I know how to use the RK 4 method for a first order differential equation of the form:
$$y' = f(x, y(x))$$
$$y_{k+1} = y_k + (G_1 + 2G_2 + 2G_3 + G_4)*dx/6$$
where $G_1 + 2G_2 + 2G_3 + G_4$ are calculated as:
$$G_1 = f(t_k, x_k)$$ $$G_2 = f(t_k + 0.5*dx, x_k + G_1*dx)$$ $$G_3 = f(t_k + 0.5*dx, x_k + G_2*dx)$$ $$G_4 = f(t_k + dx, x_k + G_1*dx)$$
But what to do if the differential equation is of $n th$ order and it being written as a function of $y'$ contains other derivatives of y as well ?
Lets say the $n th$ order differential equation is written as follows:
$$D_ny = f(x, D_1y, D_2y, ..., D_{n-1}y)$$
where $D_n$ denotes the $n th$ derivative of $y$ with respect to $x$
How do you go about calculating $G_1, G_2, G_3, G_4$ ?
I am aware of the method of decomposing an nth order DE into n first order equations but I am also aware that it can be solved as it is numerically as shown in here : https://www.sciencedirect.com/science/article/abs/pii/004579499290121F
But I am having difficulty understanding precisely how the algorithm works as I can understand exactly how the $G_i$'s are calculated and then are used to find the state variables of the next step.