Question

a) Determine the constants a,\beta ,y in the differentiation formula y'(x_0)=ay(x_0-h)+\beta y(x_0)+\gamma y(x_0+h)so that the method is of the highest possible order. Find the order and the error term of the method.

12 Mar 2024
Answer :
Word Count : 471
We are given a differentiation formula: \[ y'(x_0) = a y(x_0 - h) + \beta y(x_0) + \gamma y(x_0 + h) \] ### Step 1: Express \( y(x) \) using a Taylor Series Expansion Expanding \( y(x_0 - h) \) and \( y(x_0 + h) \) in Taylor series about \( x_0 \): \[ y(x_0 - h) = y(x_0) - h y'(x_0) + \frac{h^2}{2} y''(x_0) - \frac{h^3}{6} y'''(x_0) + O(h^4) \] \[ y(x_0 + h) = y(x_0) + _____ ________ ________ ________ _____ ___ _______ _____ ______ ______.
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