Finding the Jacobian of a system of 1st-order ODEs I am trying to find the Jacobian matrix of the following system of 1st-order ODEs.
My system is: $\dfrac{dx}{dt} = \left(x-3\right)\!\left(y+x\right) \\ \dfrac{dy}{dt} = \left(x+4\right)\!\left(y-2x\right)$
Since $(x-3)(y+x) = xy+x^2 - 3y - 3x$ and $(x+4)(y-2x) = xy-2x^2+4y-8x$, I get a matrix like this after taking the partial-derivatives. However, it's apparently incorrect and I'm not understanding where I went wrong. Any help is appreciated.
$\left[\begin{array}{cc}
y+2x-3 &x-2\cr
y-4x-8 &x+4\cr
\end{array}\right]$

Note that
$\dfrac{\partial (xy + x^2 - 3x - 3y)}{\partial y}$
$ = x - 3, \tag{1}$
not $x- 2$; the correct matrix is thus
$\begin{bmatrix} y + 2x -3 & x - 3 \\ y - 4x - 8 & x + 4 \end{bmatrix}. \tag{2}$
