The Taylor Series for a variable ($F$) can be written as :
$F = F_0+\frac{\partial F}{\partial t} \Delta t + \frac{1}{2!}\frac{\partial^2 F}{\partial t^2} \Delta t^2+\cdots$
The variables $F_x$ and $F_y$ are functions to $(x, y, \dot{x}, \dot{y})$, By using Taylor series the expansion yield:
where: $k_{xx} = \frac{\partial F_x}{\partial x}, k_{xy} = \frac{\partial F_x}{\partial y},\cdots, k_{x20}^{xx} = \frac{\partial^2 F_x}{\partial x^2},k_{x20}^{xy} = \frac{\partial^2 F_x}{\partial x\partial y},\cdots$ .
$~~~~~~~~~~~~d_{xx} = \frac{\partial F_x}{\partial \dot{x}}, d_{xy} = \frac{\partial F_x}{\partial \dot{y}},\cdots, d_{x20}^{xx} = \frac{\partial^2 F_x}{\partial \dot{x}^2},d_{x20}^{xy} = \frac{\partial^2 F_x}{\partial \dot{x} \partial \dot{y}},\cdots$ .
I tried to get the previous results but I couldn't, especially for the term $k_{x20}^{xx}$ which does not multiplied by 2 and $d_{x20}^{xx}$ does!!!
I need to know, how the expansion done for $F_x$ and $ F_y$?