I do not follow in the Butcher's book on the page $112$ what is $y(x_n)$ and $y(x_{n-1})$ in $241c$ in the snippet below. How should I plug in the step $h$ to get, say, $y(x_n)$?
I have also a problem to simply expand for the Taylor formula $$y(x_n)-y(x_{n-1})-h(\beta_0y'(x_n)+\beta_1y'(x_{n-1})+\beta_2y'(x_{n-2})+\ldots+\beta_ky'(x_{n-k}))$$ to obtain for $k=4$ this coefficients $C_1$ and $C_2$:
$C_1=(1-\beta_0-\beta_1-\beta_2-\beta_3-\beta_4)$
$C_2=(-\frac{1}{2}+\beta_1+2\beta_2+3\beta_3+4\beta_4)$
In equation (241c), we have the following expression:
y(xn) - y(xn-1) - h(β0y'(xn) + β1y'(xn-1) + β2y'(xn-2) + ... + βky'(xn-k))
Here's what each part means:
y(xn) represents the value of the function y at a specific point called xn. y(xn-1) represents the value of the function y at the previous point, which is one step before xn. h is the step size, which tells us how big the steps are between the points xn and xn-1. β0, β1, β2, ..., βk are some numbers called coefficients. These coefficients help us calculate the derivatives of y at different points. To find y(xn), we need to know the starting value of y, which is usually given as y(x0). So, we need an initial condition, like y(x0) = y0, where x0 is the starting point and y0 is the corresponding value of the function at that point.
To calculate y(xn), we use a numerical method that works step by step. We start with the initial condition y(x0) = y0 and use the given coefficients and step size to find the value of y at the next point, xn. Then we repeat this process to find y at subsequent points.
Now, regarding the expansion of the Taylor formula for k = 4 and the coefficients C1 and C2. We expand the expression like this:
y(xn) - y(xn-1) - h(β0y'(xn) + β1y'(xn-1) + β2y'(xn-2) + β3y'(xn-3) + β4y'(xn-4))
Expanding the terms and grouping similar terms together, we get:
y(xn) - y(xn-1) - h(β0y'(xn)) - h(β1y'(xn-1)) - h(β2y'(xn-2)) - h(β3y'(xn-3)) - h(β4y'(xn-4))
Now look at the coefficients C1 and C2:
C1 = 1 - β0 - β1 - β2 - β3 - β4
C2 = -1/2 + β1 + 2β2 + 3β3 + 4β4
These coefficients represent the sums of the coefficients associated with each term in the expansion. For example, C1 is the sum of the coefficients multiplying y'(xn), y'(xn-1), y'(xn-2), y'(xn-3), and y'(xn-4).
The specific values of β0, β1, β2, β3, and β4 will depend on the particular numerical method you're using. These coefficients are determined by the method's rules and may be different for different methods.