Are those formulas the same? When I am programming for Adams Moulton fourth order methods I use the 2nd formula which I mentioned in the 2nd image. Is it correct?
If correct, please explain me.
This Formula is Adams Moulton fourth order (I find this on my book)
The 5th order Adams-Moulton formula is $$ y_{n+1}=y_n+\frac{h}{720}\left[\begin{aligned} &251·f(x_{n+1},y_{n+1})+ 646·f(x_n,y_n)\\ &-264·f(x_{n-1},y_{n-1})+ 106·f(x_{n-2},y_{n-2}) \\&-19·f(x_{n-3},y_{n-3}) \end{aligned}\right] $$ This is an implicit equation for $y_{n+1}$. One could iterate it directly as $$ y_{n+1}^{[k+1]}=y_n+\frac{h}{720}[ 251·f(x_{n+1},y_{n+1}^{[k]})+ R_n], \\~\\ R_n=\begin{aligned}[t]&646·f(x_n,y_n) -264·f(x_{n-1},y_{n-1})\\&+ 106·f(x_{n-2},y_{n-2}) -19·f(x_{n-3},y_{n-3})\end{aligned} $$ or try to accelerate this by capturing the majority of the linear behavior of $f$ close to $(x_n,y_n)$ in some Jacobian $J_n$, to get the linear system $$ \left(I-h·\frac{251}{720}·J_n\right)y^{[k+1]}_{n+1}=y_n+\frac{h}{720}\Bigl[ 251·\bigl[f(x_{n+1},y_{n+1}^{[k]})-J_n·y_{n+1}^{[k]}\bigr]+ R_n\Bigr] $$ For some special case of $f$ this might result in your second formula.