# ( Proof Explanation ) Modified Euler scheme preserves the weighted area $dx \wedge dy/xy$

I already showed that the Lotka Volterra Equations preserve the weighted area $$dx \wedge dy/xy$$ see here. Now I need that a modification of the Forward Euler Method preserves the same weighted area. Here is the modification:

$$\frac{x_{n+1}-x_n}{\Delta t} = x_n -x_ny_n$$ $$\frac{y_{n+1}-y_n}{\Delta t} = -y_n -x_{n+1}y_n$$

side-question: why is this Euler scheme still explicite?

Proof that this modification preserves the weighted area $$dx \wedge dy/xy$$:

to simplify notitation :

$$X = \Delta tx +x - \Delta txy$$ and $$Y = -\Delta ty+ y + \Delta tXy$$ where we set $$X:=x_{n+1}, Y:=y_{n+1}$$, $$x:= x_n$$ and $$y := y_n$$ solved for the unknown $$X$$ and $$Y$$.

now I showed $$dX \wedge dX = 0$$, $$dY \wedge dY = 0$$ and compute $$dX \wedge dY$$ and compute $$XY$$.

I found out that:

$$\frac{1}{XY}{dX \wedge dY} = \frac{1}{xy}{dx \wedge dy}$$

Question: why is this enough to say that the modification preserves the weighted area above?