I have a problem on difference method for initial value problem .

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

is a third order method for the DE above.

I know that by Taylor's expansion, where

This gives

The difference method above also give

I am able to find y', y'' and y''' in terms of partial derivatives of f, but what am I supposed to do with and ?

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!