})+\Delta ty'(y_{n})+O(\Delta t^{2})+f(y_{n})]f(y_{n+1})[/tex]

This expression is, as far as I can see, not equal to the expression of the solution.

Any help would be appreciated!