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To solve for the constants \( a \), \( \beta \), and \( \gamma \) in the differentiation formula, we need to derive them so that the method yields the highest possible order. The formula you're dealing with is: \[ Y'(x_0) = a y(x_0 - h) + \beta y(x_0) + \gamma y(x_0 + h) \] We want to determine the constants \( a \), \( \beta \), and \( \gamma \) such that the method is of the highest possible order. ### Step 1: Taylor Expansion of the Terms We can use the Taylor series expansions for \( y(x_0 - h) \) and \( y(x_0 + h) \) around \( x_0 \): - \( y(x_0 - h) = y(x_0) - h y'(x_0) + \frac{h^2}{2} y''(x_0) - \frac{h^3}{6} y^{(3)}(x_0) + \dots \) - \( y(x_0 + h) = y(x_0) + h y'(x_0) + \frac{h^2}{2} y''(x_0) + \frac{h^3}{6} y^{(3)}(x_0) + \dots \) ### Step ___ ______ _________ ___ _____ ____ _____ __________ _________.
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