Question

Determine the constants \alpha ,\beta,\gamma in the differentiation formula
y'(x_{0}) = \alpha y(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.

14 Feb 2018
Answer :
Word Count : 495
To determine the constants \(\alpha\), \(\beta\), and \(\gamma\) in the differentiation formula for the highest possible order, we will use a Taylor series expansion. Given the formula: \[ y'(x_{0}) = \alpha y(x_{0} - h) + \beta y(x_{0}) + \gamma y(x_{0} + h) \] We will expand each of the function values using the Taylor series around \(x_0\): 1. \(y(x_0 - h)\) in terms of \(y(x_0)\), its derivatives, and powers of \(h\). 2. \(y(x_0 + h)\) similarly. ### Step 1: Taylor Expansions For \(y(x_0 - h)\), we use the Taylor series expansion around \(x_0\): \[ y(x_0 - h) = y(x_0) - h y'(x_0) + \frac{h^2}{2} y''(x_0) - \frac{h^3}{6} y^{(3)}(x_0) + \dots \] For \(y(x_0 + h)\), the Taylor expansion is: \[ y(x_0 + h) = y(x_0) + h y'(x_0) + \frac{h^2}{2} y''(x_0) + \frac{h^3}{6} y^{(3)}(x_0) + \dots \] ### Step 2: Substitute into the Differentiation Formula Now, substitute these expansions into the original formula: \[ y'(x_0) = \alpha \left( y(x_0) - h y'(x_0) + \frac{h^2}{2} y''(x_0) - \frac{h^3}{6} y^{(3)}(x_0) + \dots \right) \] \[
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