## Finite difference method

I have a problem on difference method for initial value problem $\displaystyle y'(t)=f(t,y)$.

I am supposed to determine the coefficients b0, b1, b2 such that the difference method

$\displaystyle y_{n+1}=y_{n}+k[b_{0}f(t_{n+1},y_{n+1})+b_{1}f(t_{n},y_{n})+b_{2}f (t_{n-1},y_{n-1})], n=1,2,...$

is a third order method for the DE above.

I know that by Taylor's expansion, $\displaystyle y(t_{n+1})=y(t_{n})+ky'(t_{n})+\frac{k^2}{2!}y''(t _{n})+\frac{k^3}{3!}y'''(t_{n})+...$ where $\displaystyle k=t_{n+1}-t_{n}$

This gives $\displaystyle \frac{y(t_{n+1})-y(t_{n})}{k}=y'(t_{n})+\frac{k}{2}y''(t_{n})+\frac {k^2}{6}y'''(t_{n})+...$

The difference method above also give $\displaystyle \frac{y(t_{n+1})-y(t_{n})}{k}=b_{0}f(t_{n+1},y_{n+1})+b_{1}f(t_{n}, y_{n})+b_{2}f(t_{n-1},y_{n-1})$

I am able to find y', y'' and y''' in terms of partial derivatives of f, but what am I supposed to do with $\displaystyle f(t_{n+1},y_{n+1})$ and $\displaystyle f(t_{n-1},y_{n-1})$?

I tried expanding using Taylor series but not too sure how to handle the increment in y. How do I obtain equations to find b0, b1, b2?

How many terms must I keep in the Taylor's expansion for 3rd order accuracy?

Appreciate if you can help!